Vector Calculus IA

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Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. Abstract: Allanach C. B. IA Calculus Vector JHEP to submission for Prepared etrCluu ore hr r oreape hesfrt for sheets complete. examples much pretty four are are There course. Calculus Vector uuelcues laefe ret s/oiyti resou this use/modify append to free feel please lecturers: Future a E-mail: C Road, Wilberforce Physics Cambridge, Kingdom Theoretical of University and Sciences, Mathematics cal Applied of Department • • Books Mteaia ehd o hsc n niern” U 200 CUP Engineering”, Bence and and Physics Hobson for Methods “Mathematical Vco nlssadCreinTnos,Bun n Kendall and Bourne Thomas Tensors”, Cartesian and Analysis “Vector ornm otecretato list. author current the to name your [email protected] e h ceue o it u particularly: but list, a for schedules the See hs r etr oe o h abig ahmtc tripo mathematics Cambridge the for notes lecture are These £ a 30. ..Evans J.M. £ 28. a c.I o otog,please though, do you If rce. mrdeC30A United 0WA, CB3 ambridge etefrMathemati- for Centre , i ore hs notes These course. his 99b Nelson by 1999 yRiley, by 2 atIA Part s Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. nerto in Integration 3 Integrals Line and Curves 2 Coordinates and Derivatives 1 Contents . ieetainUigCodnt Notation Coordinate Using Differentiation 1.2 Notation Vector Using Differentiation 1.1 . ute eeaiain n Comments and Generalisations Further 3.4 in to Integral Generalisation an for Variables of 3.3 Change 3.2 of subsets over Integrals 3.1 Energy Potential and Work 2.5 Differentials Exact and Gradients Integrals and Curves of 2.4 Sums 2.3 Fields Vector of Integrals Length Line Arc and Tangents 2.2 Curves, Parameterised 2.1 Systems Coordinate 1.3 .. etrvle integrals case valued Vector the for variables of 3.4.3 Change in 3.4.2 Integration 3.4.1 Examples in Variables of 3.3.3 Change 3.3.2 Definitions 3.3.1 Comments integrals sum multiple a as 3.1.3 Evaluation of limit the as 3.1.2 Definition 3.1.1 Differentials Gradients and Integrals 2.4.2 Line 2.4.1 Comments Examples and 2.2.2 Definitions 2.2.1 version functions Inverse general - rule chain 1.2.3 The functions 1.2.2 Differentiable case 1.2.1 particular a directiona rule: chain and gradient The position; of function 1.1.3 Scalar scalar a of function 1.1.2 Vector 1.1.1 ℜ 2 and ℜ 3 ℜ 3 ℜ n ℜ 2 ℜ – i – ℜ 3 n ℜ → ℜ m n 2 1 = derivatives l 14 23 23 22 22 21 20 19 19 17 16 15 14 14 13 13 12 12 10 10 6 1 8 8 6 5 4 3 3 3 3 2 1 1 Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. 0Gus a n oso’ Equation Poisson’s and Law Gauss’ 10 Coordinates Curvilinear Othorgonal 9 Theorems Integral of Applications Some 8 Theorems Integral 7 Curl and Div Grad, 6 Surfaces and Curves of Geometry 5 Integrals Surface and Surfaces 4 . ufcsadItiscGoer (non-examinable) Geometry Intrinsic and Surfaces Normals and 5.2 Curvature Curves, 5.1 Integrals Volume and Surface Line, Comparing Fields Vector of Integrals 4.4 Surface Area and 4.3 Surfaces Parameterised Normals and 4.2 Surfaces 4.1 02Lw fElectrostatics of Laws 10.2 Gravitation of Laws 10.1 Curl and Div Grad, Elements Volume and Area 9.2 Line, 9.1 Poten Laws Scalar Conservation and Fields, Irrotational 8.3 Curl and Conservative and Div of 8.2 Expressions Integral 8.1 Theorems Integral the Proving and Relating 7.2 Examples and Statements 7.1 Derivatives Order Second Properties 6.3 Leibniz Notation and 6.2 Definitions 6.1 .. rvn h iegne(rGus)termi d outl 3d: in theorem Gauss’) (or divergence the Proving theorem 7.2.4 Green’s T Divergence 2d 7.2.3 the Proving by theorem 2d Green’s the Proving or theorem Stokes’ 7.2.2 from theorem Green’s Proving 7.2.1 theorem Gauss’ or Divergence, theorem 7.1.3 Stokes’ plane) the (in theorem 7.1.2 Green’s 7.1.1 in variables of orientations Change and integrals surface and 4.4.2 Line 4.4.1 rem theorem vergence ⇒ tks theorem Stokes’ ℜ i– ii – 2 and ℜ 3 revisited tials ine heo- di- 36 33 31 24 55 51 47 36 36 35 33 32 31 30 30 30 27 26 24 57 55 52 51 50 48 47 46 45 42 41 41 40 38 36 Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. 4Tnoso ak2 Rank of Tensors 14 Fields Tensor and Tensors 13 Equations Maxwell’s 12 Equations Poisson’s and Laplace’s for Results General 11 13Itga ouino oso’ Equation Poisson’s of Solution Integral 11.3 Functions Harmonic and Equation Laplace’s 11.2 Theorems Uniqueness 11.1 Equation Laplace’s and Equation Poisson’s 10.3 43Daoaiaino ymti eodRn Tensor Rank Second Symmetric a of Diagonalisation 14.3 Tensor Tensor Inertia Rank The Second 14.2 a of Decomposition 14.1 Calculus Tensor 13.4 Rule Quotient the and Maps Multi-linear Tensors, 13.3 Algebra Tensor 13.2 Examples and Definitions 13.1 Waves Electromagnetic Currents Steady 12.3 and Charges Static 12.2 Electromagnetism of Laws 12.1 342Itgasadtetno iegnetheorem divergence tensor the and Integrals derivatives 13.4.2 and fields Tensor 13.4.1 rule quotient The maps 13.3.2 multi-linear as Tensors 13.3.1 tensors antisymmetric and Symmetric 13.2.4 Contractions 13.2.3 products Tensor multiplication 13.2.2 scalar and Addition 13.2.1 matrices and tensors 2 Rank 13.1.3 examples Basic 13.1.2 rule transformation Tensor 13.1.1 and sources Point derivation 11.3.2 informal and Statement 11.3.1 Principle Minimum) (or Maximum The Property 11.2.2 Value Mean The 11.2.1 Comments 11.1.2 proof and Statement 11.1.1 δ − ucin (non-examinable) functions i – iii – 68 66 61 73 67 67 66 65 64 64 64 63 63 62 61 61 58 74 73 73 72 71 71 71 70 70 70 69 69 69 69 69 68 68 68 Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. dF a vrrpae indices). repeated over n h derivative the and .. etrfnto fascalar function a vector of A function Vector Notation Vector 1.1.1 Using Differentiation 1.1 Coordinates and Derivatives 1 Tensors Isotropic and Invariant 15 tors rvddtebasis the provided ihsalbtfiiechanges. finite but small with etr a edffrnitdcmoetb component: by component differentiated be can Vectors Example L hn˙ then bu h origin the about r ˙ ˙ ( ( h t = a = ) 53ASec faPofo lsicto eut o Rank for Results Classification of Proof a of Sketch A Integrals 15.3 Invariant to Application Results 15.2 Classification and Definitions 15.1 ,aclrto ¨ acceleration ), = iiso etr r enduigtenr(egh so norm(length) the using defined are vectors of Limits h ento ftedrvtv a eabeitduigdiff using abbreviated be can derivative the of definition The ebi identities Leibniz oeta eiaie ihrsetto respect with derivatives that Note F r p ( F ∧ u b = ( ′ ) F ( on atceo mass of particle point A : h G , u + ) m rtetru of torque the or , ) ( r ¨ du u n ): = δF o ( nwhich in fF ( F h L F = r { stevector the is F iff ) ( ( ) = e ( r t ′ ′ F u i sNwo’ eodLw edfieteaglrmomentum angular the define We Law. Second Newton’s is ) n momentum and ) ( } = odfraporaepout fsaa functions scalar of products appropriate for hold u s‘ieetal’at ‘differentiable’ is ) r ( | u = ) sidpnetof independent is a ∧ f F ( + ′ o h p F ( − ) u δu dF du = + F = ) − em r upesd oprdt h eainabove relation the to compared suppressed, are terms ) mr fF b bu h origin. the about − lim = ( F h F ∧ ′ i ) , m ( | ( u r t u ˙ = ) δu a position has r ovninlydntdb osisedo dashes. of instead dots by denoted conventionally are = ) ⇒ e → o i ( – 1 – p ( F ( ⇒ 0 L h F u ˙ = F δu ,for ), 1 ∧ = nt h mlctsmainconvention summation implicit the (note F · ′ m ( u G G [ u ′ F m ( r if ) ˙ ) ) u ( δu ( ′ ′ r = ) t ˙ n u = = .If ). ∧ r + + oeui vector. unit some ( F F r ˙ t F δu o afnto ftime of function (a ) + ′ ′ i ( ′ F ∧ · ( δu ) u G mr ( − G ) r as ) e + stefreo particle, a on force the is ) F i + ∧ v F ( F r u ¨ δu → sn ebi.Hence Leibniz. using , · )] ∧ G n . → c ′ G ≤ , iff ′ rnilnotation erential 0 . 3 , | v f − ( u t c ,velocity ), → | n vec- and ) and 0 75 76 75 75 a Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. how h ietoa derivative directional the as on directiona and gradient function position; scalar of A function Scalar 1.1.2 ec h rdeti atsa oriae is coordinates Cartesian in gradient the Hence ihti hieo ai n oriae,tedfiiino d of definition the coordinates, and basis of choice this With ∇ n Example ofis re,teeis there order, first To + thus f √ δx sgvnb h ietoa derivative directional the by given is δr 1 2 savco,tegradient the vector, a is Taking Let f (1 pt smaller to up , ∇ = hne sw oeaa from away move we as changes , f 0 r √ : , (1 = = δx )addcessms ail for rapidly most decreases and 1) f ∇ δr ( x i ,y z y, x, f δx i e , = δf = i i 0 δf with → hn , f = = ) ( )a ( at 1) ( e r ∂f ∂x .I ieeta oain esuppress we notation, differential In 0. r with f i o + sdifferentiable is ) ∇ · − ( { x r , e no δr terms. i + + ∂f f } ∂y of n ) ,y z y, x, hnein change hn e rhnra.Setting orthonormal. lim = − f ntvector, unit a , xy ⇒ of ) f along ∂f ∂z sin df − ( n f (0 = ) δf r h ( = ) ∇ · = f z → at ( = 1+ (1 = 0 t( at r n ∇ ∇ f h 1 ( = ) r hs h rdetcnan nomto about information contains gradient the Thus, . f ∂x f f h ento asthat says definition The . ∇ ∂f , lim = [ r 1 f at ,y z y, x, if · – 2 – = f , i ( n ofis re ntedisplacement. the in order first to dr ) h aeo hneof change of rate The 0). δx ) r n ∇ r ∇ · ∂x ye · ∂f + ⊥ i = f δr if n h + xy (0 = ) → i ) he f (1 e = ∂x + · ∂f 0 So . o i sin i . ( h , 1 ( ) − hn o 0 n i δx − [ dx ( , ,xe z, f √ | , = 1). 1 + ) ) δr ( f 2 1 f i r (1 . , e ( | 0). nrae otrpdyfor rapidly most increases r as ) + i o , ]= )] xy o fixed for 0 ( hn h , sin )a rate a at 1) ) | ) δr o ∂x ∂f − − ffrniblt becomes ifferentiability ,e z, → | terms: i f . δf ( i xy r 0 gives )] derivatives l . cos f eed linearly depends , ln direction along n z ∇ · ) , f = ± (1.1) n √ = 2. Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. change ihterne ftelabels the of ranges the with f hssosthat shows This in .. h hi ue-gnrlversion general functions - the rule Consider chain The 1.2.2 xetat except osdracmoiino ieetal functions differentiable case of particular composition a a rule: Consider chain The 1.1.3 ucini smooth is function A ℜ F δy rule chain a ilb mohecp o hr hnsovosyg rn,eg wrong, go obviously things where for except smooth be will n rpthe drop and a h functions The arx rvco,gain ..( i.e. gradient vector, or matrix, .. ieetal functions Differentiable Notation Coordinate 1.2.1 Using Differentiation 1.2 eiaie xs,frexample for exist, derivatives r 3 ◦ m r ( ,j ,... . . k, j, i, x ℜ → g = ngeneral, In ovnetabeito ftedefinition: the of abbreviation convenient A i ihderivative with , × where ) M n δr ( hl etrfield vector a while f arxo ata eiaie.Freape for example, For derivatives. partial of matrix x = ) nCreincodnts(setting coordinates Cartesian In . ri 0. = o example, For r δx o ′ r δu − F f i em,wihaeunderstood are which terms, ( 1 = f + : u + ℜ and ) sdffrnibea ucinof function a as differentiable is δf o o { n ( m , . . . , ( x δx fi a edffrnitdaynme ftms ..i l par all if i.e. times, of number any differentiated be can it if δu i = } as ) ℜ → and ) ∇ f coordinates ( f ∂x r ∂ M icse nsections in discussed ) δx 2 dy · F i { m and ∂x F y δr ( r ( ,i r i, a, r r ∂x j f r = } ∂ ∂f = ∂x safunction a is ) + 2 seuvln oasto functions of set a to equivalent is i = ◦ ∂x F √ i r , o g M du df j δx ∂x ( 1 = ) ℜ , nesod If understood. ∂ ∂f ∂y | ra ( { δr 2 j n i ∂x f F ℜ = ∂x u , δx = r ℜ → ) | – 3 – a n , . . . , i p = ) ∂ i ri ∂x ∂f ∂z { } ∇ i h ucin htw ilcnie here consider will we that functions The . 3 M dx F → j ℜ → f ). r ∂x f g m ( ∇ i . · f k ,wt derivative with 0, = du f dr ) h function The . x t,adteeaettlysymmetric totally are these and etc, ℜ ri r elc ml hne ydifferentials by changes small replace · n ∂y i ∂x M , 3 { } = r ′ ℜ → r f ( ℜ → i 1.1.1 ( u x u dx g ( f i ) r ) e and δu ia ( i and i . u 3 ), y , , . m r 1.1.2 ) change A )). + } du df ℜ , g o ( 3 = δu r moh hns is so then smooth, are r maps are ℜ → f ∂x ∂f f ) M ( . y x i sdifferentiable is r dx ( 1 = ) du f egta1 a get we , = i ) . ri F δu /x = r ℜ → ℜ ( rdcsa produces x ∂y ∂x ssmooth is 1 x , . . . , r i = 3 ∂F and × ∂x tial n j r if 3 = ) , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. rmti multiplication matrix or For ihtentto ftels eto,take section, last the of notation the With functions Inverse 1.2.3 ohsot,with smooth, both ecncmuedirectly compute can we For opr ihsection with Compare ϕ deriva the compose also we functions, the maps. linear compare we if Thus, Both eed on depends n noeao form, operator In h eain a eivre i.e. inverted be can relations The Equivalently, × sntdfie at defined not is n > n o xml:for example: For o example, For n f ,w e h aiirresult familiar the get we 1, = dniymatrices. identity dx dy ◦ M ,w utinvert must we 1, g r i ( = = f and u = ) a ∂u M ∂u M through a b ( ( g

g f = y ◦ ) ) ∂u y ia ρ ∂x ∂x = ri ∂ϕ ∂ρ ∂ ∂u f ∂x r M n du a dx n ti endu oamlil f2 of multiple a to up deﬁned is it and 0 = 1 1 = f b i 1.1.3 r dniyfntoson functions identity are ,w write we 2, = ( = ∂u a ( ∂x ∂x ∂x i x ∂ϕ ∂ρ g x u m i = ) a 2 2 i i . r ∂u ∂y ( ∂u : ∂x matrices ! M × = u . obn n opr with compare and combine r a a )) i

p = ( δ ∂x f ab ∂ = ) i ∂x ∂x M 1 ∂ρ ∂ρ ; hc od hnatn nayfnto which function any on acting when holds which , ai × du dy 1 2 ∂x ∂x ρ ( m ℜ → ℜ g and { ∂x ∂x j 1 1 orelate to = i ∂ϕ ∂ϕ ∂y ∂x ) ℜ u u × − du dx = 2 1 = 1 r a i n p 1 { } ! n n = – 4 – M = = ∂u ∂x x → ← m f g = ∂x ρ 3 ∂y 1 2 × a j (

∂u dx/du ∂x , × g ∂u + ℜ → ∂x i = u 3 3 1 ) a i − ℜ x ∂u a ∂x ia i 2 p x cos sin n cos n i ρ 1 2 2 = = } a i . arxtype matrix dx × du r nes arcso ahother. each of matrices inverse are sin = and ℜ and i ρ ϕ ϕ δ ϕ ϕ 1 ij n p ϕ n let and . . − . n let and ϕ ρ ∂u ∂x ρ 1 M cos sin sin tan = cos dy a i ( . f ϕ r ϕ ϕ x ϕ = ◦ ,g f, 1 ! − g = M ) . 1 ia ( eivrefunctions inverse be π ρ x ( ie mtie)as (matrices) tives f 2 cos and for /x ◦ ( ρ g 1 ,x ϕ, ρ ecp that (except ) ) 0) 6= M ra ) Then, 0). 6= du 2 ( g = . a ◦ . ρ f sin are ) ϕ . , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. usto t a eivril ucin fsm atsa c Cartesian some of functions invertible be can it) of subset u det fe oeueu.I eea,tecoordinates bu the coordinates, general, Cartesian In use coor to useful. of choose more changes always to often section can last We the of space. results the apply we Now, Systems Coordinate 1.3 onsi h ln,btte r not are they but plane, the in points edfie.Teevcosvr ihpsto n r ill-defin are and position with vary vectors These fixed. held x r ntvcosi h ietoso increasing of directions the in vectors unit are si Section in as φ 1 a e ( { 1 n2dmnin,w en ln oa coordinates polar plane define we dimensions, 2 In n3dmnin,w en h oriae si Table in as coordinates the define we dimensions, 3 In ngnrl o netbesot as saoe det above, as maps, smooth invertible for general, In x + ∂u ∂x i } iue1 Figure x a i and ) 2 e 2 r aldteJacobians the called are e is ρ det x .Nvrhls,w a soit ai etr ihteeco these with vectors basis associate can we Nevertheless, ). 1.2.3 ˆ = i ( ln oa coordinates. polar Plane . { ρ u cos = e Fig. see , a ∂u ∂x } r smooth. are ) a i ϕe det x 1 1 sin + 1 = . ∂u ∂x ρ ρ ρ a i cos ϕe hyaennzr,with non-zero, are They . and 1 = 2 ,x ϕ, e , opnnso oiinvco (whereas vector position a of components – 5 – ϕ , > ρ r oriae o aaees labelling parameters) (or coordinates are n umto convention). summation (no ϕ n 0 and 0 2 ˆ = ρ = ϕ { and ρ u x e = a sin 2 2 } ≤ − ϕ nEciensae(rsome (or space Euclidean on ϕ, < ϕ epciey ihteother the with respectively, , M ,ϕ ρ, sin ( g oordinates ϕe 2 1 e det = ) by π . 1 1 da h origin. the at ed 1 o uniqueness. for cos + iae nEuclidean on dinates te hie are choices other t e ϕe ∂u φ ∂x { a x i 2 i } det , ordinates: uhthat such e ρ M ( r f x = = ) Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. interval . aaeeie uvs agnsadAcLength curve Arc parameterised and A Tangents Curves, Parameterised Integrals 2.1 Line and Curves 2 2 iue2 Figure π n 0 and u ≤ yidia oa oriae n peia oa coordi polar spherical and coordinates polar Cylindrical . ℜ ⊆ θ e z ≤ al 1 Table . π . e e ϕ e φ ρ = yidia polars cylindrical e cos = ρ − -iesoa yidia n oa coordinates. polar and cylindrical 3-dimensional . x x C 1 2 sin x e x = ,ϕ z ϕ, ρ, 1 = sgvnb mohfunction smooth a by given is z ϕe 3 ϕe = ρ ρ = 1 cos sin 1 e sin + z 3 cos + z ϕ ϕ φ θ x 3 ρ ai vectors Basis ϕe ϕe 2 r – 6 – 2 x e e 2 ϕ θ e = r = = x peia polars spherical cos = r 1 − − x sin = sin x sin 3 2 r ,θ ϕ θ, r, = θ θe sin θe ϕe sin r z z cos 1 r sin + θ cos + ,r ϕ, ( cos + cos u θ with ) ϕ θe nates. = θe ϕe ρ | ρ r 2 . u | eogt some to belong r = | r e φ | e 0 , r e θ ≤ < ϕ Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. o eo h aaeeiaini aldregular called is parameterisation The zero. not derivative The emnso hc h aaeeiaini regular. ca is curve parameterisation the the that which so on points, segments isolated at perhaps except holds hsi aaeeiainof parameterisation a is This 2. 1. Examples r r ( ( u u cos 2 = ) = ) : u ˆ i + r u u ′ ˆ 2 i ( ˆ j u sin + savco agn otecrea ahpit rvddi is it provided point, each at curve the to tangent vector a is ) + u 3 u k ˆ ˆ j r(u) for 3 + −∞ δ k ˆ x r 0 2 hr 0 where / r(u+ u) + 4 < u < r’(u) y δ 2 – 7 – 1 = ≤ ∞ u uv ihu end-points. without curve a , y , ≤ π ≥ uv ihend-points. with curve a , if 0 z , r ′ ( u 3. = ) .W ilasm this assume will We 0. 6= lasb iie into divided be always n C Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. vra nevlwt ends with interval an over .. entosadExamples Let and Definitions Fields Vector of 2.2.1 Integrals Line 2.2 moh(ahi mohfnto fteohr.Tecanrul chain The other). the of function smooth a is (each smooth ie agn vector tangent a gives C otetnetvco hne ie u o ieto.Choos direction. not but size, changes vector tangent the so endby defined h infie h ieto fmeasuring of direction the fixes sign The Example s r du ds (2 ( h ieintegral line The . u π ( ) = C h aaeeiaino uv a ecagd take changed: be can curve a of parameterisation The length arc The ds 10 = ) > u easot curve smooth a be | = r ′ helix A : ( ±| u )is 0) ) π r | . ′ ( = u ) s | √ ( du u r 3 5 = ) s ( 2 sasaa ieelement line scalar a is u ⇒ fasot etrfield vector smooth a of 4 + esrdalong measured 3cos (3 = ) t Z = u du ds 2 δs r C ec,tedsac from distance the Hence, . α ( .Tu h r egh esrdfrom measured length, arc the Thus 5. = dr ds F u = = and ihedpoints end with ) ( fui length unit of r | ± ) δr u, · β

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= o u ˜ C = ( | Z = δr – 8 – rmsm xdpitsatisfies point fixed some from , u, ±| α β | s du dr = ) r o nraigo decreasing or increasing for , F 4 ′ ln h ieto fincreasing of direction the along u on ( / ( u ) r du d | F r ) ( ⇒ r u ˜ C | u ( ′ . ( ( α )) r . u r = ) along ) ) ′ · ( δu r u r ′ ( = ) ( 0 3 = (0) a | u + and ) du. o C − ( sadirection a is n ˜ ing δu r sin 3 ihti retto is orientation this with ˆ i ( u (2.1) ) β to u mle that implies e = ) ↔ = u, r (2 s u b ˜ cos 3 h r length, arc the , π othat so , netbeand invertible 3 = ) r a u 0 3 = (0) u, . → ˆ i 8 + 4). s b u . along (2.3) (2.2) runs π ˆ i k ˆ to is Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. dr ahtr is term Each h ethn iede o eeduo h parameterisation the upon depend not does Eq. side hand left The sn opnns with components, Using odvdn h nevlfor interval the dividing to R edefine We Examples For aibe.On variable). ( α ue β = F u hscnb eadda h ii fasmoe ml displacem small over sum a of limit the as regarded be can This 2.3 i 2 C r dx du u , ′ 2 ( hscnas ecekdb h hi rule: chain the by checked be also can This . i u : Z du 6 ) u , r C : du ( 1 . F t Z C F a ( = ) 3 C steln element line the is ( .Hence ). 1 r 2 · C ( = ) F dr 2 : , · ,t t t, t, F r = dr ( xe ( u Z r a=(0,0,0) ( = ) = ( 0 y ) t z , 1 )=( = )) ⇒ Z F F F r 0 2 C 1 ( · u xy , r ( r F = r te ′ ,u u, ) nosalsegments. small into ( ′ ( te ( t · t r u (1 = ) x C ,so ), δr 2 + t ) ) on i 2 t , e · 2 u , du i 2 = du dr t t , C 3 and 2 ) R ) = F 2 . du , u=0 C .Hence ). dt 1 b ( ⇒ – 9 – Z F r , = F 0 ( )(eaenwusing now are (we 1) C [ = u 1 · b=(1,1,1) F )) 1 ue r dr = a ′ te ( ( · u r u = t 2 F du ) dr ] (1 = ) 0 1 2 + i · R e − d C dr i + , u ˜ u Z xe o h ieitga is integral line the d 7 0 ( u. ˜ 1 y 3 + δu , e dx 2 F t ) u, u . C dt u=1 + 5 3 2 + du u z 2 2 n onihrdoes neither so and , .On ). t / = dy 5 = 3 steindependent the as nscorresponding ents e/ + 1 + 2 ydz. xy / b 3 C . R 1 / C , 4 F . F i dx ( r i = ) = Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. . uso uvsadIntegrals smooth and piecewise Curves A of Sums 2.3 h ieitga vrapeeiesot curve smooth piecewise a over integral line The circulation the called is and .. Comments 2.2.2 C h n-onso ucsfiesget onie n hr i there and coincide, orientations: segments successfive of end-points The C: 2 2. .Teoinaino uv sas qiaett hieof choice a to equivalent also is curve a of orientation The 4. If 3. 1. + closed A · · · h eghof length the ensaui agn vector tangent unit a defines R tation β > α R R esrn h r length arc the Measuring ergre stelmto u fterms of sum a of limit the as regarded be C C C C C f F F 1 + ( stecrewt orientation with curve the is s · · C . ) dr dr b n uv has curve ds → hr each where , = eed on depends Z sawy vlae ntesneo increasing of sense the in evaluated always is C a R and , α C C F β C 3 2 F curve · n sgetrta 0. than greater is and dr · b r a=r( ) R ′ = − = ( u C C C Z ) a α C F of C h ieitga ssmtmste written then sometimes is integral line The . du ossso ubro emns ewrite we segments; of number a of consists ngnrladntjs pnteedpoints end the upon just not and general in i 1 · s F F sasot uv iharglrparameterisation. regular a with curve smooth a is dr ln h ieto ie yteorientation the by given direction the along o orientation for · around = dr t R and + β α 0– 10 – Z a F F C C → R 2 · C . C F r ′ F b ( ewrite we , · u dr · ) a C dr du → + F is = . . . = b · hshlswhether holds This . δr R − + C − R = F Z C C b=r( ) C · optbecoc of choice compatible a s F F n t o uv ihorien- with curve a for F · · ds β dr s t nttnetvector. tangent unit · δs othat so , h nerlcan integral The . dr . C yconvention, By . . n−1 C R a dr β < α H C C and = 1 = C F C ds n t · 1 b dr ds . or + is Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. h aaeeiain neach on parameterisations The or cancelling orientations opposite the construc with useful segments some for on allows rely This connected. not are they R Example lsdcrea hw with shown as curve closed C C F ti ovnett xedsm oayst fpeeiesmoot piecewise of sets any to sums extend to convenient is It · dr seSection (see : = I R C C xe F y · dr dx ( = = + 2.2.1 I Z z C a=(0,0,0) e/ C 2 1 F ) 1 + 2 C dy F · 3 dr · C + = dr i / ydz xy − = 4) a ecoe independently. chosen be can + C C I − Z 2 C 3 rmorpeiu results, previous our From . C 1– 11 – 1 5 where , 3 F / F = 3 · · dr dr e/ C + b=(1,1,1) = 2 C 1 I − Z C = 2 C 17 1 F C F / · 1 12 dr · + dr . ): C . C in,freape(these example for tions, 2 − 3 sapeeiesmooth piecewise a is Z C 2 F uvs vnif even curves, h · dr C 1 Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. o any for .. ieItgasadGradients and If Integrals Line Differentials Exact 2.4.1 and Gradients 2.4 o n aaeeiainof parameterisation any for srequired. as ytecanrl,sesection see rule, chain the by F • • • Proof oethat Note = lentv entoswihaeeuvln wt suitabl (with equivalent later. are which definitions alternative h uv onn them. joining curve the If When When ∇ curve F f : = o oesaa field scalar some for F C C C ∇ sagain,teln nerldpnsol nteedpoint end the on only depends integral line the gradient, a is sacoe curve, closed a is 1 C f from , 2 F Z scle conservative a called is a C to F b · . C dr [ = a 1.1.3 Z with = = f C f H ( Z F ( r Z C r , α ( C then ) β · F u a ∇ dr ))] du = d · f 2– 12 – dr α β = r ( · = ( f dr if 0 = α f C ( ( r and ) f b ( = ( ) u b etrfil.Teeaeanme of number a are There field. vector − ))) ) Z − α F f β b ( f du ∇ sagradient. a is = a ( ) a f , b r ) · ( , β du dr ). du supin)-see - assumptions) e C ,nton not s, (2.4) Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. neatdffrnil Eq. differential, exact an sn hs,i a epsil ofind to possible be may it these, Using o constants for ftept fapril sacurve a is particle a of path the If under moving particle point a of position the Consider Energy If Potential and Work 2.5 differentials with work to convenient often is It Differentials 2.4.2 fti od,w a s h eesr condition necessary the use can we holds, this if eitgae ln uvs uhadffrnili aldexa called is differential a Such curves. along integrated be stelmto u fterms of sum a of limit the is df eodlaw: second ∂x Example if ∂ i F 2 Z ∂x a = f C ieetascnb aiuae using manipulated be can Differentials ( (0 = j r (3 ∇ safrethen force a is ) o nc’fntos.W ilseltrta,lcly this locally, that, later see will We functions). ‘nice’ for x f 2 , : y · 0 dr sin , )and 0) m K µ = dx z ¨ r ( and = β ∂x ∂f ) F i + − b λ ( dx (1 = r K x hr sas ebi rule Leibniz a also is There . K .Tekntceeg ftepril is particle the of energy kinetic The ). 3 i R ( So . ( sin C t α = ) 2.4 , Z F = ) 1 C dy z π/ , · F F is dr 2 1 d F Z d m ( · ) o instance. for 2), α = ( λf + dr · β stework the is fg r ˙ dr 2 ∇ dt x C d + ( = ) = ⇒ 3 ..(h opnn of component (the i.e. , f K y from µg Z cos ⇔ ( dt 3– 13 – C d f t = ) df ) df K yinspection. by F dz z dt ) a F i g ( = λdf t = = h oc osaogtecurve the along does force the = δ + = ) = ) f r r Z f ∂x ( ∂f + ( b α F ( m α ) i dg β ∂x ∂F Z µdg to ) − · r F ⇔ ˙ C j i ) dr · . f d = · r ¨ F ( ( r ˙ b = x a = i F ∂F ∂x dt 3 = ) dx y . F ( F j i r r sin = i codn oNewton’s to according ) i snebt r qa to equal are both (since ( · F dx β = r Z ˙ ( z ,then ), ct C sas ucet For sufficient. also is r i = ) df along ) F sojcsta can that objects as fi a h form the has it if · seat otest To exact. is dr x 3 y δr sin ) ×| z C δr b a This . 1 = | . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. uv)idpneto l ftecocsw aemd eoet before made have we choices the of all of independent curve) field scalar or function ..tettlenergy total the i.e. .. ento stelmto sum a of limit the Let as Definition 3.1.1 and oriae.Cnie approximating Consider coordinates. . nerl vrsbesof subsets over Integrals 3.1 oepitwti ahsalst easm httelmtexi limit the that assume We set. small functions each within point some easm htas that assume We done work the to equal is energy kinetic the in change the i.e. raitgas eern otergo,o dmi’o inte of ‘domain’ or region, the to referring integrals, area nptnileeg.S,i hscase, this in So, energy. potential in f nerto in Integration 3 motion). ..tinlso aallgas aeldby labelled parallelograms, or triangles e.g. ecnandwti ico diameter of disc a within contained be = Taking o osraieforce conservative a For D N − V ∞ → easbe of subset a be nEq. in f ie moh and smooth) (i.e. f . yields 1 = 2.4 ), l → K y R C ℜ f + and 0 K F ( ℜ R 2 r euetepsto vector position the use We . D ( V · ,w define we ), 2 β dr , dA + ) sconserved is and F D N = ae of =area = freapeteitro fannitretn closed non-intersecting a of interior the example (for V ∞ → V ( −∇ ℜ r ( ℜ ( a 2 β 3 ) D V )= )) − 4– 14 – h no falo h ml sets small the of all of union the R l where , . by D D V i te od,i scntn uigthe during constant is it words, other (in f ( nerl with Integrals . K N ( b I r δ ,o h okdn seult h loss the to equal is done work the or ), ( ) ml,dson ust fsml shape, simple of subsets disjoint small, α dA iharea with A + ) V ( lim = r I V steptnileeg (setting energy potential the is ) ( r ( l α → gration, δA D r )) 0 P , f ( = I x ahsaleog to enough small each : r lokonas known also are 1 6= I kn h limit the aking yteforce. the by f ,y x, t o well-behaved for sts ( r D I ∗ ihCartesian with ) ) . δA I → where D o a For . l r → I ∗ is 0 Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. D δx nCreincoordinates. Cartesian in R ecnsmaiealo hsb h ttmn htteae ele area the that statement the by this of all summarise can We vertic narrow a in subsets over sum first can we Alternatively, over integrals successive as expressed be can integrals integrals multiple Area as Evaluation 3.1.2 y δA { xdadtaking and fixed R X D x } sflos hoetesalst ntedfiiint erecta be to definition the in sets small the choose follows: as I f hn esmoe l uhsrp n take and strips such all over sum we Then, . xdadtake and fixed ( : ( R = ,y x, Y ,y x, x δxδy f ) ( ) ,y x, dA ∈ umn vrsbesi arwhrzna ti with strip horizontal narrow a in subsets over Summing . ) = D dy R } Y hn umn vralsc tisadtaking and strips such all over summing Then, . δx dx δy R Y Y where , → x → y x f ie contribution a gives 0 ( y oget to 0 ,y x, y y y+ y X ) δ dx sterneo all of range the is δx x+x x dy R 5– 15 – Y where x δ f X x ( ,y x, y Y ) δy sterneo all of range the is dy x δx R in ihrange with x y → f D ( : ,y x, oget to 0 x ) x h coordinates the dx ment lsrpwith strip al Y ge,ec fsize of each ngles, x ihrange with y R = D in is δy f { ( D y dA ,y x, → : ( : y = ) and ,y x, gives 0 x x dA xdy dx x y and and ) δy = = ∈ Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. .. Comments 3.1.3 Example Alternatively, Z D .Tersl htw a nert over integrate can we that result The 2. range the with notation a adopted have We 1. f odmr eeal,frexample: for generally, more hold set a by thlsif holds It [ ubro nevl.Tesml itrsw rwt illustra to drew we pictures Section simple in The intervals. of number ne oemr eea odtosto nsm cases, some In too. conditions general more some under a eoeifiie and infinite, become can eswl eae xmlsd xs hr ieetase i condition in answer like different just other, a the where or order exist one in do integrating examples behaved well Less basis. case-by-case ( a ,y x, 1 b , : Z ) 1 f ] D dA ( ∪ ,y x, f hc a oss fdsonce nevl.Frexample, For intervals. disconnected of consist may which , [ ( a = = 0 y ,y x, 2 = ) 3.1.2 f b , Z 15 2 ) sacniuu ucinand function continuous a is 0 2 ] 1 x dA . ⇒ 2 D nov agswihaesnl nevl,btteargument the but intervals, single are which ranges involve y Z . = = R 0 X 2 − Z 1 2 f 2 0 y Z ( 2 x x 0

R ) 2 2 dx dx y D Z x 0 2 dA f 1 0 1 (1 y = − x/ − R D 2 a 6– 16 – b dy 1 1 x x 2 tl xss u hs utb ade na on handled be must these but exists, still f 2 ) x dy y 2 ( = x and dx ) 2 Z dx 0 ! = 1 y + y D x faoedmninlitga given integral dimensional one a of dx x nete re scle uiistheorem Fubini’s called is order either in 15 2 R x sacmatsbe of subset compact a is a 3 . b = 2 3 2 f Y Z 2 0 ( − 0 , x x 2 2 ) y x dx these intervals the unionof 2 dy n iial o any for similarly and , lycnegn series. convergent ally y 2 = f 2 a ieg,or diverge, can 3 8 0 1 etearguments the te − Z x/ 0 bandon obtained s 1 2 y dx (1 − ℜ 2 y and , X ) 3 dy D = s . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. area steJacobian the is ﬁeld) scalar (or function a for Then in Integral an for Variables of Change 3.2 nte( the in where osdrasot netbetasomto ( transformation invertible smooth a Consider noet-n correspondence: one-to-one in y .I h pca case special the In 3. .B ltigtegahof graph the plotting By 4. ro noutline in Proof A R R D D = ,v u, f f | ( ( J ,y x, ,y x, ln.Ec etnl smpe prxmtl oaparalle a to approximately mapped is rectangle Each plane. ) | δuδv ) ) nteformula, the In . xdy dx dA nte( the in write : a eitrrtda h ouebnahit: beneath volume the as interpreted be can = D f R ( ,y x, a ,y x, b R g D x ( ln,sneu to up since plane, ) ′ x = ) δx δy stelmto h u frcage farea of rectangles of sum the of limit the as x J z ) dx | = = J g f | R ( f ∂ ∂ = x c , stemdlso h Jacobian. the of modulus the is d ( ( ( ) R ,y x, ,y x, ,v u, h h D 7– 17 – ( ( z y f ∂u ∂u ∂x y ∂y D ntredmnin,w e ufc and surface a get we dimensions, three in ) ) ) ) and ) ( dy ,y x, = ∂x ∂y ∂v ∂v .

) ,y x, ∂u ∂u ∂x ∂y D xdy dx o = ∂x ∂y ∂v ∂v ) δu δv − ℜ ↔ terms, { v 2

( = ,y x, ( , ,v u, R D : ) ′ ihregions with ) f y ( a x ( ≤ ,v u, x D’ ) ≤ y , ,c b, ( ,v u, δA D ≤ ormof logram )) ′ y | = and J ≤ | δuδv udv du d D u } , ′ , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. h orc rafrec aallga.Tersl a esu be can result The parallelogram. each for area correct the and h ( the Z Example hslmt 0 limit, this o hsfnto aGusa) ecntake can we Gaussian), (a function this For ec,w eietewl nw eutfrteGusa integr Gaussian the for result known well the derive we Hence, p nti ipecs,w e htsalchanges small that see we case, simple this In xy pl hst regions to this apply Z π/ dA f D ln fapoiaesize approximate of plane | ,y x, J 2. dA f | x stefco ywihteaesaerltd umn vrpar over Summing related. are areas the which by factor the is y ln ie h nerlover integral the gives plane ) = atsa oriae ( coordinates Cartesian : y = = Z = D ρ ρ D ≤ Z cos sin ′ : ρd dϕ dρ fρ x ,y< y x, =0 ∞ x ϕ ϕ 2 y Z + y and D =0 ∞ y D ∞ 2 e ≤ = J − and so ( x R = Z 2 δρ ρ + 2 =0 R ; ∂ D ∂ y δA 2 ( ( ,y x, dA ρδφ ) ′ ,ϕ ρ, Z ,y x, / for 2 ϕ x = π/ =0 ,y x, xdy dx = ) ≥ ) 2 x ρδϕ δρ ρ f =

e 0 ) ∂ ∂ = − D 8– 18 – ↔ ( (

ρ ,y x, ,v u, 2 e cos = sin / scamd ihteJcba atrgiving factor Jacobian the with claimed, as − 2 ln oa oriae ( coordinates polar plane ρdϕ dρ ρ ( D oﬁmn h aoinfco.Nwwe Now factor. Jacobian the conﬁrming , x ) ) ρ ϕ ϕ 2 Z

R + π ′ 0 − 0 : y δρ udv. du ∞ /2 2 ∞ → ρ ) δφ cos / e φ ≤ sin 2 and − = = x φ ρ ϕ 2 ϕ D’ h / e ≤ n h nerlsileit.In exists. still integral the and δϕ 2 −

− dx ρ e = R 2 − rdc ml rai the in area small a produce / 0 ; ρ 2 ρ 2 : R / ⇒ 2 mrsdas mmarised ≤ i Z 0 R dA 0 ϕ ∞ [ ϕ al δρ ρ ≤ e ] = 0 ,ϕ ρ, π − 2 R π/ y ρdϕ. dρ ρ = 0 2 leorm in allelograms ∞ / ). 2 2 π 2 . e dy − (1 ρ x 2 − / 2 e = − dx R π 2 2 / . = 2 ) . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. o example for uod fvolume of cuboids volume over N to Generalisation 3.3 ucin(rsaa field) scalar (or function a l where e volume a Let Definitions 3.3.1 hr ehv taken have we where → ml ijitsbeso ipesae(o xml,cuboid example, (for shape simple of subsets disjoint small h ento a erdcdt ucsieitgainover integration successive to reduced be can definition The and 0 D r : I ∗ δV x saycoe on nec ml subset. small each in point chosen any is I N ahcnandwti oi peeo diameter of sphere solid a within contained each , ∞ → V ℜ ∈ z δV h no ftesalsbestnsto tends subsets small the of union the , Z I V 3 δz = f n oiinvector position a and ( → xδ δz δy δx Z r V ) ℜ f rt then first, 0 dV f ( 3 r ( over ) r δ δ δ ) A= xy = dV V n taking and Z D lim = V

9– 19 – D, rangeofallx,yinV Z sdefined is z δx x y l → f → 0 ( δx ,y z y, x, X r y and 0 I {z:(x,y,z) Vforfixedx,y} range ofzforfixedx,y: → ( = f ( 0, ) r ,y z y, x, I ∗ dz δy δy ) δV ! → ε → I xdy, dx .W approximate We ). , ogta raintegral area an get to 0 0, V hn h nerlof integral the Then, . δz l )lble by labelled s) easm htas that assume We . → ,y z y, x, nsm order, some in 0 ytaking by I V with by Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. R osdra netbetasomto ( transformation invertible an Consider hc means which as ihregions with ways, .. hneo aibe in Variables of Change 3.3.2 ntedfiiino h nerl ec h result. the hence integral, the of definition the in rtt e nae nerloe slice over integral area an get to first nerl.Sc nerl rs with arise integrals Such integrals. nSection in eaie fvolume of lelapiped h uniyi ml volume small denoted a in often quantity probability, the or charge mass, example steJacobian. the is in ml uodwt sides with cuboid small a V V xd dz dy dx f motn pca ae r yidia oa rspherical or polar cylindrical are cases special Important Taking Alternatively, h utfiain(nifra ro)i iia ote2-dim the to similar is proof) informal (an justification The osmaie h oueelement volume the summarise: To . x 1.3 f o yidia oa coordinates polar cylindrical For . = gives 1 = R V R z ′ V | V f J f | J | ( δuδvδw and r | ) ∂ ∂ ud dw dv du R u v δw δv, δu, dV ( ( V ,ϕ z ϕ, ρ, ,y z y, x, J V dV x = = ′ δV = n( in n1:1crepnec.Te,frafunction a for Then, correspondence. 1 : 1 in R ∂ V vlm of =volume ∂ ρ ) ) z ( at ( ( cos ,v w v, u, ,y z y, x, = ,y z y, x, ℜ R where epciey egta prxmt ml paral- small approximate an get we respectively, D r 3 f ρ . z D ,y ϕ, D ( 0– 20 – f R ⇒ δ z r V ( ,y z y, x, hnsmoe lcsadtake and slices over sum then , ) ’,therangeofall(x,y)suchthat z’ z en h density the being ) pc,adw a s hs ssalsets small as these use can we and space, ) ) ,y z y, x, ρ nCreincoordinates Cartesian in dV = (x,y,z) Vforfixedz. = ( r

) V ρ = ) ∂u ∂u ∂u ∂x ∂y ∂z dV ) sin y . ↔ d ϕdz. dϕ ρdρ xdy dx ε ∂x ∂y ∂v ∂v ∂v ∂z f stettlaon ftequantity the of amount total the is ,z, ϕ, ( range ofallzinV r lokona volume as known also are 1 6= ρ ,v w v, u, ∂w ∂w ∂w ∂x ∂y ∂z Then, . ) dz

..take i.e. , ,wihi mohboth smooth is which ), o oeqatt,for quantity, some for ρ oa oriae,as coordinates, polar ( ninlcs:given case: ensional r steaon of amount the is ) δx dV and = δz xd dz dy dx δy → ozero to 0: (3.1) f . , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. .. Examples 3.3.3 i.e. oueo pee 4 sphere, a of volume sflgnrlrsl.W a nesadi stesmo sph of sum the as it understand can We result. thickness general useful a 0 b nclnrclplrcoordinates polar cylindrical In Example Example Z Z V eoe (where removed ≤ V dV nteohrhn,i peia oa coordinates, polar spherical in hand, other the On nsmay o eea oriaesse,tevlm elem volume the system, coordinate general a for summary, In dV f ϕ ≤ = osdrtevlm ihnashr fradius of sphere a within volume the Consider : : 2 Z x δr π = = ρ f = b and ( Z Z a n oue4 volume and r 0 r Z sshrclysmercand symmetric spherically is ) =0 a a ϕ rr dr 2 =0 − z π Z a < b p x θ ∂ ∂ Z =0 π ( ( z 2 = a ,y z y, x, ,θ ϕ θ, r, = πa f z 2 Z = − ( r ϕ − .Thus ). √ r √ 3 2 =0 sin )[ / π a ρ a 3. − 2 2 πr ) ) 2 dV f − − θ ( ρ ≤ cos = ρ 2 cos r 2 2 δr ) = ρd dz dϕ dρ ρ z r r V θ 2 taking , 2 ,y ϕ, ≤

] ,ϕ z ϕ, ρ, sin sin 0 π ∂ ∂ : [ p ( ϕ ( x ,v w v, u, ,y z y, x, θ rd dϕ dθ dr θ 2 ] a 0 2 = 1– 21 – + π ⇒ 2 , y r 4 = − y δr dV dV sin 2 ) 2 = ) ρ +

2 π → . θ = V = Z ud dw. dv du z π sin 2 0 .Nt httaking that Note 0. r Z a sashr fradius of sphere a is = a ≤ ρd dz dϕ dρ ρ 2 ρ b f ,z ϕ, sin a ( Z a dρ r 0 2 ) a d θdϕ. dθ θdr r and dr 2 2 = ρ dr, p Z r a x 0 cos z=(a −) a π ihaclne fradius of cylinder a with 2 2 dθ + and − θ. y Z ρ 2 0 2 n is: ent 2 V 2 2 π ≥ = f ϕr dϕ a rclsel of shells erical b ρ : 4 . 2 ie the gives 1 = 3 π . b ( 1/2 2 a ≤ f 2 ( − r ρ sin ) b (3.2) ≤ 2 ) 3 θ / a 2 , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. coordinates hc ie h correct the gives which h entoso nerl in integrals of definitions The .. nerto in Integration Comments 3.4.1 and Generalisations Further 3.4 vrteCreincoordinates evaluated Cartesian are the integrals over The ‘volume’. and length Euclidean variables with frdu ,with a, radius of hcigtedmnin forase,w aeavlm times volume a have we answer, correct). is our (which of dimensions the Checking nshrclplrcoordinates, polar spherical In Example h oa hrein charge total The dV Q = = = J Z ρ a r H 0 = 2 Z dV ρ osdrtedniyo lcrccharge electric of density the Consider : sin { ∂ ∂ 0 x ( ( a u x i u rd dϕ dθ dr θ r 1 1 { ↔ } ,...,u ,...,x a 3 = eaiet h product the to relative dr n n Z ) ) z Z r u det = =0 0 a ≥ a π/ H } Z 2 and 0 Z ihteregions the with . θ sin is D Z =0 π/ n ℜ D f − 2 n θ ′ ∂x ( u Z iesoa ulda ouefrsalcagsi the in changes small for volume Euclidean dimensional f x a cos ρ ϕ i ( 1 2 =0 0 { x x , π h aoin sbefore, As Jacobian. the , ℜ x x osat hti h oa hreof charge total the is What constant. a dθ θ i 2 1 ρ 2 a ( r x , . . . , x , 0 { H and r u ≤ 2 Z cos a x , . . . , } 0 z 2 ) δu 2– 22 – a ℜ π } D θr n 0 , ) 3 dϕ ) 1 | 2 J ↔ δu . . . dx xedesl to easily extend n sin | = ≤ : du 1 D dV rd dϕ dθ dr θ dx ρ ′ a ϕ 1 n 0 n11correspondence, 1:1 in . du 2 = ≤ dx . . . r 4 ρ 2 4 dx ( du . . . 2 y r π 0 a = ) 1 n dx n 0 and 2 1 | = J n 2 ρ sin yscesv integration successive by ℜ | 0 · · · nesa cl factor scale a as enters z/a n 2 θ with dx ≤ nahemisphere a in 0 π/ n . θ 2 hredensity charge a o hneof change a For n [ ϕ ≤ − ] H 0 2 dimensional π π/ ? = .Also, 2. 4 1 πa 3 ρ H 0 . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. .. hneo aibe o h case the for variables of Change 3.4.2 o igevariable single a For stecnr fmass of centre the is and mass total the is components Cartesian With umes. to way similar a in ihrwy h oml yields formula the way, Either ..w a nert opnn ycomponent. by component integrate can we i.e. positions R Then, Let h ouu of modulus The mohi ohdirections, both in smooth eas iisapyt etr swl ssaas ecndefi can we scalars, as well as vectors to apply limits Because integrals valued Vector 3.4.3 reads vrsubsets over D ′ f aetecs of case the Take o xml,suppose example, For u | du dx = | du ,β α, r α ( D = α orsodto correspond and R aestepril,nttecmoethere): component the not particle, the labels dx/du α β f R V D du dx n hsgnrlssdfiiin o e fpitprilsat particles point of set a for definitions generalises This . f ′ ρ . scretbcueo u aua ovninaotintegrat about convention natural our of because correct is ,teJcba sjust is Jacobian the 1, = ( du ( M r r ) en asdniyfrabd cuyn volume occupying body a for density mass a being ) Z dV D or D = f du dx x X steinterval the is stelmto u vrcnrbtosfo ml vol- small from contributions over sum a of limit the as β > α ( α R x = a R ) .Either 0. 6= b dx m f M ,b a, = F ( α x = R , ( = ) M and r epciey Since respectively. , 1 = ) Z 3– 23 – dx Z D Z V ′ V = du dx f F ρ r ( ( i a R = ρ ( x r < r α n ( ) ( β ≤ r ) dV β < α u M f e 1 = ) 1 ,so 0, )) J i dV ( x , x X

ehave we = ( ≤ du dx α u dx/du )) b R m

D du. and du dx so a ′ r f α du R R | efrinstance for ne . x du dx D V n h eea formula general the and du dx aiirresult. familiar a , f F | ↔ ( du ( > x r u ) ) dV = ,i hc case which in 0, dx sivril and invertible is R = = β α f R R R a ( b V V − f F F ( du dx x i ( ( ) ) r r ) ) du dx ing dV dV V . . . e i , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. f ytecanrule, chain the By function smooth appropriate an Normals For and Surfaces Integrals 4.1 Surface and Surfaces 4 needn agn directions tangent independent f Y Example ..tecnr fms sat is mass of centre the i.e. ρ ( h oa mass total The . r osntcag ln t tany At it. along change not does .I h egbuho fti on,teeuto nedde indeed equation the point, this of neighbourhood the In 1. 2. nasmlrway. similar a in 0 = = ) ∇ directions). tangent independent two has c f ensasot surface smooth a defines osdrasldhemisphere solid a Consider : snormal is Z X = = = M ρ M M M du otesraea hspoint. this at surface the to Z ρ 1 d 0 f = a Z Z ( r 0 H r R a 3 ( ρdV xρ H u r dr 3 )= )) dV ρ R dr Z 0 (0 = π/ Z ⊥ = ∇ 0 2 2 = π/ S to sin f , M osdraycurve any Consider . 2 ρ 0 f · πa ∇ sin r , θ f du dr 3 Z on 4– 24 – H f on cos a/ 3 r 2 hsi eo ic h uv sin is curve the since zero, is This . Hence . =0 ρ/ a dθ θ with ℜ 8). S .I eput we If 3. θdθ Z 3 θ n utbeconstant suitable a and with =0 ¨ π/ Z r Z 0 ¨ 2 2 0 ≤ Z π du dr 2 ¨ π ϕ cos ∇ a 2 =0 ¨ dϕ π f and ¨ xr dϕ ϕ = ,teeeittolinearly two exist there 0, 6= ¨ R 2 ¨ z sin M r ( = ρ ( ≥ 0 = u a rd dϕ dθ dr θ in ) 4 ,Y Z Y, X, 0 4 , . 2 1 ihuiomdensity uniform with 2 cie ufc (it surface a scribes S π . = ): c h equation the , 3 8 a , S and Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. and ufcs hc ue u xmlslk h Möbius band: the like examples out rules which surfaces, unbounded and sphere, solid some suiu pt in h ufc scle orientable called is surface normal The unit sign. of a to up unique is closed called is Defining curve. closed smooth For Taking Examples ∇ oei h eod(ti iglra h px0 apex the at singular is (it second the in cone f surface A tec on nasrae hr saui normal unit a is there surface, a on point each At z c 2( = ,tesraedgnrtst igepit0 point single a to degenerates surface the 0, = ntefis w ae.Asraei bounded is surface A cases. two first the in 0 = f ,y z y, x, ( : r = ) f ( r . 2 = ) S = ) x n 2 a edfie ohv boundary a have to defined be can hc aissotl.W hl elecuieywt orien with exclusively deal shall We smoothly. varies which + x r 2 y . 2 + − y z 2 2 + = z 2 S c = eut nahproodfor hyperboloid a in results saoebtwith but above as x c x tews.Abuddsraewt oboundary no with surface bounded A otherwise. x For . 5– 25 – z z z > c ,ti ecie peeo radius of sphere a describes this 0, f c=0 where , y y y f z n ≥ ntefis ae n oadouble a to and case, first the in anra fui egh that length) unit of normal (a ∂S fteei ossetchoice consistent a is there if fi a ecnandwithin contained be can it if 0, ∇ f ∂S ossigo piecewise a of consisting 0 = > c stecircle the is . 0. ∇ f 2( = x 2 ,y, x, + y 2 table − = √ z c c ). . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. ipeeape,w a pcf h retto yreferrin by orientation the specify can we examples, simple odfieasrae ene w ieryidpnetdirecti parameterisation independent regular linearly two a need require we surface, a define To n hsdfie ntnormal unit a defines this and Area surface and A Surfaces Parameterised 4.2 normal: aallga on parallelogram h parameters the hieo in h hc fnra scle h orientatio the called is normal of choce The sign. of choice a hr the where ∂u ∂r and o noinal ufc,teui omli hndetermine then is normal unit the surface, orientable an For ∂v ∂r S δS r agn etr ocre on curves to vectors tangent are a lob endb oiinvector position a by defined be also can = u

∂u ∂r S and pto up , ∧ ∂v v ∂r δu .

∂u S ∂r uδv δu o δr ssetotas out swept is − δ δ em ihvco area vector with terms = u r δS stesaa area. scalar the is n ∂u ∂r = : on n δu ∂u ∂r ∂u ∂r n S + 6– 26 – ∧ h ml changes small The . ∧ S ∂v ∂r ∂v ∂r ∂v ∂r δ δ u v r δv 0 6= S and = + with Sn δS o v -terms S ayi oeregion some in vary δv v , r ∂v ∂r ( and ,v u, . oteoutward the to g eedn mohyon smoothly depending ) u n tec on,s we so point, each at ons u δv δu, osat respectively. constant, n ftesrae For surface. the of everywhere d rdc small a produce D . rinward or by Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. . ufc nerl fVco Fields Vector of Let Integrals Surface 4.3 for F parameters the of order the by ( S where section nashr fae 4 area of sphere a in hcigti result, this Checking hnteae is area the Then Example lmn is element cos ,v u, ∂θ ∂r ( is r S = over ) S ϕ oeta ihteecnetos h hieo h ntnorma unit the of choice the conventions, these with that Note aaee pc.Aan h re fteprmtr speciﬁe parameters the of order the Again, space. parameter ) ˆ j qiaett hieo ntnormal unit of choice a to equivalent ae easot ufc endby deﬁned surface smooth a be dS Thus . θ 1.3 : , = S e S ). dS θ

ihti retto sdﬁe by deﬁned is orientation this with en ato peeo radius of sphere a of part being cos = ∂u ∂r ∂θ ∂r = ∧ r ∧ a ( Z ,ϕ θ, 2 ∂v ∂r ∂ϕ S ∂r θ sin (cos F

πa = udv du α = ) ( Z θdϕ dθ θ r 2 a 0 = ) ϕ 2 2 bt fwihaekonresults). known are which of (both · ˆ π a i π/ sin dS +sin dϕ Z stesaa raelement area scalar the is cos S eut nahmshr fae 2 area of hemisphere a in results 2 θe = = Z dS ups that Suppose . u 0 r ϕ ϕ Z Z where , α ˆ j and = sin S D dθa ) F − F Z θi ( D sin ( 2 r v r r 7– 27 – sin ( ysmigadtkn iis h raof area the limits, taking and summing By . )

sin + ( ,v u, ∂u e ∂r ,v u, · θ r n k ˆ θ = α with ) ∧ and n dS )) 2 = a θ h ufc integral surface The . n aaersdb oa angles polar by parametrised , S ∂v ∂r · sin h uwr oml h clrarea scalar The normal. outward the , πa a stergo 0 region the is ∂ϕ ∂r

∂u ∂r ϕj udv, du 2 S (1 = en h prpit einin region appropriate the being cos + ∧ . a − ∂v ∂r sin cos θe θk α ϕ udv du ) where , . = πa ≤ ae θ 2 and ≤ r norientation an s favco field vector a of . sdetermined is l e α ϕ 0 , α = = ≤ − π ,ϕ θ, ϕ sin results ≤ ϕ (see 2 ˆ i π + . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. of u over integral the writing By nerlover integral fteprmtrsto,fragvnoinain ti call is It orientation. given a for parameterisation, the of ntnormal unit element area vector the is rmtegoerclntr ftelf adsd,teintegr the side, hand left the of nature geometrical the From a Example ∂ϕ ∂r ( 2 r S sin fafli ie h oino n ml oueo udat fluid of volume small any of motion the gives fluid a of ) h udflxtruhasraepoie hscleape T example. physical a provides surface a through flux fluid The of orientation the changing that Note hc sas qiaett hnigtesg fayflxinte flux any of sign the changing to equivalent also is which , = θe a r sin o peeo radius of sphere a For : θdϕ dθ θe n S hc seuvln ocagn h re of order the changing to equivalent is which , ϕ F h etrae lmn sthen is element area vector The . eoe h ii fasmo contributions of sum a of limit the becomes where , ( n r ) · δS n = = . = F F dS F e D ( ( r r r = ( ) = stelmto u vrsalrectangles small over sum a of limit the as ,v u, · δ a r n n S /a efudfrom found we S dS ) δS ) h uwr normal. outward the , · = 8– 28 – ∂u ∂u ∂r ∂r n δ δ S δ u v S seuvln ocagn h ino the of sign the changing to equivalent is ∧ ∧ u(r) ∂v ∂r ∂v ∂r F r udv du ( ,ϕ θ, dS uδv δu nsection in ) = dteflux the ed + al n u dS o R and -terms S r F = . D · v gral. dS 4.2 ∂θ ∂r ntedefinition the in of evlct field velocity he . sindependent is ∧ that F ∂ϕ ∂r through δuδv θdϕ dθ ∂θ ∂r = the , ae S = θ . , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. u upon smoothly depends it that assume We eua aaeeiain o h surface the for parameterisations regular δt e nttime). unit per hs ml that imply These eea) o n ml area small any For general). time fradius of atsa oriae.W have We coordinates. Cartesian steflux the is dS Example cos δt R Z S 2 · n hnei aaeeiaini oeepiil.Let explicitly. done is parameterisation in change Any = u θ δt n u ,so ), · δS ntesnegvnb normal by given sense the in , · a dS dS 2 : sin a = ⇒ of = = = u ih0 with θn δtu h u of flux the u ( = Z Z Z 0 0 0 through θdϕ dθ α α α − · θa dθ θa dθ dθ ≤ δS x, Z 0 ϕ otevlm rsigtewoesurface whole the crossing volume the so , 0 3 3 z , where , 2 π ≤ sin π u S nCreincodnts and coordinates, Cartesian in ) 3cos (3 dϕa , 2 o eoiyfield velocity a for , R θ ∂u ∂r ∂v π ∂r ∂u ∂r S δS n 0 and 3 π u 2 = = sin (cos ∧ · n n θ dS δ ntesurface the on − n ∂ ∂ ∂v ∂r ∂r ∂r steotadnormal, outward the is S θ n u u 2 sterate the is , ˜ ˜ · ≤ )sin 1) θ (cos u ∂ ∂ ∂u ∂v = n − u u θ ˜ ˜ = 9– 29 – stevlm fteclne shown: cylinder the of volume the is , ∂ ∂ ≤ + + )+2 + 1) 2 ( (˜ r S θ θ ,v u, u, α ∂ ∂ ∂r ∂r ytecanrule chain the By . · r = − si h xml nsection in example the in as , v v ˜ ˜ u v ˜ nt hti ilas eedupon depend also will it that (note /a ) ) πa ∂ ∂v ∂u ∂ )cos 1) π u ∂ fvlm crossing volume of ∂r v v ˜ ˜ S . S cos 3 u ˜ . ( = [cos h oueo udcosn tin it crossing fluid of volume the , ∧ 2 2 − θ ∂ ∂r ϕ and θ x v ˜ u t cos + − , 2 . δ + since S cos n r z stescino sphere a of section the is ( 2 2 3 ,v u, ) = θ θ a ] Z 0 α = 0 r and ) 2 = /a S π a cos ( πa S ie h amount the (i.e. − ( = r 3 2 ntime in sin (˜ cos dϕ ϕ u, ,y z y, x, 2 4.2 v ˜ θ α etwo be ) cos sin . = ) (4.1) /a δt 2 2 π t ϕ α in is in + Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. osdrteseilcs fasurface a of case special the Consider .. hneo aibe in variables of Change 4.4.2 parallelepiped: a aua order natural h ml changes small The hnigvralsin variables changing h nerl a ergre ietya iiso us oe To sums. of limits as use directly parameterisations, regarded other be can integrals The F element area scalar The since nsection in dS functi scalar of integrals orientations for definitions and analogous integrals have surface We and Line Integrals Volume and Surface 4.4.1 Line, Comparing 4.4 ( hycnb eue opeiu ae,frinstance for cases, previous to reduced be can They rvddta ( that provided r eesr.Ohrknso nerl a ecniee,fre for considered, be can integrals of kinds Other necessary. ∂ ∂ x ( (˜ ( u, u,v ,v w v, u, ( ln curve a along ,v u, v ˜ = h etrae element area vector The h eut for results The ) )

> ) ∂ ∂ ∂u ∂r y , ( (˜ f orientation of independent R R 0. othat so ) u, u,v S C v ˜ ie egho area or length gives 1 = ∧ ( ) ) dS f ds f 4.2 ,v u,

∂v ∂r udv du , R ) α R ,v u, β udv du , S C with 0) f = n (˜ and ) u v δw δv, δu, roe surface a over or ( ℜ r ⇒ δr dS d ) ycneto,telmt on limits the convention, By . 3 ℜ dS d u β < α ˜ dS 2 a edrvdb trigfo mohparameterisation smooth a from starting by derived be can = ∂u = ∂r stherefore is on nsection in found v ˜ = u, = ∂u ∂r .

∧ ∂u ∂r v ˜ R is ∂u ∂r except ds δu orsodt uodi aaee pc,producing space, parameter in cuboid a to correspond orsodt h aeoinainon orientation same the to correspond ) ∂v D ∂r ∧ f + ∧ = (0 = ( ∂v ∂r r ℜ ∂v ∂r ∂v ∂r (

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δw revisited ∂ ∂r nerl eas allow also we integral: ∂ ∂r = u ˜ R R u ˜ C + S ∧ J ∧ du dr F F S o n ercvrtefruafor formula the recover we and ∂ ∂r t a du ∂ ∂r = · − v · ˜ → rnormal or · v ˜ u t n d R ,

terms. d u ds ∂ ∂ ˜ C ,D D, dD dS d and ( ( u,v x,y d u F ˜ xample ons v ˜ ) ) = ds rmtedeﬁnition the From . v ˜ v → f

= nerl r nthe in are integrals ∂u ∂r aut hmusing them valuate n rvco functions vector or D . h choice the R R ∧ C ′ C and ( ∂v ∂r F a β > α S ·

ds ic then since , F dS udv du ) , r R ( ds → hr if there ,v u, S . dS f dA and = ) ). . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. lmn is element opr hswt h etreuto o icepsigth passing circle a for equation vector the with this Compare ice oscn re naclnt,b taking by length, arc in order second to circle, Normals Let and Curvature Curves, Surfaces and 5.1 Curves of Geometry 5 volume of h point the a aldtecurvature the called Let eiaino h euti section in result the of derivation agn vector, tangent for ntepaedﬁe by deﬁned plane the in , n ycmaio ihEq. with comparison By ups httecrecnb alrepne around expanded Taylor be can curve the that Suppose t r .Teeqatte a ayfo on opito h curve. the on point to point from vary can quantities These ). ′ ( s = eacreprmtrsdb h r length arc the by parameterised curve a be ) κn s dV δV 0. = hr h ntvector unit the where = = t 2 |

J 1 = ∂u ∂r | ud dw dv du δu δw w a make can (we r ( ⇒ s · ∂w ∂r = ) ∂v ∂r t = δv t δv · and t r r ∂v ∂r ′ where 0 + (0) + (0) ∧ pcﬁsadrcinnormal direction a speciﬁes 0 = 5.1 ∂w δu ∂r n eseta ecnmthagnrlcret a to curve general a match can we that see we , sshown: as , 3.3 δw ∂u ∂r st sr J κ n .

| r ′ ( s = oiieb hoiga prpit direction appropriate an choosing by positive 0 + (0) s =0 = 1– 31 – (0) scle h rnia normal principal the called is ) n ∂u ∂r + | J · | 2 1 2 1 uδ δw δv δu ∂v ∂r θ s s 2 2 ∧ a κ r a ′′ | ∂w ∂r s 0 + (0) =0 1 = = t n eddc httevolume the that deduce We . /κ + ∂ ∂ s O ( ( Since . u,v,w h aiso curvature of radius the , O x,y,z ( s ( 3 s ) ) ) 3 s hsi nalternative an is This . ) rough . 0: = otecreif curve the to t ( s = ) r 0 ihradius with (0) dr ds and saunit a is κ t ′ ( (5.1) s 0. 6= is ) at Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. ecnsuytegoer fsrae hog h uvswhic curves the through surfaces point of given geometry a the (non-examinable) study Geometry can Intrinsic We and Surfaces neg and 5.2 helix, right-handed a for z positive is torsion one). is the it non-zero), is (and curvature the torsion (and plane a in is iet hr re nisTyo xaso,adhwtecurve the how and expansion, Taylor its in order third to like fw oetepae eas oetecreCpoue ytein the by produced C curve the move also we plane, the move we If torsi of radius a defines one curvature, to Analogously plane. curvat the changes length: basis arc this of how functions τ in scalar two encoded by is specified curve be the of geometry The xrse ntrso egh,age t.wihaemeasure are which etc. angles lengths, of terms in expressed range a derivatives the to expression K of intersection binormal t etecraueof curvature the be ,n s, , ( s = = ) [ The Given curve a for practice, In o-xmnbeadntlectured: not and Non-examinable κ κ , min − κ ytkn derivatives taking by b hoeaEgregium Theorema κ obe to min ′ t max · ( n s and ) . ≤ P S scle h asincurvature Gaussian the called is b κ nasurface a on ihapaecontaining plane a with = ≤ n t C ( ∧ κ s at max ecnetn oa rhnra ai ydefining by basis orthonormal an to extend can we ) n P where endaoe stepaei oae about rotated is plane the As above. defined , r ( u d/ds sthat is ie ntrso oeparameter some of terms in given ) S d/du ihnormal with , κ afterwards. min n hnuigteCanRl ocnetthe convert to Rule Chain the using then and K h oso eemnswa h uv looks curve the what determines torsion The ] and 2– 32 – P sintrinsic is n n n : κ b max t n . osdrtecredfie ythe by defined curve the consider , r h rnia curvatures principal the are C s otesurface the to r.Ahlxhsconstant has helix A ero. nieyo h surface. the on entirely d ure tv o left-handed a for ative on κ σ it u fthe of out lifts ( u i nte.At them. on lie h ln t hscan This it. along s ecncalculate can we , 1 = escin Let tersection. n h torsion the and ) S ..i a be can it i.e. , /τ facurve a If . n efind we , b ( Then . s ,the ), t n , κ Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. Since ra ice.I’ ayt e that see to easy It’s circles. great ehv aiyo edsctriangles geodesic of family a have We S θ ecncekti hoe when theorem this check can We geodesi a consider geometry: intrinsic using example an For extrinsic ergr h gradient the regard Notation We and Definitions Curl 6.1 and Div Grad, 6 o n atsa coordinates Cartesian any for Then u fatinl ocre space. curved to triangle a of sum Let eemnslnt n nlso t u hnw td nygeo e only this relativity. study with general we away and do then geometry we but Riemannian Eventually, it, structure. on intrinsic this angles and length determines 1 hsmasthat means This . + θ hsi h tr fitiscgoer medn surface a embedding - geometry intrinsic of start the is This θ 1 θ 2 θ , K 1 + 2 + θ , emtyi od ihhwsrae r meddi h space. the in embedded are surfaces how with do to is geometry scntn over constant is θ θ 3 3 2 = + eteitro nls(enduigsaa rdcso tang of products scalar using (defined angles interior the be θ π 3 + = α π ∂D geigwt h rdcino h hoe.B otat t contrast, By theorem. the of prediction the with agreeing , + ∇ R ∇ ossso geodesics of consists D = f dS K S sotie rmasaa field scalar a from obtained as i ∂x , ∂ x R D + i steGauss-Bonnet the is n rhnra basis orthonormal and dA K j S ∂y ∂ κ sashr fradius of sphere a is min + 3– 33 – D = k = sson with shown, as , ∂z ∂ h hretcre ewe w points. two between curves shortest the : K κ ( = max × rao = D of area ∂x ∂ 1 = , hoe,gnrlsn h angle the generalising theorem, ∂y ∂ /a , e f i ∂z so a ∂ ( sesection (see o hc edsc are geodesics which for , triangle c r θ bdigatgte,in altogether, mbedding ) yapplying by ) K 1 . = a 1 1 = 2 ( ,θ α, nEcienspace Euclidean in a ercapcsof aspects metric 2 /a α D = ) 2 2 constant. a , 1.1 = n vectors). ent nasurface a on ,or ), θ ∇ α 3 Then . = = e π/ i ∂x he ∂ 2. i Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. rdiv or n o n constants any for right-hand and orthonormal are that vectors basis use us Let e ieeta prtrand operator differential n savco field vector a is and field scalar a is and field vector xml utabove, just example Example i u uli pcfi to specific is curl but , ∧ h curl The rd i n ulaelna ieeta operators: differential linear are curl and div Grad, ∇ Since e j (nabla of = ∇ : F ǫ ∇ ijk ∇ F ∧ F is: · e rdel or , = of ( F F sa prtr reigi motn,so important, is ordering operator, an is k r . = ) xe = = F λ z = sdfie obe to defined is ˆ ∂x i and . F ∂ tcnas ewitnas written be also can It . sbt noeao n etr ecnteeoeapyi to it apply therefore can We vector. a and operator an both is ) + ∂x ∂y F i xz ∂ ∂ ( ( r y xe ∇ · ∇ ˆ ( ( ∇ i ) µ 2 xyz y ∇ e n ( + z F rdaddvcnb nlgul endi n dimension any in defined analogously be can div and Grad . sin ∧ i 2 + ) ∧ eas tue h etrproduct. vector the uses it because 3 = ∇ = · sn ihrtesaa rvco rdc.Tedivergence The product. vector or scalar the either using sin ∇ ∇ ∧ F ) ( ( xe x λF λF ( · xe − ˆ j ∂y λf ( = x F z ∂ + ) z = − ∂z ∂ + + ( ∂z − ( = ∂ + e xyz y yz e i ( 2

µG µG k + ∂x ∂y µg y ∂ sin e ∂x ǫ ∂ F e ) 2 ∂ i k ijk ˆ ˆ j 1 y 1 i = ) = ) = ) sin 1 ∂x ( . 2 ) 4– 34 – xe + ∂ x F sin ∂x ∧ F e + ) i ∂ i x y z 2 ) 2 ∂x 2 λ λ λ ) ( 2 ) ∂ · F x ∇ ∇ ∇ j cos ∂x ( F e ∂z j ∂y ∂ ∂ F ∂ k ˆ ˆ 3 savco ieeta prtr nthe In operator. diﬀerential vector a is i f e 3 3 ∧ · j j + ( F x

e + + = ) xyz F j . k ˆ = ) + xyz µ . + ∂z ∇ ǫ ∂ = ) µ ijk µ g ∇ ∂F ∂x ∂z ( ∂ ∇ xe ∂F ∂x . · e i i ∧ z G F z j i ) 2 + G e − ∇ · k y ∂x ∂ ed: sin = ( xyz x F e + i i ) ∂x · ∂ xy. e i j ˆ j sascalar a is = + δ ij and a , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. epoe sn ne oain h properties The notation. index using proved be e stk clrfields scalar take us Let Properties Leibniz 6.2 sn h ebi rpris ecnetn u eut osom to results our extend can we properties, Leibniz the Using r rqetyueu.Teeaeas oels omnyuse commonly less some also are There useful. frequently are ih adsd fEq. of side hand right Examples Examples Examples 2. .[ 2. 1. 1. 3. 1. 2. aigEq. Taking ∇ ∇ ∇ ∇ ∇ ∇ F F j ∇ r ∧ · · ∧ · ∂G ∇ ∂x α ∧ ( ( r : : Consider : ∧ r F ( r i j = · ∇ ( = r α ∇ = − ( ( α r ∧ F F ( e r = ) ∂x ∂x F i F ǫ = ) ∧ G ∂x ∧ ∧ ijk 6.2 i i ∂ j · = ) G i ∂G ∂x = G G G ∇ ( ∂x ∇ r )] ∂ j ( = ) = ) = ) i aogwt the with (along ( δ α i . i r ( ii ∂x ( = ) r ∂ Hence α x r = 6.1 α 3. = i ∇ ) j f = F F ( ∇ ) ) · ∇ F e ǫ e ∧ ∧ ( r and ijk ilsislf adsd,ie h oml for formula the i.e. side, hand left its yields k [ ∧ x ∇ i · ∇ [ ∇ αr ∧ ∧ + F r i ( ( 0 = e ( ( F fF fF ( · fg + ∇ G i F α r F ∧ j , G g − α ( ) ) r . r ∧ ∇ ( = ) ( = ) ( = ) ( i ∇ 1 · ) n etrfields vector and 2 · α ∇ ∂x G = − ∂r G G ∇ = ∧ · )] i ∧ + ) r G ∂x − ∧ x i = G ∂ F ∇ ∇ ∇ G ( = ( i i = r ) F x ∇ ( 5– 35 – f f f e )] G i ↔ ǫ i i = ) ) ) ijk ∂F αr i ∂x αr · · = ⇒ g ∧ · ∧ ( = F αr F + F G ∇ α α j i F ( ǫ ( + ) 2 j − − F G ∇ kij α f + G r ∧ onepr n usiuigit the into substituting and counterpart 2 1 j + − j ( r ∂x x ∂r k ∂G ∧ ǫ ∂x G ∇ f 2 ) + i = ) kpq f r i /r G ( · F ) F g ∇ ( F 2 = ∧ i . j r ) ∇ F ∇ · ( + ) G , , j = − + ǫ r · j ∂G ∂x kij ∧ ∂G F ∂x x 0 = hi ebi rpriscan properties Leibniz Their . αr ) ( r i F F and ) F α p ∂F i or , q ∂x F j 3 ( = (3) α . ) − ∇ · . j i − ( = ∇ · G 2 ( r ∂x k ∂r ) F ) rpristoo: properties d G δ = − i G ip ∇ · rvosexamples: previous e i α = δ αr F ( + jq 3) + j x ) ǫ α G i − jik G − /r 1 r δ ∇ · r ˆ ∂G . ∂x iq α . i δ k ) jp . F ) F j ∂G ∂x (6.1) (6.2) p q = Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. nasaa field, scalar a On . eodOdrDerivativesIf Order Second 6.3 .. re’ hoe i h plane) functions the smooth (in For theorem Green’s Examples 7.1.1 and of kind Statements right the match 7.1 to derivative of kind gener To right the differentiation. need of we inverse the is integration Fund that the generalise results closely-related following The Theorems Integral 7 components. using by checked be may this Again, o oevco potential vector some for some for derivativ partial mixed of symmetry the from follow results mle ust selater). (see subsets smaller ident vanish curl and div grad, field of combinations different Two h ovreo ahrsl lohlsfrfilsdfie nall on defined fields for holds also result each of converse The F 1 esalsei section in see shall We If h alca prtri endby defined is operator Laplacian The = f F , ∇ ∇ or h edi aldconservative called is field the , f ∧ clrpotential scalar a , H ( ∇ r endol nsbesof subsets on only defined are f 0 = ) ∇ ∇ 2 f 2 P Z = = 8.2 n o n etrfield vector any for and A ( ǫ ,y x, ( kij ∇ ∇ ∇ A ∂Q ∂x o hscncag o edwihi o endo l of all on defined not is which field a for change can this how 2 nti case, this In . ∂x · ∇ · and ) A ( ∇ ∂ lo gi for again Also, . ∇ − i 2 ∇ = ∂x f = · f ∂P H ∂y hra navco field, vector a on whereas ) ∇ ∧ j ∂x Q ( 0 = F 0 = ) ∇ ( ∂ i ,y x, ∂x dA 2 0 = · ǫ , 6– 36 – If . A ⇔ i ): = ) r( or ⇔ H H ∇ − ∇ Z ℜ scle solenoidal called is C F ∧ kij = ∂x 3 ( ∂ H F then , dx P = ∧ 2 ∇ ∂x A 2 ∂ endon defined 0 = ( ∇ , + 2 ∧ i ∇ A ∂x ∇ f A ∂y + k edi aldo irrotational or called is field a , ∧ ∂ f j · , 2 mna hoe fcalculus: of Theorem amental dy Q A 2 ( or 0 = ∇ + ) ls ohge dimensions, higher to alise . A ∧ . ∂z es: 1 ∂ ) a edfie nyon only defined be may A , ℜ 2 2 of .Bt fthese of Both 0. = ) cly o n scalar any for ically: 3 ) . integral. , . ℜ 3 : ℜ 3 . . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. n lcws rettoso h neirbudr.Frex For boundary. interior the on orientations clockwise orientati and anti-clockwise with curve) closed intersecting itn fanme fdisconnected of number a of sisting where moh o-nescigcoe uv,taesdanti-cl traversed curve, closed non-intersecting smooth, Examples 1. .If 2. re’ hoe lohlsfrabuddregion bounded a for holds also theorem Green’s P meitl oteFnaetlTermo aclsfrinte for Calculus R of mension: Theorem Fundamental the to immediately Also, h etclsget otiue digteetointegral two theorem. these the Adding contribute. segments vertical the nytehrzna emnscontribute. segments horizontal the only parabola nerlaoegives above integral A A A sabuddrgo nte( the in region bounded a is = − : ∂P sarcagewt 0 with rectangle a is ∂y x R 2 A y dA ∂Q ∂x and y 2 = dA 4 = Q R = 0 a ax = dx R 0 xy b n h line the and R 105 104 dy 0 b 2 dy a R ⇒ 4 0 a ( seEape he I). Sheet Examples (see − dx ≤ R b y A ∂P ∂y opnns(ahbigapeeiesot,non- smooth, piecewise a being (each components ( x y ( C ∂Q ∂x y ,y x, = ) 2 ≤ 2 7– 37 – = ) y =4ax − x ln ihboundary with plane ) a, 2 R = x 0 0 R a 2 C A 0 ) dx a b ≤ [ ohwt 2 with both , A Q dA [ y − ( ,x a, ≤ P = a a ( C b ,b x, ) R hnGenstermreduces theorem Green’s then , − C C A n nteetro boundary exterior the on ons x ockwise. x x 3 + ) Q C ihteboundary the with 2 1 (0 dx y P a y , ample, ( ≥ ]= )] x, + ∂A ) = 0)] y xy ie h eutin result the gives s − ≥ R = C 2 Qdy rl noedi- one in grals dy C R 2 C piecewise a , If . a dx P hneach then , ic only since , ∂A C since , sthe is con- Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. uv,adwhere and curve, ic h oiotlparts horizontal the since single a of case the to related be can These where section from tion t mohcoe uvs ihtesm uedtriigteori the determining rule same the ponent. with curves, closed smooth that ( o mohvco field vector smooth a For theorem Stokes’ 7.1.2 together). close brought dS ∧ n h hoe lohlsif holds also theorem The = S onsotof out points S n so h left the on is dS sabuddsot ufc ihboundary with surface smooth bounded a is ), t stnetto tangent is S 2.3 S . and : lentvl,mvn around moving Alternatively, . C cancel F aecmail rettos ndetail: In orientations. compatible have ( r Z C ): S ∇ ∂S ( nteopst ietos(ntelmtta hyare they that limit the (in directions opposite the in dr S ∧ = ossso ubro icnetdpiecewise disconnected of number a of consists F t · 8– 38 – dS ds ). onaycmoetuigteconstruc- the using component boundary = t Z and C F n · dr aecmail rettosif orientations compatible have C n , ∂S t ndirection in n = t C C naino ahcom- each on entation icws smooth piecewise a , n t with snra to normal is n ‘up’ ⇒ S Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. r dS Example u ( ϕ = ( = ) = ∇ a 2 ∧ a Take : sin sin F θe Thus . Z α C r cos F θdϕ dθ S · ,a ϕ, ob h eto fashr fradius of sphere a of section the be to dr R , S = = sin F ∇ a Z (0 = 3 0 ∧ α 2 sin π sin F a | 2 xz, , sin · α ,a ϕ, C S dS cos {z α x 0) cos cos = α ∇ ⇒ 9– 39 – Z πa ϕ α 0 } 2 ) a | α π 3 , cos cos cos {z ∧ 0 z ≤ F 2 α α } a dϕ ϕ ϕ a | n=e ( = sin sin ≤ α − 2 α = r h onayis boundary The . x, π {z cos dy . πa 0 z , a 3 dϕ ϕ 0 : sin ,a nsection in as ), ≤ } 2 α θ cos ≤ α α. : ∂S 4.3 = with C : Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. o mohvco field vector smooth theorem a Gauss’ For or Divergence, 7.1.3 > a > b b, ( h boundary The Example and o iceo radius of circle a For ( R hc ie h normal the gives which − − C b R x, 0 F n ≤ sin 0 ∂ρ ∂r · z , onsupwards. points ϕ dr ,R ϕ, ⇒ ). Take : ∧ ⇒ ) nclnrclplrcoordinates polar cylindrical In 0). ≤ − ∂ϕ ∂r dS 2 Z R cos π S C . a ( = (cos = ∇ ∇ ∂S F ϕ, S ∧ ∧ − · ) Then 0). ob h eto ftecone the of section the be to F F dr = ρ cos · · ϕ, C = R dS dS n sin b , ϕ, ssono h iga.Taking diagram. the on shown as π F − C = ( = ( ( ϕ, b − R r R C 3 Z Z ), ρ C C C ρ 1) − a : a C R b sin 2 hr h ice aeteoinain sshown. as orientations the have circles the where , b b ∇ x F cos a dρ ∧ r 3 C ,ρ ϕ, ) · · ( a ( = Z . 2 − dr F S 0 ϕ o h rettos elo onthe down look we orientations, the For ρ z 2 ) 0– 40 – dV π = + R sin ρdϕ, dρ dϕρ n φ C cos R ρ = ,ρ ϕ, 0 2 a 2 ) π 2 Z ,R ϕ, (3 ρdϕ dρ R S r cos 3 / ( F ,ϕ ρ, + 2 cos sin · ϕ, z dS 2 2 ( = ) o 2 cos )=( = 0) ,R ϕ, dϕ ϕ = y x ϕ/ 0 ) ρ 2 = − cos F + )= 2) ρ πR ≤ (0 = y cos ,ρ ϕ, 2 , 3 ϕ π hence , a ϕ, xz, , ( sin ≤ b ≤ 3 − − 2 ,ρ ϕ, ρ z 0) π sin a R ≤ 3 ⇒ ∇ ⇒ ), ∂S ) ,ρ ϕ, . b a F r z (where ′ ≤ ( ) ∧ · − ϕ dr F axis ρ = ) ≤ = = Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. omlto normal t 2d the or theorem Stokes’ from theorem Green’s Proving Theorems Integral the Proving 7.2.1 and Relating 7.2 where ufc ihnormal with surface egh ihtesneson ( shown: sense the with length, Let Hence ftetheorem. the of F Example On S On hemisphere. ( = 1 hmshr)+ (hemisphere) (0 = A S S xd,dy/ds, dx/ds, 1 2 eargo nte( the in region a be V : R : ec theorem gence , dS S Take : 0 1 sabuddvlm ihboundary with volume bounded a is dS z , C ⇒ F = · out + Z = dS n S a 1 dS V n ) of + F F )i h nttnetto tangent unit the is 0) ∇ ⇒ S dS ob h oi hemisphere solid the be to R · · ( = A 2 S n dS dS (disc). . 2 = onigoutwards. pointing F ,y z y, x, n 2 = = = · · − = F dS ,y x, Z z (0 t πa ( 0 .Then, 1. = ) z ∧ , 2 5 = adS /a π ln ihboundary with plane ) 0 3 + k , x dϕa 1) where , ( a / − s ) 3 ) adS /a dS 1 3 πa 3 y , . Z cos ( . 3 0 S s 1– 41 – π/ − ) F 3 2 k S , 2 θ = R πa n 0): 1 dθ (0 = · V − C dS a 3 ∇ sin cos 2 1 and 2 = , x · = cos ∂V 0 n 2 θ n F , θ / (cos + 1). − n (cos 2 3 dV πa = θ dS a y ( = 2 k 2 C 0 π/ 3 S θ θ + 2 = geigwt h prediction the with agreeing , t dy/ds, icws mohclosed smooth piecewise a , 2 1) + cos + C = z = n so and 2 ∂A / ≤ 3 5 3 a πa πa 2 θ − a aaeeie yarc by parameterised sin ) 2 3 dx/ds, 3 z , h oueo the of volume the , . R θdϕ. dθ θ S 2 ≥ F 0. )i h unit the is 0) · dS ∂V = diver- = − πa S = 3 . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. and and eea etrfil rmalna obnto fteto The two. the of combination linear a from field vector general ese opoete2 iegnetheorem divergence Th 2d Divergence the 2d prove the to Proving by seek theorem We Green’s Proving 7.2.2 consider Now gi,Genstermeutstergthn ie fEqs. of sides hand right the equates theorem Green’s Again, hsa plcto fSoe’termfor theorem Stokes’ of application an Thus plane range Eqs. h direction the (0 eae h ethn sides: left-hand the relates vec a for theorem divergence the of version two-dimensional ehv hsnti re fitgainbecause integration of order this chosen have We n , steotadnra.I ucst consider to suffices It normal. outward the is 0 ie any Given , 7.1 ∂Q ∂x Y x , − a easnl nevlo no fintervals: of union a or interval single a be may 7.2 ∂P ∂y hra restermeutstetorgthn ie.Th sides. right-hand two the equates theorem Greens whereas ) i G . P a etetdsmlry exchanging similarly, treated be can Then, ( ( = ,y x, re’ theorem Green’s Q, and ) tks theorem Stokes’ Z − Z A P, Z A ( ∇ A ∇ Q 0) Z Z ( Z ∇ C ( ∧ C A · ,y x, ∇ ⇒ G G F G ∇ · G ) · · dA ,cnie h etrfield vector the consider ), · n · t G dA k · ds ds = G dA dA ⇔ = = = Z 2– 42 – = ddvrec theorem divergence 2d X ⇒ = = Z Z Z ∂Q ∂x A C C re’ theorem Green’s Z Z ( Z dx P C A dx P ∂Q − ∂x Y = ( F x R ∂Q ∂A ∂x ∂P ∂y A ∂G ∂y − G qae h w ethn ie of sides left-hand two the equates ∇ G + ehave We . + ∇ − ∂P ∂y = · dy dy Q · dy Q · x n ∂P G ∂y G G ) ds and ( dA. dA = ,y x, ) dx. dA. ∂G F ∂y y = ) o field tor n hnw bana obtain we then and , j eedn on Depending . ( = eas field a because , n, R C 7.3 ,Q, P, = ∂A , 7.4 G G 0) · hra the whereas nte( the in n ∇ ⇒ ds erefore, eorem where , ∧ x H (7.1) (7.3) (7.4) (7.2) (7.5) the , ,y x, F in = ) Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. Consider o region a For segments boundary the to corresponding o any for Y x y Hence C x − x in nsm interval some in C I A . Z Z Y f‘ipeshape’ ‘simple of x I ∂G ∂y ∂G ∂y X δ δ δ δ dy s x s x n dy = n n Z A y I dx − y j + ( uhthat such x ( ) x = = ) ∂G Z Z ∂y C I + ( C x G dy 3– 43 – G + ( ,y x, Y C · = x n ± G On On sasnl interval single a is + ds sw aedanabove. drawn have we as ( ( x ,y x, + )) C C − + Z + − : : C ( − x G n n )) G ( · · ,y x, j j − · C n ≤ ≥ G + C − ds. and 0 and 0 ( ( − x ,y x, ))) n y − + I j j ( dx ( dx dx x x n )) ) = = ≥ − n y n · ≥ j · ds j y ds − ( x x ), Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. sson etake we shown, as give .Poeddrcl,ee if even directly, Proceed 2. .Divide 1. o regions For R emnscrepnigto corresponding segments od nec subset each on holds C G · n A ds rvn h result. the proving , nosmlrsbeswihd aetepoet bv.Tere The above. property the have do which subsets simpler into A fmr eea hp,either: shape, general more of I = X Z and I A i y Z Y ⇒ ∂A Y x x A A sasmo nevl n e u vrall over sum a get and intervals of sum a is x thlson holds it ∂G ∂y 3 1 = nterange the in C A C − dy + 4– 44 – C C C 2 1 + 3 X A C 2 dx I − C A = + n h nerl over integrals the and I X i A Z x 4 A C i 5 G · n ds. C ± obn to combine boundary sult Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. ,v u, Then, .. re’ theorem Green’s 7.2.3 osdraprmtrsdsraegvnby given surface parameterised a Consider o parameter for ntebudr.Itgaigwt epc to respect with Integrating boundary. the on Eq. shown have we therefore field vector smooth a For yteCanRule, Chain the By oso hs ent that note we this, show To h ih-adsd fEq. of side right-hand The hs eoti that obtain we Thus, yuigan using by enwitgaeec ieo Eq. of side each integrate now We ln.Spoetebudre r ie by given are boundaries the Suppose plane. ( ∇ ∧ F S ǫǫ ) · t dniy eseta h ih-adsdso Eq. of sides right-hand the that see We identity. ( . ∂u ∂r ∂F − ∂u v ∂v ∂r Z ∂F = ∂u A ∂F = ) ∂u F v ∂u F ∂ v ∂S 7.6 Z − ( ∂F ⇒ u ∂u ǫ r − A 7.6 kij ,w define we ), = ∂F F F v ∂v is n tks theorem Stokes’ F ∂F u ∂F ∂v u ∂x . F − i u du du dt ∂x ∂v u · j i = ∂F ∂v ǫ ∂u ∂r ( = + i + kpq ∂x u ∂u F F 5– 45 – F , 7.6 ∇ = ∂x v ∂u v j dv dA dv ∂x dt ∧ ∂v ∂x ∂F r p ihrsetto respect with ( ∂x = F i ∂v ,v u, = = j i t ) v ∂x Z q eobtain: we , ∂u F · Z ∂x ∂F = ∂S ,crepnigt region a to corresponding ), = ( ∂A s k · ∂u ∂r ∇ F j F i ∂x ∂v dr ∂x dt ∂u v − : ∧ · · − i j u dr ∂v ∂r ∂F + F ∂x ∂A ∂x ( ∂v ∂v ∂r t A . . ) F · j i v , i dS i ) udv du ∂u∂v . ∂ ( . t . ∂x ∂F 2 and ) x 7.8 i j i : . − and ∂S ∂F ∂x j i 7.7 : r ( , u r equal, are u ( A t ) (7.10) nthe in v , (7.9) (7.6) (7.7) (7.8) ( t )) Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. F h onaysurface boundary The Eqs. of sides left-hand the equates theorem Green’s .. rvn h iegne(rGus)termi d outl 3d: section in in theorem approach Gauss’) the (or extend divergence We the Proving 7.2.4 iegnetermi ieri h etrfil,w consider we field, vector the in linear is theorem divergence sides. right-hand their S si them With treat case). and general directions the get other to in combinations fields vector provide we where no fdson intervals). disjoint of union S ± i − , S F sgvnby given is h ocuini httelf adsdsbigeul(..G (i.e. equal being sides hand left the that is conclusion The + v Z on xy A sterneo the of range the is δ k n k ⇔ δ A δ S S z tks hoe for theorem Stokes’ n = z ± ytefc that fact the by R n h nerlo hsover this of integral the and of diagram, the From S ( ,y x, − S x ∂F ∂z Z k F = V tteed fsm nevlin interval some of ends the at ) hngives then ∂V ∇ · z n · z F a edvddit surfaces into divided be can dS nerto o fixed for integration dV ri total in or , V F D F ( 7.2.2 n = ,y z y, x, on 6– 46 – sawy h outward the always is Z δA D S rmtodmnin otre ic the Since three. to dimensions two from +

). = ( (x,y) Z ∇ ,y x, Z R ± xy ∂V · k F )) D ∂F z ∂z · F xy = − n 7.9 dS a ewritten be can ˙ ,y x, ! δS F ∂F ∂z , srequired. as , ( 7.10 dA, ,y z y, x, ecnie h integral the consider we , on i eea hswl ethe be will this general (in Z y S hra tks equates Stokes’ whereas S F xy ± ± oml.Teintegral The normal). − . ial,tkn linear taking milarly, ( = nwhich on tesgsaegiven are signs (the ,y x, enstermfor theorem reen’s F ( )) R ,y z y, x, S + k F ine ± ) k k · n · (again, n dS ≥ + 0. Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. At at limit the taking on exact becomes This field vector a Curl for and theorem Div divergence of The Expressions Integral Theorems Integral 8.1 of Applications Some 8 n iial,Soe’termapidt ml ato plane a of part small a to applied theorem Stokes’ Similarly, hs r aietycodnt needn entosof definitions independent coordinate manifestly are These point a gives r 0 , r 0 gives Z ∂A F Z ∂V · dr F n = · ∇ ∇ · dS Z · A F ∧ = ( ∇ F lim = Z ∧ r A V lim = ∇ F n 7– 47 – ) V · · → o F r A n F 0 0 r → V dV 1 0 dA ple oasalvolume small a to applied 0 : R A 1 ∂V ≈ ≈ R F F ∂A F ( ( ∇ n F · ∇ · dS · · F dr . )( ∧ . r F 0 i n curl. and div ) iharea with )( V. r 0 ) A. A V n normal and containing Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. osdrtrewy odfieavco field vector Poten a Scalar define and to ways Fields, three Irrotational Consider and Conservative 8.2 f on conservative a of A,()ad()aealeuvln for equivalent all are (C) and (B) (A), Examples eoiyfil,dsrbn udflow. fluid describing field, velocity a ( a ℜ .Divergence 1. .Curl 2. a hw nsection in shown was ) esalnwso ht(C) that show now shall We 3 (B) (A) (C) tion. let , xdtm interval, time have fixed therefore we fluid, of quantity δS otebudr 2 = boundary the to R fluid. of quantity example, material For fixed a by occupied volume of frotation. of around velocity angular local the stevlm cuidb h xdmtra uniychanges quantity material fixed the at by above, occupied volume the as at oueicessa h aecntn eaiert everywh rate relative constant same the at increases volume ihcntn nua velocity angular constant with ∂A r ie e nttm) Taking time). unit per (i.e. R ∇ F h xrsin bv lo nepeain of interpretations allow above expressions The : u 0 C aetepaearea plane the Take : C ..tietelclrt frtto.Frexample, For rotation. of rate local the twice i.e. , · ∧ = F dr and F ∇ · = dr r f 0. = eko rmsection from know we : etrfil,ad()dfie nirrotational an defines (C) and field, vector 0 C ˜ R , o oesaa field scalar some for sidpnetof independent is ∂A ∇ eaytocre from curves two any be u ( u · r u · = ) t lim = ds 2.4 πa δt n αr V (2 = ∇ · (A) . ˙ 2 ω V udflw u fteorigin. the of out flows fluid , steapoiaevlm htcossi time in crosses that volume approximate the is hr edefine we where , ⇒ → ∧ A πa 0 B n (B) and (B) ⇒ u /V V ob ico radius of disc a be to ˙ ) C C a hw nsection in shown was (C) lim = V × ω F o oegvnedpit n orienta- and points end given some for , 8– 48 – bu 0 about ob h oueocpe yafie material fixed a by occupied volume the be to h vrg of average the f h oa ie at (i.e. local the , ( n 4.3 ( r Then . V r on ) ˙ a n ). F → a = that 0 A n B r qiaetdefinitions equivalent are (B) and (A) , → πa ⇒ R ℜ and , 1 ∂S 3 2 A in (A) b R (A) . (2 ω : u S πa 1 = · u u t ∇ dS · u 2 ⇒ dS ∧ ω ℜ /a · fraboundary a (for t B with (B) 2 = ) ( 3 a h opnn tangential component the , ω sterate the is Given . r × ∇ . 0 ∧ eaiert fchange of rate relative ) teaeaeof average (the · field: u r ω u 6.3 2 = ) = ∇ and (via ω R F · ere. C u ∧ ω ( ∇ tials ffli crossing fluid of F r wc h rate the twice , r .Te,from Then, ). 3 = byn (C) obeying ) ∇ ii motion rigid , ∧ · dr ∇ ∧ ∂V u α when = ..the i.e. , u f tsome at f · 0 = ( t b ,as ), ) u δt − . ) is , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. audsaa field scalar valued dr oasml once subset connected simply a to and from change small a For (C) of R If and have we But any eomto eeae mohsurface smooth a generates deformation a tl odo fi ssml connected simply is it if D on hold still S ′ S C ∇ ⇒ o n curve any for If oso (B) show To If δr C saysraewt boundary with surface any is ˜ ∧ f()hls othat so holds, (B) if B od saoead(B) and above as holds (B) F ehv that have we , D ihfie n-onscnb mohydfre nooeanoth one into deformed smoothly be can end-points fixed with F ( r · sntsml once,te h osrcinaoeproduc above construction the then connected, simply not is sdfie n aifis()ol nasubset a on only (C) satisfies ans defined is ) dS δf ySoe’term ec,(C) Hence, theorem. Stokes’ by ⇒ = f C ( A,w utdefine must we (A), r f ∇ from ( ⇒ + f r F on ) a δr · δf δr = r a = ) f ∇ to = = to + D ( r f o f F Z swl defined. well is ) r r C D C ngnrl oee,w a lascos orestrict to choose always can we However, general. in ( ~ C ( n ec (B) hence and + ( | ( r 0 δr + r r + ) ) δr δC ′ ⇒ ∈ | sadmyvral) h nerli independent is integral The variable). dummy a is · rmtedfiiino rd ic hshlsfor holds this Since grad. of definition the from ) ∂S hr sasaletninof extension small a is there , A tl follows. still (A) D δr r C F b F · ( = uhthat such + ydfiiin hsmasta n curves any that means this definition, By . f r 9– 49 – dr ) ( o C r ′ · ( .Ltu fix us Let ). | = δr S − δr ⇒ with Z a + C | ˜ ⇒ ) C δ . A,a claimed. as (A), , o F (B). C f R ( | C ( ∂S S δr · r F dr ilb igevle on single-valued be will ) ∂S | r+ r ) = · ′ . D a + dr δ C of n define and Z − δC − b ℜ R 3 C F C ˜ ˜ hn()ad()will (A) and (B) then , oteagmn for argument the so , F · dr C · dr ′ f by ( = r rin er δC = ) R samulti- a es ∂S sshown, as R D F C Such . D F · dr (8.1) 0 ( . r ′ ) C = · Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. Q fixed at theorem divergence the by fields vector and scalar time-dependent Consider Laws Conservation 8.3 Let with R fqatt leaving quantity of D F Example D V ( 0 t ρ = stettlaon fsm uniyin quantity some of amount total the is ) = ( = V r ρ ℜ t , ( ∇ ℜ r eafixed a be 3 ) 3 t , f { − V : dV { − ) F where δV z-axis ( = afplane half satisfies h muti ml volume small a in amount the Q(t) − f } y/ dQ t dt hc sntsml connected. simply not is which , V − tan = ( needn ouewt boundary with volume independent codn oteeuto above. equation the to according , x = 2 x + Z ≥ V − y 2 1 0 ∂ρ ∂t ) ( y , x/ , y/x C dV ~ ( 0 = = ) x ∂ρ ∂t 2 = t . + } + nohrwrssml once on connected simply words other in , − ϕ y 0– 50 – 2 ∇ Z smlivle on multi-valued is ) V , φ, · )i eldfie n obeys and well-defined is 0) C ∇ j V angle usual polar δV 0 = · ; j ρ h u of flux the ; ρ ( dV . ( r r t , t , stedniyfrti quantity, this for density the is ) = and ) − Z S j S ( j j D r · = t , u of out u igevle on single-valued but , dS byn h osrainequation conservation the obeying ) ∂V Then . S n ie h rate the gives ∇ ∧ F D Q 0 0 = ( . t = ) on j Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. Q sashr and sphere a is . ie raadVlm Elements Volume coordinates and For Area Line, Coordinates 9.1 Curvilinear Othorgonal 9 conserved. is charge independent ∂v ∂r Examples = dv .Cnevto feeti charge. electric of Conservation 1. .Cnevto fms o fluid. a for mass of Conservation 2. eunn otegnrlcs,take case, general the to Returning R + ℜ o h aei which in case the For nsection in and density, current u h osraineuto nti aeis case this in equation conservation The hsbecomes this hrei ml volume small a in charge dQ dt 3 ( ∂w ∂r ρ r = ( t , : r dw en udvlct ed yetnino h icsino discussion the of extension By field. velocity fluid a being ) t , − ) o odprmtrsto,w require we parameterisation, good a For . R dV S ,v w v, u, | j 4.3 r · | sconstant is dS ∇ , = ρu · on oside ml osraino mass. of conservation imply indeed does R u δt If . :tefli ste aldincompressible called then is fluid the 0: = ℜ · j 3 n ehv mohfunction smooth a have we , ρ · δS ⇔ δS | ⇒ scntn n nfr ie needn or independent (i.e. uniform and constant is j δV → | ∂u ∂r stems ffli htcrosses that fluid of mass the is stecag oigthrough flowing charge the is Q . ˙ δ Q n o o xml,tettlms rtotal or mass total the example, for so, and 0 = S · ∂ρ ∂t ucetyrpdyas rapidly sufficiently 0 ( n t δ 1– 51 – stettlcag in charge total the is ) ∂v ∂r S + ρ V ρ ( r ( ∇ ∧ r ob oi peeo radius of sphere solid a be to t , t , n · stecag density, charge the is ) ∂w ∂r ( stems est and density mass the is ) ρu 0 = ) 0 6= . u t δ r ∂u ∂r j ( ,v w v, u, , ∂v ∂r V | and r δS . | with ) j = ( e nttime. unit per δS r ∂w ∂r t , . R ρ steelectric the is ) ( ntime in ob linearly be to udvolume fluid f dr r ∞ → j t , R = ) = , δV r S ∂u ρu ∂r then , and = δt sthe is du with ∂V so , + t ), Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. raeeetis element area h . rd i n Curl and Consider Div Grad, 9.2 h h e section in condition Jacobian the (see oriae prpitl) nteecodnts h lin the coordinates, these In appropriately). coordinates h bv r rhgnl hn ecnset can we Then, orthogonal. are above by parameterised ai rgthne meaning (right-handed basis h in changes from resulting direction factors orthogonal scale positive the and Examples w u w u u u δu h h δw snra otesraeadw a e that see can we and surface the to normal is > .Creincoordinates Cartesian 1. .Shrclplrcoordinates polar Spherical 3. coordinates polar Cylindrical 2. v v osdrasraewith surface a Consider esythat say We h e and approximately. , w w 0, e h 1 y h , r udv du ud dw dv du h = 1 = f h ϕ : v ( v ˆ j = r δv > , h , ( o rhgnlcoordinates. orthogonal for , ,v w v, u, e ρ prxmtl.Tevlm element volume The approximately. , 0, z θ . dS = = u e h hc orsod oasalcbi ihsides with cuboid small a to corresponds which , u ρ , | w , k ˆ ,h r, , δr v )adcompare and )) = . e v , > ϕ | w 2 ϕ and and ∂u ∂r = and 0 = dr r rhgnlcurvilinear orthogonal are h ∧ r w = u 2 df e sin w ( ∂v ∂r z r δu hnteohrtoaehl fixed. held are two other the when e h h ( = e r si section in as are osat(a) aaeeie by parameterised (say), constant u u ,y z y, x, u θ u udv du ) e ∧ h , . , 2 ∂f ∂u u e e + e v du r r v v h , du , , ( h r = ) ,θ ϕ θ, r, = e 4.4.2 ( v 2 e w + e θ ,ϕ z ϕ, ρ, agnrlfrua = formula) general (a ( 2– 52 – w + , w δv e eemn h hne nlnt ln each along length in changes the determine h e w omn northonormal an forming x ∂f v ∂v ) ϕ hc a eahee yodrn the ordering by achieved be may which , ∂u ∂r .Teevcosaetnett h curves the to tangent are vectors These ). ˆ e i = ) 2 v r si section in as are + + = ) dv dv = h y e u + 1.3 r ˆ j w 2 h , + v (cos ρ ( + u v δS e ∂w δw ∂f (cos . e h odntsi h agn vectors tangent the if coodinates u and dV u z w , ) k ˆ ϕ sasalrcagewt sides with rectangle small a is dw e 2 w . ∂v ϕ ∂r sin o-terms + = ˆ w i lmn is element e dw h sin + = : x ∂u ∂r θ ˆ i = · h 1.3 sin + v h h e ∂v ∂r ϕ u v . u y ˆ j , e ∧ + ) u = and n right-handed and ∂w ϕ ∂r ∧ ∂w ∂r sin h h z h z = u v k ˆ v u h δu, h vector The . θ 1, = e , ud dw dv du h ˆ j v h w cos + ρ udv du e v w = δv e with , x h and = θ z k ˆ = = = ˆ ), i , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. with and ple osm etrfield vector some to applied various coordina the curvilinear know orthogonal to general in gradient the is s see grad, of definition independent coordinate the is (this xrsinfor expression hr r ubro asteecnb derived be can these ways of number a are There ooti i rcr.Terslsaesmlrt tnadCar standard to similar are results The extra curl. with or div obtain to Example 2 .Uea leri prah rt apply first, approach: algebraic an Use 2. .Apply 1. l fwihaenon-examinable. are which of All o osdrtedffrniloperator differential the consider Now sflwe ehv xlctfrua o h ai etr ( vectors basis the for formulae III). explicit have we when useful rtwriting first ∇ · : ∇ ( ∇ ∇ f h ∧ f ∧ = ∇ atr pern eeadthere. and here appearing factors F dr ∇ · f cos = sin = r 0 = ) = = or sin · bv,adsnetebssvcosaeorthonormal, are vectors basis the since and above, F F h h ∇ θ u v h θ ntrsof terms in = ϕ 1 . h ∧ h cos cos 1 (sin factors. w v ∇ h h n ieetaetebssvcosepiil.Ti smost is This explicitly. vectors basis the differentiate and ∇ u ϕ w f ϕe h ∂v θe 1 ∂ =

v nshrclplrcodnts Then, coordinates. polar spherical in = h F r h h r ( w h u + ∂u u h h cos + 1 ∂ F = 1 e u w u u u e 1 ∇ r F e ∂u F u ∂ ( u h w u h r ∂u u ∂f ∂u ∂ v ) ∂v , v e ( ∂ θe cos ( = F e h u − ∇ v v + v ϕ + + v h 3– 53 – ) ∂w ∇ θ h h , ∂ w F h − w cos w 1 h ∂w f ∇ F 1 ∂ v v F e v ( ) e e sin u w w h w e v v · tosmlrterms) similar (two + ) ϕ v v nasial a n using and way suitable a in , dr ∂v +

F ∂ ∂f ) ∂v ϕe ∇ , e v F . θ ) · + ϕ w + + 2 or . e : e h w u 1 r h w 1 ∇ w sin sn clro etrproducts vector or scalar using tosmlrterms) similar (two + e 1 ∧ e w w θ ∂w e.T pl hs eneed we this, apply To tes. u aclt h eutby result the calculate but ∂w ∂ ∂f ( − r ection sin einfrua but formulae tesian θ e xml sheet example see sin 1.1 . ϕ .Fo the From ). ) e ϕ ∇ ∇ ∧ (9.1) (9.2) f = Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. endb h changes the by defined C h onayo nae fapoiaesize approximate of area an of boundary the h tnadapoiaeepesosfritgasaoge along integrals for expressions approximate standard the oflo 3 o ul osdrteae endb changes by defined area the consider curl, for (3) follow To iiigb raadtkn h ii,w e,fo section from get, we limit, the taking and area by Dividing scamdi Eq. in claimed as atthat fact re odvd yteae n aetelmt n ow edto need we so and limit) the take and area of the arguments by divide to order Z stesalcrutshown, circuit small the is Z C .Ueitga xrsin o i n ula nsection in as curl and div for expressions integral Use 3. S etk iia prahfrdv osdra prxmt c approximate an consider div: for approach similar a take We F F odgoercinsight. geometric good · · dr dS h u ≈ ≈ δu ≈ ≈ F h and u [ ∂u u ∂w h ∂ h , ∂ h u u ( 9.1 h v ( ( h h e v ,v u, h and v F w v u . F δ h w ∇ · v v ) v ( u v δw δv, δu, ) δu ,v w v, u, F r h egh ftesget,adwr oorder to work and segments, the of lengths the are F − w ∧ u + ) , ∂v δuδvδw F ∂ F F e v ( = v u + h emnsof segments the : h h u e v h δw w F ihboundary with ( u u 1 u h w iia terms. similar two + ) ) + w e − w δuδv. u v δu, 4– 54 – e h ∂u ∂ u v e h ( u v ) h δv F v v u w h F C − ( u v ,v w v, u, h ) C S F v − δuδv . u ntr are: turn in h e v+ v ∂v ∂ v u u+ u )] ( δ ,v u, ihanormal a with , ( δuδv h δ u F + 8.1 c segment ach 8.1 δu u ) w iia terms similar two + δv hsapoc gives approach This . and , ) δu eptako the of track keep δv − bi volume uboid F with v e h C w v n the and , euse We . ( δuδv ,v u, w fixed. (9.3) ) δv (in Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. h rvttoa field gravitational the o n lsdcurve closed any for function. osat hs qain determine equations These constant. sGus Law Gauss’ is osdradsrbto fms rdcn rvttoa f gravitational a producing mass of distribution a Gravitation Consider of Laws Equation Poisson’s 10.1 and Law Gauss’ 10 by dividing Then, g volume ∇ in n spootoa to proportional is and tion, m systems. dinate ftems otie ihnaradius a within contained mass the flu of the for symmetry. Law about assumption Gauss’ an from with follows together above, gravitation of law Newton’s conservative: is field gravitational Example ( r ∇ · at = ) F osdrattlmass total a Consider apply to how know to need we course, this In ∧ r above. A h oa oc sasmo otiuin rmec ato th of part each from contributions of sum a is force total The . g V ( : = r with )ˆ A r r ihˆ with , = 2 sin 1 nitga omfraycoe surface closed any for form integral in M 1 r tan θ en h oa ascnandin contained mass total the being δV

r C e ∂r 2 θ 0 0 ∂ = rfreprui as raclrto u ogaiy The gravity. to due acceleration or mass, unit per force or , r e . = φ re r ∂θ ∂ ( / h θ θ | u M r r 6= h | sin m v en ailui etrand vector unit radial a being itiue ihshrclsmer bu 0 about symmetry spherical with distributed h π Setting . w nshrclpolars. spherical in ) Z θ r δuδvδw S ( sin 1 r g ∂ϕ Z tan( ∂ · C θe g dS ( g 5– 55 – ϕ r θ/ · rmtems itiuin nparticular, In distribution. mass the from ) r = n aigtelmtgvstefruafor formula the gives limit the taking and F dr r=a 2)) = ( − r 0 = 4 = ) S:r=R

n=r a πGM = peia ymtyipisthat implies symmetry Spherical . r mg hs omlei atclrcoor- particular in formulae these 2 sin 1 V ( r . S ensavco field vector a defines ) θ G hc stebudr of boundary the is which , e r orce sNwo’ gravitational Newton’s is ∂θ ∂ g ( F r sin 2 en oescalar some being ) ( r of x napitmass point a on ) 2 ( θ/ g 2) distribu- e swritten as = n all and , r 1 2 g e ( r r . ), Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. rvttoa potential gravitational with omli ˆ is normal oso’ qain ntepeiu aewt peia sym spherical with case previous the ϕ In equation. Poisson’s as’lwi ieeta form differential in law Gauss’ osdrGus a with Law Gauss’ Consider ups httemass the that Suppose point-like). not certainly a pulls is mo that earth the one the keeps the with that point-like) field approximately gravitational is earth the identify can we that o any for h limit the suigshrclsmer adn asin mass no (and symmetry spherical assuming masses. point for Gravitation of noraoeeapewt peia symmetry, spherical with example above our In any for g hence yuigshrclsmer.Te,uigGus Law, Gauss’ using Then, symmetry. spherical using by ( ( r r rmGus a nitga om u tcnb ecs ndiff in re-cast be can it but form, integral in Law Gauss’ from ) = ) h rvttoa oc namass a on force gravitational The h condition The ftems itiuini o ucetysmercl ti it symmetrical, sufficiently not is distribution mass the If o any For S − = a > R S a GM/r r with ∂V → . ,tefil n h oc for force the and field the a, or , egtapitmass point a get we 0 Z yGus theorem Gauss’ by ∂S S Z for S g g · = R a > r dS C · C dS g 4 ϕ πR = M · ySoe’term Since theorem. Stokes’ by ( dr = toestebudr condition boundary the obeys It . r − 2 with ) rssfo asdensity mass a from arises g Z 4 o n closed any for 0 = S g ( S πGM R ( g utemr,since Furthermore, . r en peeo radius of sphere a being = ) ( = ) R ∇ ⇒ g )ˆ ⇒ r ⇒ ∇ − = − M · 4 2 GM R r −∇ ˆ · ϕ Z πGM r dS V 6– 56 – g m V tteoii 0 origin the at 2 4 = ( = ∇ ∇ ϕ a > r r ˆ at = − · πGρ, · n hnGus a becomes law Gauss’ then and ⇒ Z g g r 4 S dV πGρ, a > r 4 + stherefore is C g g ( ∇ eed nyo h oa mass total the on only depends ( a > r ( R R = a ere-written be can πGρ ∧ S = ) ) ∇ − .Ti ipebhvormeans behaviour simple This ). dS g sabtay this arbitrary, is ρ 4 ) ∧ n ercvrNwo’ Law Newton’s recover we and 0 = ) . ( πG − 4 = r g r dV .Then ). GM pl rmate (where tree a from apple n ,w a nrdc the introduce can we 0, = R Z = automatically. F πR 2 V ϕ o any for 0 = ( a > R ni ri weethe (where orbit in on ρ er,w a choose can we metry, r → 2 icl oslefor solve to difficult s ( = ) g r ( ) as 0 R dV ) − R h outward The . GMm S r rnilform. erential ∇ ⇔ ∇ ∞ → volume ∧ g r ˆ ∧ /r dS . g 2 0. = In . 0 = M V , , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. and as’Lwi nerlfr,js si h rvttoa cas gravitational the in as just form, integral in Law Gauss’ hrecnandwithin contained charge avco ed,o oc e ntcharge. unit per force or field), vector pr (a They rest. at charges a electric of distribution a Electrostatics Consider of Laws 10.2 a er-xrse as re-expressed be can surface closed any for Law: Gauss’ obeys it Also, vative. olm’ a o h oc naohrpitcharge point another on force the for Law Coulomb’s ∇ where form), differential in Law (Gauss’ in form differential in re-expressed be can above laws the and for eoti on charge point a obtain we 0 2 trs at rest at n l ftecag en otie ihnaradius a within contained being charge the of all and ϕ a > r ahmtclysekn,ti sietclt h gravitat the to identical is this speaking, Mathematically h lcrcfil obeys field electric The o xml:tk oa charge total a take example: For = − ρ/ǫ and r 0 oso’ equation. Poisson’s , hc spootoa to proportional is which S E hc stebudr favolume a of boundary the is which V t0 at . = R ǫ 0 C −∇ ntelimit the in stepritvt ffe pc,acntn fnature. of constant a space, free of permittivity the is E F ϕ · dr E with , ϕ Z = S ( ( ∇ r r o n lsdcurve closed any for 0 = qE E Q = ) = ) q 7– 57 – · : · E itiue ihshrclsmer about symmetry spherical with distributed = dS F ϕ ρ 4 4 a = ( ( πǫ πǫ 4 Q Q = r r qQ → = πǫ en h lcrsai potential electrostatic the being ) ǫ steeeti hredniy these density. charge electric the is ) ρ 0 0 0 qE 0 ǫ Q 1 r r .fo hsrsl for result this from 0. r ˆ 0 2 r . r ˆ ( 2 r . ,where ), q at r dc oc nacharge a on force a oduce r V = oa aei section in case ional h aeway: same the : .Thus, e. E with , a ( C esleti using this solve We . r steeeti field electric the is ) i.e. , Q E en h total the being E ( r erecover we , sconser- is ) ∇ ∧ E 10.1 and 0 = , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. npyisadmaths. and physics in 03PisnsEuto n alc’ Equation Laplace’s and Consider Equation Poisson’s 10.3 ieeta qainfor equation differential odtos eswti bv for above this saw We conditions. rsin for pressions satisfies ∇ ttc) For statics). complicated. ymtyand symmetry rclnrclplrcodnts hntesolution the then coordinates, polar cylindrical or ymtyi w iesos e suse us t Let equivalent dimensions. being two latter in symmetry the dimensions, three in symmetry Example esaluse shall we dinates. eghaogthe along length hr hr sn otiuinfo h n-as Thus, end-caps. the from contribution no is there where h lcrcfil u oaln fcharge. of line a to due field electric The nerlfr oaclne fradius of cylinder a to form integral 2 3 .Frclnrclsymmetry, cylindrical For 2. .Frshrclsymmetry, spherical For 1. ϕ oethat Note n xml sirttoa n nopesbefli o.Th flow. fluid incompressible and irrotational is example One tas cusi w iesoso rae hntredimens three than greater or dimensions two in occurs also It esalmil ecnendwt xlctsltoswt sp with solutions explicit with concerned be mainly shall We 0. = stegnrlsolution. general the is ∇ : Z ∇ E S ∧ 2 E ( ∇ ϕ ρ r ∇ u r 2 = ) · ,w e alc’ equation Laplace’s get we 0, = = ∇ 2 0 = = dS o h ailcodnt nclnrclplrcoor- polar cylindrical in coordinate radial the for r vial o o-atsa oriae,btte a b can they but coordinates, non-Cartesian for available are ϕ − ∂x z 2 = − = E ∂ ρ i ⇒ E ∂x 2 eamt ov for solve to aim we : xs ehv yidia ymty u here but symmetry, cylindrical have We axis. ( ϕ ( i r πRlE r u )ˆ ′ ϕ o atsa oriae in coordinates Cartesian for ( = ) r ϕ ϕ r ′′ = )ˆ ′′ ( ysmer.W pl as’lwin law Gauss’ apply We symmetry. by + r r + alc’ equation Laplace’s . ∇ ): 2 ( 1 r R πǫ σ ϕ 2 r ϕ = ) ϕ where , 0 ′ ′ = 1 r = r ˆ ρ oa hreisd cylinder inside charge total 1 r r − → 1 ( R 2 rϕ 8– 58 – ϕ n length and ( r r ′ > r ) 2 4 o h ailcodnt nete spherical either in coordinate radial the for stevlct potential velocity the is ′ ϕ |{z} πGρ σ ϕ 0 = σl ′ ( 0) stecag e unit per charge the is ′ r ∇ 0 = given ) . ⇒ gaiain and (gravitation) ℜ 2 ∇ ϕ n . 2 ϕ ⇒ l ϕ ,wihcosu aytimes many up crops which 0, = ϕ = sshown. as ( hnbcmsa ordinary an becomes then 0 = r ϕ ρ /ǫ a peia rcylindrical or spherical has ) A ( = r 0 log ,wt utbeboundary suitable with ), , A r ouin ihcircular with solutions o r + + B eia rcylindrical or herical B ρ and osa well as ions → l velocity e ρ/ǫ ∇ n n · 0 z u (electro- rather e R 0 = 3 u Ex- ( ⇒ n=r r ) ^ Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. h ieeta qaindrcl)o yadn particular adding similar by above. a or in directly) obtained equation differential be the can equation Poisson’s to Solutions a loseiydffrn onaycniin ndffrn b different on con conditions boundary physical different the specify also on can depends condition boundary specifying of choice The If conditions. boundary onaycniinon condition boundary If htfo section from that Example − ϕ 4 6 1 .Specify 2. .Specify 1. oietenwntto fadrvtv ihrsett vect a to respect with derivative a of notation new the Notice ρ o nqeslto oPisns(rLpaes equation, Laplace’s) (or Poisson’s to solution unique a For 0 saslto nabuddvolume bounded a on solution a is stegnrlsolution. general the is ∂V r 2 h eea ouinwt peia ymtyi then is symmetry spherical with solution general The . . osdrshrclysmercsltosof solutions symmetric spherically Consider : ∂ϕ/∂n 4 ϕ ∂ϕ ∂n iihe condition Dirichlet a : ϕ = ( orsod oseiyn h omlcmoetof component normal the specifying to corresponds 6.2 r n = ) , ∇ · ∇ ∂V ϕ eea ouino alc’ equation Laplace’s of solution general ϕ 2 ( emn condition Neumann a : sdfie nalof all in defined is : r α = ) α | A ( r α {z + 1) + 9– 59 – B } r V α hnteeaetocmo id of kinds common two are there then , − ℜ 2 3 so eotnrequire often we , where , α − ie h atclrintegral particular the gives 2 = atclrintegral particular ∇ or! 2 nerl otediscussions the to integrals n ϕ | 6 1 ρ sa uwr omlon normal outward an is {z = udr components. oundary a ie yintegrating by (i.e. way 0 r − } 2 ems pcf the specify must we ρ ϕ 0 et o example for tent, osat Note constant. a , → . as 0 g or | r ∞ → | E We . . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. ntelf-adpnl ehv fixed have we panel, left-hand the In Example where h rvttoa field gravitational The r otnosat continuous are lnto osatdensity constant of planet a that is interpretation physical The ovnin yreplacing by conventions M h acigat matching The 4 = ρ / 0 3 ese peia symmetric spherical a seek We : πa sacntn and constant a is 3 ρ 0 . A r r and ⇒ = = g a ϕ ( a D r g rmaoe h eea ouinis solution general the above, From . ( implies = ) ϕ r = r xdt ezr ytebudr odtosat conditions boundary the by zero be to fixed are = ) ρ = ϕ ϕ −∇ 0 − ρ ∇ snnsnua at non-singular is 0 : − → ϕ radius , ϕ 2 ¨ A/r ϕ ′ B GM/ ϕ ( ′ r ¨ + : ¨ = + = ) = ϕ 4 + g πGρ 6 1 (2 ( stegaiainlptnilisd n outside and inside potential gravitational the is 3 1 GM/a C/r 4 r B 0– 60 – a − 4 ) πGρ a e πGρ − (( ) − ρ n oa mass total and 0 − r 0 ϕ GMr/a + 0 Mrr>a > r GM/r with n rt h eut bv ntrsof terms in above results the write and 6 1 − ( r/a 0 ρ r

a 0 a > r D

ntevria axis. vertical the in 1 = GM/r with ) 0 a 2 r ) 2 = = r 2 r a > r − 0, = 3 ≤ − C/a 2 3) C/a a r ϕ , ≤ 2 ( M r a > r r ≤ a ersoegravitational restore We . ∞ → r a ≤ . a ) → ,and 0, r ϕ 0 = and , ∞ ϕ . ′ Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. h u of flux the hsi aldthe called is This c,w a solve can we ics, where that Suppose volume 111Saeetadproof Consider and Statement 11.1.1 Equations Theorems Poisson’s Uniqueness and Laplace’s 11.1 for Results General 11 For Example of sidpneto o h asi itiue,poie ther provided distributed, is mass the how of independent is sw on rmPisnsequation. Poisson’s from found we as 2. 1. 1. 2. S ngnrl vni h rbe a ohn od ihgravita with do to nothing has problem the if even general, In peeo radius of sphere a , R ϕ ϕ ϕ ∂ϕ f ∂n ≤ ( ( ( ( ( V r r r r r natraieslto st pl as’Lwfraflxof flux a for Law Gauss’ apply to is solution alternative An : suiu pt h replacement the to up unique is ) or unique, is ) = ) ) ϕ a or ) Z ihboundary with eget we , = ∇ ( r S ϕ aifigPisnsequation Poisson’s satisfying ) f n g ϕ g ( u fasial surface suitable a of out ∇ · · r ( aifiseither satisfies dS r ∇ on ) r ie.Te either Then given. are ) as lxMethod Flux Gauss ϕ 2 ϕ 4 = = S = g πR Drcltcniin D) condition, (Dirichlet − R ( r ρ . Nuancniin N) condition, (Neumann ) 2 S R g with ( = R Z ≥ S = ) ∂V a ∇ ρ a led encvrdi section in covered been already has and ϕ − en lsdsrae ihotadnormal outward with surface, closed a being · 4 . dS GM πG S ϕ 1– 61 – S xiiigeog ymtyb considering by symmetry enough exhibiting = asin mass | R = ϕ − ∇ ( ∂V R/a {z → Z a 2 V ϕ R : ϕ dV. ρ ≤ ) = } 3 a + ⇒ − c ρ where , g o some for ( R = ) sshrclsymmetry). spherical is e c − saconstant. a is GMR/a ρ ino electrostat- or tion ( r nabounded a on ) 10.1 g = 3 , teresult (the g ( r ) e r out n . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. ∇ R yete N r() Since (D). or (N) either by n (D) and odtos()o 2.Then (2). or (1) conditions 112Comments 11.1.2 So (N) (D) V ψ 5 ∇ • • • • e atI Methods. IB Part See ro:Let Proof: 0 = ϕ ψ · hs odtoson conditions These but and ϕ only the is it know we then means, symmetry O h nqeesrsl a rcia motne fw a fin can we if importance: practical has result uniqueness The h hoe a esae n rvdsmlryfrrgosin regions for similarly proved and stated be can theorem The onaycniin as conditions boundary h eutsy ohn bu xsec.Ide,for Indeed, existence. about nothing says result The ssdfiiin fga n i,adtedvrec theorem. divergence the and div, and grad of definitions uses with h eut xedt none oan,freapeaspher a example for domains, unbounded to extend results The 2 ∂ϕ ∂n on 0 = ( ( ψ r (1 Z = ⇒ = ) ψ = ∇ S /R ρ X ∇ ψ ψ on 0 = | g ψ ψ and ψ ( ∂ψ ) ) ∂n ϕ on x ( = ( S → 1 dV r ) r ϕ ( ) Y ) r c | ⇒ g ∂V dS ϕ 1 | osat on constant, a , + ) as 0 ( ( = = ( y = Z r xdt ei ih oEq. so with, begin to fixed are r S r c or ) and ) V ≤ ,o oegnrly iercmiain fsprbesolu separable of combinations linear generally, more or ), R Nuan hr a nyb ouinif solution a be only can there (Neumann) O c and 0 = O r(N) or ∇ S ∇ R max (1 sclaimed. as (1 ψ 2 R · dV ϕ ∞ → /R ∇ /R ϕ ( ( ψ Z r ψ | 2 r ψ and ) ( V ) ψ ∇ 2 | ∂ψ/∂n |∇ = r Y · .Teeoe h ro xed to extends proof the Therefore, ). ( |∇ aif oso’ qain ahoeigteboundary the obeying each equation, Poisson’s satisfy ) R hsls eutflosbecause follows result last This . ϕ ψ dS R r nue ytkn o example for taking by ensured are r = ( ψ | 1 θ = ) = ) ψ ψ = ). | But . Z = 2 ∂ψ V ∂n | | ∂V 2 | ≥ ∂ψ ∂n ϕ ∇ ϕ So: . r = dV ∞ → | × | 2 2 ∂ϕ ∂n ( ψ ,teitga a nyvns if vanish only can integral the 0, ( r n on r 2– 62 – | ∇ · = ) ∇ · 4 = dS − ouin o xml:fidaslto with solution a find example: For solution. πR V Z ψ ψ ϕ S or , O ( ⇔ 2 + 1 S ψ on 0 = (1 ( ≤ r ∂ψ | ψ ∂n ste“peea infinity”). at “sphere the is Z nV on 0 /R satisfies ) ∇ constant V {z 11.1 2 dV ρ 2 dS ψ for ) } S = 0 = a o eesrl etrue. be necessarily not may ytedvrec theorem, divergence the By . + |∇ | × r ∇ Z ψ | ∂V 2 ( | = ψ 2 R ∇ 1 . dS g on 0 = R 2 R 1 ϕ V 2 ouinb any by solution a d | ⇒ )4 | = ψ = 0 = πR ( − r ℜ V fradius of e R ℜ ) ρ S , 2 |∇ | 3 2 ylinearity by , ψ = 3 on = ihthese with ,... ψ ∂ψ ∂n O | tjust it : O V 0 = ( dS (11.1) (1 tions R 1 with /R ) | . R ⇒ = 5 ) , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. So ihboundary with But 121TeMa au Property Value Suppose Mean The 11.2.1 v some have and mathematics and physics ties. of areas many in arise ouin fLpaesequation Functions Laplace’s Harmonic of and Solutions Equation Laplace’s 11.2 R h vrg of average the ( ,θ χ θ, u, 2 sin ϕ • • ro:Nt that Note Proof: ( ∂ϕ ∂u R oeie aldGensScn Identity. Second Green’s called sometimes h ro ssaseilcs fteresult the of case special a uses proof The bound the of parts different being N kind or various D and with equations perhaps related ditions, to apply results Similar oeie aldGensFrtIett.Antisymmetrising Identity. First Green’s called sometimes θdχ dθ θ ete on centred ) sidpnetof independent is ) = ϕ e ( u r shroi naregion a in harmonic is ) ∇ · ⇒ ⇒ ϕ S ϕ d R over dS/R ϕ = dR : ( r R n | Z Z r S ) = ∇ · S ϕ S 2 − R = ( ( ( . sidpnetof independent is u a R dR u ϕ a d hn h clrae lmn on element area scalar the Then, . ∇ ∇ 4 | R ) = πR = v ϕ v 1 n h eutfollows. result the and , ϕ → ) ( ( − ∂ϕ ∂n R · R 2 R dS Then . ϕ ∇ Z v = ) = ) on S ∇ ( 2 R a ϕ = u as ) ∇ S 4 ) 4 r locle harmonic called also are 0 = Z πR R πR V ϕ 1 3– 63 – dS V 1 . ϕ · ∇ R otiigasldsphere solid a containing 2 2 ( dS a = Z R u Z = ) → S S Hence . ∇ · Z = R R V ϕ ∂ϕ ∂u .Tk peia oa coordinates polar spherical Take 0. ϕ 4 dV v ( πR ( u r 1 R

) ∇ u where ) 2 = dS, 2 or R v + Z V − Z R dS. V v ∇ ∇ u 2 ∇ dV ϕ 2 u 2 dV, v S R dV, fbudr con- boundary of s in r pca proper- special ery 0 = s( is V ucin.They functions. u R ary: . and u : = | r v − R gives ) a ≤ | dS = R Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. iswithin lies osmoe l uhcnrbtos ereplace we contributions, such all over sum To Principle that Minimum) say (or We Maximum The 11.2.2 region osdrteptnilfrapitsuc fstrength of source point a for potential derivation the informal Consider and Statement Equation Poisson’s of 11.3.1 Solution Integral 11.3 fo analogously works Everything consistent maxima. principle. is maximum analysis the standard as, This inconclusive. is this hc ugssthat suggests which λ ∇ ⇒ 0 eea sources several function. harmonic a for property value mean uinfo n ml ouea position at volume small any from bution to.Spoeat Suppose ation. 0 o osdragnrldsrbto fsources of distribution general a consider Now ie hr sn oa aiu rlclmnmm nesall unless minimum) local o or can maximum point local no stationary is the there hence and (i.e. sign, each of eigenvalue otetaeo h esa matrix Hessian the of trace the to < ǫ < R < = ro:If Proof: h euti ossetwt h sa nlsso stationa of analysis usual the with consistent is result The | − r 2 ϕ 4 − V πGM a = then , | , V ǫ < ∂x ϕ ∂ ϕ ϕ i ( ti spsil,since possible, is (this 2 o on as or mass, point a for ∂x h aiu rnil ttsta fafunction a if that states principle maximum The . ϕ r ( a oa aiu at maximum local a has ϕ λ r ) i a oa aiu at maximum local a has ) α ϕ < anthv oa aiu ta neirpoint interior an at maximum local a have cannot = r tpositions at = H ( a a ii for ) , = ∇ ϕ P | ϕ 0 = r i ϕ ( λ r − r ρ ( α i = ) r (r’)dV’ h oeta sasmo terms of sum a is potential the , and a = ) .(oeta h u fegnausi equal is eigenvalues of sum the that (Note 0. = λ ψ | H = a = = 4 1 H X .Ti mle htteeeit tlatone least at exists there that implies This ). π sa neirpit n hn o any for then, and point) interior an is Q/ǫ 4– 64 – R a α 4 ij 1 Z π nteitro,w choose we interior, the in V ⇒ = 4 | r 0 1 ′ r’ r π ′ a | . o on hre si section in as charge, point a for − ∂x r ϕ λ | ρ r ∂ f o some for if, ( − ( i 2 a ∂x R P r − λ ϕ ρ | ′ r ) α ) ( . j r α ′ r | ϕ < r a eigenvalues has α with ) 0 dV by | . λ ′ ( R at a V .Btti otait the contradicts this But ). ′ ih u o spowerful as not but with, ρ a dV > ǫ ( : r ′ ′ ypit ydifferenti- by points ry ) l easdl point saddle a be nly λ ntefruaabove, formula the in dV 0, iiaisedof instead minima r i ϕ ,i hc case which in 0, = ϕ ′ ǫ λ ( en h contri- the being othat so shroi na on harmonic is i r a . ) ϕ of ϕ < sharmonic is V | ( . r a − 10 R when ) a (11.2) For . | with ǫ < Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. obeys ertr to return We ssnua when singular is 132Pitsucsand sources Point 11.3.2 equation Poisson’s to solution the is For o ecmaewith compare we Now region the over taken considering h nerloe l space. all over integral the the efix We Therefore, Example ieto ( direction so h ouinw bandpeiul nsection in previously obtained we solution the r of ′ = χ ouinwt onayconditions boundary with solution ρ . ( | r r ∇ h ′ ′ ≤ | o-eobtsial elbhvdas well-behaved suitably but non-zero ) p r ψ sn Eq. Using : n nrdc oa coordinates polar introduce and r = r ϕ a ′ R 2 .Then, ). ( S − and + r ψ ∇ = ) λ/ ϕ r ψ = 2 r ( (4 | r − ( r · = = ) λ/ π dS − 2 = ρ ) 11.2 V a rr ϕ 0 (4 R × r ′ ρ V ( = u h iglrt a egvnapeieitrrtto by interpretation precise a given be can singularity the but 4 ′ ′ π nwhich on 1 | r 0 π R cos ( ecncalculate can we , ∇ = ) / | ( = 0 r r − r 2 Z 2 ρ dr − − λ 0 dV ψ Z θ 0 ∇ r a 0 i R ′ a 4 2 o n peewt centre with sphere any for a δ dr a θ π 1 2 r 0 π ) r a =0 ′ − ϕ | + dr 2 / ,aptnilfrapitsuc,adnt htit that note and source, point a for potential a ), R ′ dr | ucin (non-examinable) functions + Z ρ V = r r for ′ = r ( 0 ′ ′ ′ 2 − r ′ r R π | ∇ 2 r ′ r | ′ r − 2 snnzr ( non-zero is ) r V a rr dθ − ρ a 2 1 − 0 5– 65 – dr r + ϕ | r’ 2 | ′ ′ oi peewith sphere solid a 3 ϕ ρ | Z rr h ′ = 0 r 0 θ ( r dV and p 0 ′ r 2 ′ ′ | − | π ) ϕ cos − r r | ′ dχ ( ′ where , ′ = = ρ ,χ θ, , 2 = χ r ∇ 10.3 r θ ,teslto to: solution the ), + n steitga ihrsetto respect with integral the is and | | √ ρ ρ | r r ) 2 r O − 0 0 1 ψ ≤ | | ′ r r / ∞ → | 3 1 (1 a > 2 ′ 2 r ρ . a 2 − for for 0 = ′ − 3 oteitgadi independent is integrand the so / | + outside 0 = a 1 6 ρ 1 r dV | r = r 2 r 0 r r . | 2 r rr and ) 2 ′ ′ ′ + 2 − ecnas take also can we a ′ ihrsett hsfixed this to respect with = 2 sin cos 2 yepii calculation. explicit by , ∂V 2 1 r 2 r r a ′ r rr θ 2 ′ 6= θ |∇ 2 = ′ i sin cos θ π ( ( a V ϕ =0 S r a > r h function The . ( for for ′ h divergence The . dr θ ≤ .Ti sindeed is This ). r θ . ) | a r < r r > r ) ) = ′ θdχ dθ , O V ′ ′ (1 ′ , / = , | r V ℜ | 2 r ψ 3 ). ′ , ′ : Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. volume theorem. hoe od fw take we if holds theorem h eann awl’ qain lohv nerlforms integral have also equations Maxwell’s remaining The as’Lwfor Law Gauss’ ttceeti hre neati a hti mathematic is moving that generally, way More a interactions. in gravitational interact charges electric Electromagnetism Static of Laws 12.1 Equations Maxwell’s 12 required. as npicpe hsgvsacmlt ecito falclass all of description complete a nomena. gives this principle, In lcrcadmgei ed r eemndb awl’ equ Maxwell’s by determined are fields magnetic and electric sageneralised a is − integ In charge’. ‘magnetic zero with but electrostatics), where electric by described be can that immediately: equations te r ohpyia osat) hn h oet Force Lorentz the Then, constants). physical both are (they oteefilsis fields these to dt d 6 h xrsinfor expression The If osraino lcrccag oe rmMxelsequat Maxwell’s from comes charge electric of Conservation R S ∇ ρ B ǫ ( 0 V 2 r · ψ t , stepritvt ffe pc and space free of permittivity the is dS containing ( stecag est and density charge the is ) r = ) o n surface any for ucin(rdsrbto)satisfying distribution) (or function E 4 1 π ∇ sumdfid n o ehv iia a for law similar a have we now and unmodified, is Z V 2 a ∇ ∇ ψ ′ sn hs ecnvrf h nerlsltoso Poisson’ of solutions integral the verify can we this, Using . Z ρ ∇ · · bv ae es nutvl o on oreconcentrat source point a for intuitively sense makes above S ( B E r 2 E ′ ψ ) = ∇ 0 = · = S dS 2 ǫ ρ 0 | − r ihabudr facoe curve closed a of boundary a with F = n magnetic and λδ − 1 ∂ρ ∂t = Q/ǫ ( r r ′ q + | ∇ ∇ − ( dV 6– 66 – j E 0 a ∇ ( ∧ ∧ , r where ) ′ ˙ + t , B E · = j h urn est,te h resulting the then density, current the ) r + − − 0 = ∧ Z µ ∂B S Z µ fields ∂t B 6 0 V . 0 B ǫ ) h he-ieso et function delta three-dimension the ′ 0 . R lcrccagsitrc naway a in interact charges electric stepreblt ffe space free of permeability the is ρ 0 = · ∂E V ∂t ( dS r E f ′ ) ( ( = δ 0 = r r a form: ral ( ) t , r µ δ cleetoantcphe- electromagnetic ical ( and ) 0 − o example for , . r j lyams dnia to identical almost ally napitcharge point a on ations: − , r ′ a ) ions: dV ) B dV ( ′ C r = t , = rmStokes’ from , − ). f ρ R ( B ( C a dat ed r o any for ) ) E o (in too , · q dr a due . = s Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. awl’ qain hnimply: then equations Maxwell’s ǫ n light. and h rnvremdlto ftesm eimwihi h cau the is which phenomena medium netic same the of modulation transverse the where osdrMxelseutoswith equations Maxwell’s Consider Waves Electromagnetic 12.3 Currents Steady and If Charges Static 12.2 where scalar or ∇ gto ihspeed with agation c 0 o xml h olm oc.eapetefrebtencur between force the example force. Coulomb the example Magnetostatics for ǫ section in described equation Poisson ⇒ ∇ ∇ Electrostatics 1 = 2 0 ,j ρ, obn ogv h xeietlvleo h pe flight. of speed the of value experimental the give to combine E ∧ · estesaeo lcrsai effects, electrostatic of scale the sets awl aosysi “ said famously Maxwell o awl’ qain rdc lcrmgei ae whi waves electromagnetic predict equations Maxwell’s So, E E / = E , E √ ∇ c f = = 2 + ∇ 0 = ǫ 2 and −∇ 1 = 0 ρ/ǫ ( − ecie ih oigwv and wave moving right a describes µ ∇ 0 c / 1 ϕ · 0 ≈ B 2 ( E ǫ ∂t and ∂ 0 2 r l needn ftm,then time, of independent all are ) 2 µ . .Ti a h nfiaino h hoiso lcrct,ma electricity, of theories the of unification the was This ”. 2 ∇ − 998 0 .Teeaewv qain ntredmnin eciigp describing dimensions three in equations wave are These ). ∇ E c ecmaewt h n iesoa aeeuto o a for equation wave dimensional one the with compare We . × 2 ∧ ϕ ∂x 0 = ∂ ( 10 ∇ 2 = 2 8 , ∧ − − ms n iial ecnobtain can we similarly and ecnsacl vi h neec htlgtcnit of consists light that inference the avoid scarcely can We E ρ/ǫ c 1 2 − = ) ∂t 1 0 ∂ ..teeprmnal esrdvle of values measured experimentally the i.e. , , 2 2 h scalar the ∇ ρ ∧ f and 0 = 7– 67 – ∂B 11.3 0 = ∂t f , = ambiguous: . f ∂t ∂ j − µ ⇒ wires. case. scalar general the similar to a solution has it equation: Poisson make B ∇ ∇ 0 0 = ( etmvn wave. moving left a E ∧ · ed h clrfield scalar The field. ∇ B estesaeo antceet,for effects, magnetic of scale the sets B = B and ∧ = ie mt space): empty (i.e. ∇ 0 = f B = ± ∇ · = ) ( B µ A x ∧ 0 r olne ikd since linked, longer no are ∓ j 0 = ∇ A A ∂t ∂ ct 2 u h etrptnilis potential vector the but , → eo lcrcadmag- and electric of se ( − ) ∇ ⇒ µ hmv ihspeed with move ch A 0 c 1 ǫ 2 0 + ∂t ∂E ∂ 2 ∂t A ∇ 2 2 = ) χ = χ B rdcstesame the produces a ecoe to chosen be can − µ 0 = µ 0 gnetism 0 ǫ rent-carrying µ j 0 0 , , ∂ ∂t rop- h vector the and 2 E 2 , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. hsi h esrtasomto rule transformation tensor the is This ainrl.B ento,atensor a definition, By rule. mation ihrsett ahbasis each to respect with 312Bscexamples Basic 13.1.2 bases right-handed orthonormal rule Consider transformation Tensor Examples 13.1.1 and Definitions Fields 13.1 Tensor and Tensors 13 yarotation a by atsa coordinates Cartesian basis/coordinates M n coordinates: and vector a of components The Examples ip 7 .det 2. 2. If 1. 1. M oeta ee eue eea eutfo etr n Matr and Vectors from result general a used we here, that Note esr r emti bet hc byagnrlsdform generalised a obey which objects geometric are Tensors jq δ oso hs ecektetasomto ue o example For rule. transformation the check we this, show To httercmoet r nhne ihrsett n basi any to with respect rule with unchanged are system: components their that rnfrainrl with rule transformation . h orc ue n iercmiaino uhexpressi such of example combination linear for Any rule. correct the i.e R ij M ip u kr R and R v , : ǫ jp pqr ,or 1, = w , . . . , = : ǫ T T T (det = ijk δ e R δ ij ′ i ′ frn ,ie oindices no i.e. 0, rank of frn 1, rank of frn 2, rank of T ij ′ = qi v r esr frn n epciey ihteseilpro special the with respectively, 3 and 2 rank of tensors are ij T = ′ R = R are = ij R M = qj R ip δ = sspecial is v ) u ij ǫ { = ip e i n ijk e i ′ p x u R v i and , etr,then vectors, δ i i j ′ . } = v jq ij x ( = j i ′ δ { and T v or + v pq = e R ij...k i ′ ′ i e R rnfr nasmlrwyudrti hneo basis of change this under way similar a in transform } a ip ǫ RR i ′ R orthogonal. = i ijk ′ ip { ⇒ R b rcodnt system coordinate or ip u x j = jq R x T p i ′ = o any for v } p )( R ip R i = ′ : T ǫ where R R kr . = ip ijk x T 8– 68 – R jq jp R ǫ frank of ij...k R pqr . = T v jq = q ip R T T T = ) u { R . . . x v (det = i ij e ′ ′ v , = = i p δ = e i = . ij } = i a , u R I T = n R kr nohrwords other in , i and hc stetno transformation tensor the is which R ip v b , ′ scalar a , ip T j R a components has ip R x . w . . . pq...r T i ′ R jq ) e p { ǫ i vector a , jq ( ijk e ′ u n uhbssaerelated are bases such Any . . i ′ T { p } k x pq v = ensatno frank of tensor a defines i q in } = ) omamatrix. a form ǫ ijk n ne hneof change a under and ℜ 3 R which n r lotensors, also are ons ip ihcorresponding with R R n T sorthogonal is jq fti transfor- this of ij...k 2, = csI,namely IA, ices T 7 pq s/coordinate stetensor the is , ( T n ij indices) = perty u . i v n j . . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. rank satno,w hc h rnfrainrl,freapefo example for rule, transformation the check we tensor, a is esrtasomto uefrrn .I arxnotation matrix In 2. rank for rule transformation tensor tnadtasomto o h arxo iermpunder map linear ( a basis of of matrix the for transformation standard endb ( by defined uefrvectors: for syste rule basis/coordinate of choices different to respect with 321Adto n clrmultiplication scalar Tensors and Addition 13.2.1 Algebra Tensor 13.2 vectors matrices between map and linear tensors A 2 Rank 13.1.3 ie rank a Given Contractions 13.2.3 define n r optbefrany for compatible are endsmlryfraynme ftnos o xml for example for tensors, of T number any yourself). rule for transformation similarly the defined (check tensor a is This aern,dfie by defined rank, same ( If products Tensor 13.2.2 rule). transformation the density h odciiytensor conductivity the substance, Example aee tutr)adthen and ( structure) directions layered different a in differently conduct may Substances T − T + tensor A by 2 and S n ) T + ij ′ T j = S nmn usacs napideeti field electric applied an substances, many In : R = m S T and codn otelna eain(h’ Law) (Ohm’s relation linear the to according p...q r esr frank of tensors are u σ T T endby defined = ik ij ⊗ ′ T n + + = R = v S tensor fayrn a emlile yascalar a by multiplied be can rank any of S v S − ⊗ · · · σδ δ i ) ′ ftesm akcnb added; be can rank same the of ij 1 ′ ij ij...k ). = ik T = ijp...q R ⇒ w R αT T = ip ( ip u i satmtclyatno,b h ruetgvnabove). given argument the by tensor, a automatically is (it v T by j ihcomponents with ij...k T R p = ij...k iff = ⊗ = jq T T T σE S ij...k = u R R n pq iip...q ) + ip ip j...k | . . . ij and j n αT . + and M ( S indices = {z M sntprle to parallel not is R ij...k hsi aldcontracting called is This . ij...k pq v pq u ip m } i R u a h eea form general the has v = q...r | . . . pq 9– 69 – m naycodnt ytm oso htthis that show To system. coordinate any in q j jq naycodnt ytm(tses ocheck to easy (it’s system coordinate any in hntetno product tensor the then = ) · · · indices {z S M pq T ij w ′ M } ijp...q R = k = ij jq ′ si section in as R u T edfieanwtensor new a define we , ip j ′ or ij...k R = T jq M E M + S ( ik T ′ pq...r ij ngnrl o nisotropic an For general. in .Uigtetransformation the Using m. S ′ E pq ( M = R sas esro rank of tensor a also is + nue neeti current electric an induces j etno rdc a be can product tensor he . v α jq ′ i o xml fte have they if example for i r 13.1 S R ; = n u = = pq n ip notooa change orthogonal an αT q vectors RMR .Teeexpressions These ). nteindices the on = ) R 2: = M T . σ kq ik ij satno fthe of tensor a is ⊗ M R E u S j k ip T pq where , or hc sthe is which , R satno of tensor a is u hsi the is this ; v , jq v ( i S ′ w . . . , T = frank of + M i σ S and ij ik ′ ) we pq u n is j ′ . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. ocekta otato rdcsatno,w aeteexam the take we tensor, a produces contraction that check To oethat Note Contracting esr nov diinlidcsta r nffce yth by unaffected are with that indices additional involve tensors Mlilna’masthat means ‘Multi-linear’ 324Smercadatsmerctensors tensor antisymmetric A and Symmetric 13.2.4 Tr( j ymti - nteindices the in (-) symmetric ne ar.Ti od nalcodnt ytm fi holds it if systems coordinate all in holds This pair). index neeypi fidcs o example For indices. of pair every in ℜ nismerc hr r oal ymti esr farb of tensors symmetric totally are There anti-symmetric. 331Tnosa ut-iermaps tensor multi-linear A Rule as Quotient Tensors the and Maps 13.3.1 Multi-linear Tensors, 13.3 otatn nadffrn aro nie eut nadiffer a in results indices of pair different a on Contracting . endby defined .I h rtepeso 1 sseie o all for specified is (1) expression first the If 1. .Uigtesadr eainhpfrvco components vector for relationship standard the Using 2. .Ayttlyatsmerctno frn a h form the has 3 rank of tensor antisymmetric totally Any 1. .Teeaen oal niymti esr frank of tensors antisymmetric totally no are There 2. T esri aldttlysymmetric/anti-symmetric totally called is tensor A = ) T oet ftetno r nqeydtrie b taking (by vectors). basis determined of uniquely choices are possible tensor the of ponents 1 n 2 ge iff agree (2) and (1) any hr saoet-n orsodnebtentnoso ran of tensors between correspondence one-to-one a is there iermp.Ti ie a ftikn bu esr’inde tensors’ transfo tensor about the thinking and naturally. basis, of of way choice or a system gives coordinate This maps. linear scalar esr ihalcmoet zero. components all with tensors ij T klr...s ′ eadda y3matrix, 3 by 3 a as regarded T T T a pp p ′ frank of frank of nmti language. matrix in , b λ T q ′ = . ii c . . . R frn ie scalar), a (i.e. 0 rank of ki T r ′ R ( n n iff a lj b , seuvln oamlilna a from map multi-linear a to equivalent is obeying R T c , . . . , rp pq...r ′ T R . . . T ij...k = i slna nec ftevectors the of each in linear is = ) and R T sq R ijp...q pi T pi z T a ijp...q R j ij...k p ′ Teeeit iia ento o n other any for definition similar a exists (There . R qj = δ qj R . . . ij a T 0– 70 – = b (1) ± }| i ii T q ′ b = T j ± ii R . . . ′ stetrace the is c . . . jip...q δ rk R = ji T ki R ij...k sttlysmercand symmetric totally is rk R { k ip ssi ob ymti + ranti- or (+) symmetric be to said is c lj = R r ′ R , = iq T z etetasomto ue Thus, rule. transformation the ie vectors rp pq...r T ′ T . R . . . pq pq...r ′ T fi ssymmetric/anti-symmetric is it if = a ii ′ > n (2) }| p ′ b sq δ a Tr( = q ′ a otato.Nt that Note contraction. e tayrank itrary pq p ′ v nayoe since one, any in T c . . . b , b p ′ T jip...q a xet(rval)the (trivially) except 3 q ′ n esr ngeneral. in tensor, ent pq c , . . . , = b , c . . . n mto ueemerges rule rmation T l of ple { T r ′ c , . . . , ijk = R vectors a . ′ b , = Tr( = ) pi r ′ T = hc od for holds which , v k c , . . . , pp edn fany of pendent ± i hntecom- the then λǫ ihrrank Higher . T or n T n ǫ individually. ij lkr...s ijk ′ ijk u in but , a n multi- and v TR RT b , frn 2. rank of i stotally is ob all be to o some for = c . . . , . R T pi = ) ℜ v to p 3 ′ , : Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. okmr opiae hni elyis. ind really of it than proliferation complicated the more but look directly, rule transformation 332Teqoin rule If quotient The 13.3.2 arcs o,b h hi rule chain the by Now, matrices. xetfrweetig biul al o xml where example for fail, obviously things where for except R uta ihsaaso etr,atno field tensor a vectors, or scalars with derivatives as and Just fields Tensor 13.4.1 Calculus Tensor 13.4 form special the take ∇ times of number any differentiated as write also we components the for rule e esro rank of tensor new hsi xetdfrcmoet favco,ada emphasise as and vector, a of components for expected is This nsection in T satno frank of tensor a is loatensor. a also o any for n htfraytensor any for that and i...jp...q ip T 8 = and , enn hti a eeduo l ftecoordinates the of all upon depend can it that Meaning j | . . . i osraln hns esalueteiesi section in ideas the use shall we things, streamline To ovrf h rnfrainrl,cnie h derivative the consider rule, transformation the verify To hsi h utetrule quotient the is This ovrey ups that suppose Conversely, {z n e i c ∂/∂x } vectors p q | . . . p x d . . . q 13.1.3 {z m ( = i q } = satno,btthen but tensor, a is R satno frank of tensor a is a 8 h eea aecnb rvdi iia a,cekn the checking way, similar a in proved be can case general The . b , . . . , e − T i ′ 1 n ∂/∂x ij...k n ) qi i satno rdc of product tensor a is (it + u x ( c , p...q i ′ m x i ′ u T = d . . . , l . p...q .W sueta ed r smooth are fields that assume We ). savco prtr sn hs ti ayt hc the check to easy is it this, Using operator. vector a is i...jp...q = R ehv enadpoe h pca case special the proved and seen have We . , iq ∂x c T ∂ v hsi nuht nuetetno transformation tensor the ensure to enough is This . p x i...jp...q i...j | d . . . sosre nsection in observed as , ∂x i ′ i ′ v ∂ ⇒ = i...j p n sdfie bv sas esr then tensor, a also is above defined as q . . . ∂x ∂x {z m + ∂x ∂x v = sa ra endfrec oriaesystem, coordinate each for defined array an is o n vectors any for i...j q i ′ q 1– 71 – i ′ m ∂x T ∂x = ∂ a i...jp...q ∂ and i q } q b . . . R T satno tec point each at tensor a is = iq j...k | . . . ij u oethat Note . T u R j {z p...q n p...q = iq and ∂x } T saytno frank of tensor any is ∂ , i...jp...q q c u . d , . . . , olwdb contractions). by followed , 13.3.1 T csmksteargument the makes ices a R 13.3.1 i sntdfie.Ti sa is This defined. not is ip b . . . yassumption by ; s: above. and x ota hycnbe can they that so j i ′ ti ucetto sufficient is It . c p = nsection in d R d . . . T R iq ij...k ip r constant are q x m n p ( sascalar a is T x then , = ⇒ i...jp...q ,which ), v m ∂x ∂x i...j 1 = 9.2 p i ′ = = is , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. n eoti h esrdvrec hoe,btcontracted but theorem, divergence tensor the obtain we and hc sterqie rnfrainrl alfrCartesia for (all rule constant transformation required the is which h u fvlm is volume of flux The combi linear uses just one otherwise i but derivatives points, tensor different of at definition The rule. transformation where ρu ie Since side. v ρ ihvlct field velocity with esr(ti ais otiko h nerla h ii fa of limit the as integral the of think to easiest is (it tensor h u fthe of flux the hnuei odsuscnevto asfrvco n tenso and vector for laws conservation discuss to it use then auditga ssrih owr.Frexample, For forward. th straight assumption) (by is limits integral take valued and theorem tensors divergence add can tensor we the Because and Integrals 13.4.2 ouebuddb mohsurface smooth a by bounded volume a Then ∂x l ) ∂ = i i ′ u ∂x o xml:drvtvso clrfield scalar a of derivatives example: For o hscleape ealordsuso fteflxo qu of flux the of discussion our recall example, physical a For ro:apyteuulDvrec hoe otevco field vector the to Theorem Divergence usual the apply Proof: ti loes ognrls h iegnetermfo vec from theorem divergence the generalise to easy also is It j a ∂ h u through flux The . i n j ′ b j l · · · · · · R sa uwr onigui normal. unit pointing outward an is .Frjs n eiaie ehv ( have we derivative, one just For ). a ∂x c ∂ b , k T k ′ , i ij...kl ϕ · · · th = opnn fmmnu is momentum of component u c , ( where x u R hog ml ufc lmn asmn nfr densit uniform a (assuming element surface small a through ) r rirr,te a ecneldo n h eutfollow result the and off cancelled be can they arbitrary, are · n ip Z δS ∇ S ∂x S n ∂ T a is · · = p b , ij...kl v v R u c , . . . , = = S j T n n R n ∂x ∂v ij j l l δS jq v n dS l l l j ∂x r xdcntn vectors; constant fixed are h u fms is mass of flux the , dS = = ∂ = 2– 72 – q a a . S n i i Z b b V · · · = j j · · · · · · ∂x δ ∂V ∂ S ρu c c l ∇ R k k ( and T kr ϕ T ∂x i R u ∂ ij...kl ij...kl ) V ∂x j u i l n ∂ T T = T j r ij...kl ij...k δS n ij...k ) ∂x ∂ϕ l ρu , dV, ϕ i = vle oprn tensors comparing nvolves ( oriae eae ya by related coordinates n vector. a , , x · = easot esrfield. tensor smooth a be n ain n hnlimits. then and nations T ) sum). uniis.Let quantities). r δS ij R dV with ento fatensor a of definition e n ip j = ost esr (and tensors to tors R δS n h euti a is result the and niisfrafluid a for antities jq ρu a involving , i · · · b j j n · · · v j R δS kr endby defined c whereas , l ∂x neach on ∂ p (13.1) T V ∂x ij ∂ be q = s. y · · · ∂x ∂ r ϕ Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. further: bu 0 about masses some TensorConsider Inertia The 14.2 a symmetric its of sum a as written parts: be Tensor can Rank tensor rank Second second a Any of Decomposition 2 14.1 Rank of Tensors 14 where steietatensor inertia the is a Cand IC 6 has of ncomponents, In rmtedouble- the from hr nysmercprsof parts symmetric only where novsascalar a involves Example ymti piece symmetric ie bv a ymti rcls piece traceless symmetric a has above given ∂F ∂x S k k ij osummarise, To L = . B T P = otevlcte are velocities the so , ∇ ij k ij S etrfield vector A : X ij = = · a Cwhereas IC 5 has α F = S 1 2 A r ec,acmlt ecito ftedrvtv favco fi vector a of derivative the of description complete a Hence, . ij ǫ P α ij ijk ij L + ∧ ǫ = T i ∇ A + m dniis or identities, A ij = ij 2 1 · ij n nyteatsmercprsof parts antisymmetric the only and , α bu 0 about 1 3 ( F T I v δ T = a needn opnns(C,where (IC), components independent 9 has ij ij α ij vector a , ij m ω Q ǫ = I = − j F ijk α ij where i where X A T ymti traceless symmetric ( ( B ihpositions with = oeta ngeneral in that Note . α A r ji Q ij derivative ) k v a C nfc,tesmercpr a ereduced be can part symmetric the fact, In IC. 3 has ) ij X m α = a .Teatsmercpr a ere-expressed: be can part antisymmetric The 1. has T with α = ) |{z} = α P P ∇ ǫ contribute. r m ijk ij ii α ω   ∧ α B ,or 0, = ∧ B ∧ F − k k | ( B 3– 73 – r r 0 ω B ⇔ α = α tensor traceless symmetric a and 2 T h oa nua oetmaot0 about momentum angular total The . 3 | ∧ + 2 ij B δ r − 2 1 ǫ r B ij α ǫ P ijk = k α 0 B ijk l oaigwt nua velocity angular with rotating all , 3 ij = ) P − = 1 vector |{z} ∂x ∂F ij ∂x ∂F is B ( − 2 1 j r B i j k = i X 0 esrfil.Tedecomposition The field. tensor a , B traceless ǫ α L 1 α ijk = ) 2 + i 2 1 sntprle to parallel not is (   A m ( r 3 1 − ∂x ∂F α ij δ α . 1 2 ) ij j i j ( ∇ |{z} scalar | + and , r Q T α ∧ ∂F ∂x | 2 contribute. ω F j i ) Q ) − − k danti-symmetric nd S = r n trace a and ij 1 3 α δ S ( = ij r ω ii α ∂F ∂x . 1 2 sthe is · ( k k P ω Q T nanti- an , ij ) ij k . + = . eld trace Q T T ji is kk ω F = ) Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. 43Daoaiaino ymti eodRn Tensor notation, Rank matrix Second using Symmetric a Recall, of Diagonalisation 14.3 o example, For rnia axes principal eleigenvalues real R ymti tensor symmetric rnfrainrule transformation qiaety hr xssabsso rhnra eigenvec orthonormal of basis a exists there Equivalently, Then Example bu n axis. any about for result same the get We oeta o h pca case special the for that Note rmthe from nsmay h o-eocmoet are components non-zero the summary: In x V 1 ρ ( o ii oyocpigvolume occupying body rigid a For If r ) T dV x ssmerc tcnb ignlsdb uha rhgnltr orthogonal an such by diagonalised be can it symmetric, is : I I 3 θ 11 33 I bv oobtain to above integration. 13 = = = = for I 33 ρ Z Z λ 0 − V V ( T 1 b inspection). (by = ρ ρ 1 4 λ , Z h esri ignli atsa oriae ln thes along coordinates Cartesian in diagonal is tensor the ; x 0 0 T a V 2 ( ( 2 4 ij x x R ′ ρ λ , 2 x xso symmetry. of axis omdensity form h nri esrfraclne fradius of cylinder a for tensor inertia The V 2 2 1 2 0 lπ = 1 x I + + 3 ρ I = 12 epciey h ietosdfie by defined directions The respectively. , 1 ( R + ij x r x x ip r 3 = )( I 3 2 2 2 = 1 2 22 ) ) R cos dV x l a I dV dV Z 1 2 jq = 2 13 wihhssin has (which V 2 T T + θ = π √ ,similarly. 0, = , ρ pq = = 3 2 ( ( = x x ρ 3 − r l becomes 0 2 2 2 a/ ρ ρ ( ) 3 ρ ) 0 0 n mass and = ) = V 0 4– 74 – T 2, x Z Z dV Z ij k 0 0 r ihms density mass with 0 I x a a ), ρ sin a ij k dr dr 0 dr and δ πa I T = ij 33 θ T Z Z 2 ′ Z , 2 − 0 0 2 1 θ ′ l 0 M = dV ( = 2 2 Ma = I 2 π π → x π 12 Ma dθ dθ i a 2 = TR RT dθ x 2 = T 2 2 cos j = δ Z Z ij ) ′ + Z ij 2 rd dx dθ dr r πa − − ), / 2 − dV. l l l l − ⇒ 2, l T 2 3 l θ dx dx 2 R dx where , .Also, ). l lρ R 2 I ρ L V tors 3 3 11 0 3 ( ( rr = ecos coordinates choose We . r r ρ r = . 2 ,w replace we ), = 2 ( 2 r cos R sin 1 2 = e ) a I 3 Ma ij x 1 R with , 22 e , n egt2 height and 2 1 ρ httetensor the that , θx T x 0 θ e = 2 πla 2 2 1 e , 3 = + ω e , ansformation. M dV 0 = 3 x o rotation a for R 4 2 x , e , 3 2 ( for − 3 . ) 4 1 r 1 P 3 a en the being . 2 I r the are T α axes. e ij + l m uni- , with sa is 3 1 α l 2 → ). Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. o,teegnausaecle h rnia oet fine of moments principal the called are eigenvalues the sor, ere ftebd:wt h xsw chose, we axes the with body: the of metries osdratno endby defined tensor a Integrals Invariant Consider to Application 15.2 51Dfiiin n lsicto Results tensor Classification A and Definitions Tensors 15.1 Isotropic and Invariant 15 calculation. ǫ section scle isotropic called is n etrproducts vector and r needn fteCreincodnt ytm Isotro system. coordinate Cartesian the of independent are ..ec opnn sucagd tensor A unchanged. is component each i.e. classified: o oesaa constants scalar some for Example xml,yumgttikof think might you example, ijk .Aliorpctnoso ihrrn r bandb combin by obtained are rank higher of tensors isotropic All 4. .Tems eea ak3iorpctno is tensor isotropic 3 rank general most The 3. .Tems eea ak2iorpctno is tensor isotropic 2 rank general most The 2. .Teeaen stoi esr frn vco)ecp th except (vector) 1 rank of tensors isotropic no are There 1. r stoi esr.Ti nue httecmoetdefin component the that ensures This tensors. isotropic are hsapist n ymti esr o h pca aeof case special the For tensor. symmetric any to applies This esrpout,cnrcin n iercombinations. linear and contractions products, tensor 14.2 h otgnrliorpctno frn is 4 rank of tensor isotropic general most The : T ecnotngestecretpicplae for axes principal correct the guess often can we , sinvariant is ie h aei l ietos.A oe nsection in noted As directions). all in same the (i.e. ne atclrrotation particular a under a T ij...k · ′ ,β γ β, α, T b T ijkl = ij...k = ǫ ijp δ = ij R hr r oohridpnetcmiain (for combinations independent other no are There . ǫ a = αδ ip klp i b R Z j u hsi qa to equal is this but , ij , V jq δ kl f 5– 75 – · · · ( + x R ) ( βδ x a T kr i ∧ x I ik hc sivratudrevery under invariant is which T ij j δ b pq · · · jl ) a on ob ignl ydirect by diagonal, be to found was ··· i T T + r = x ijk ij R = k γδ ǫ = dV ijk = if T il αδ ij δ a βǫ ··· jk j δ ij b k ik ijk k , i esr in tensors pic δ o oescalar some for rtia jl o oescalar some for I − ij ing seepie in exemplified As . zero e toso h scalar the of itions δ ae ntesym- the on based il δ δ h nri ten- inertia the ij jk 13.1.2 ). and vector. ℜ ǫ 3 , ijk rotation α δ a be can . β ij using . and Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. hsatv transformation active this hr sn hneo ai etr ee and here) vectors basis of change no is there 53ASec faPofo lsicto eut o Rank for Results Classification of Proof a of Sketch A 15.3 with h eut olwsrih owrl yaayigconditi analysing by through rotations forwardly specific straight follow results The rm(2) from rm(1) from otgnrlfr loe satal isotropic. actually is allowed form general most ( rmtedfiiinof definition the from osdra active an consider Then lryueu f()ad()hl o any for hold (2) and (1) if useful ularly det( Example hr sacoeyrltdrsl o h nri esro s a of tensor inertia the for result ρ related closely a is There with applies above hc sisotropic. is which ene nydtrietescalar the determine only need we R x 0 0 i a 1. 2. rmass or r or Bwt h hneo variables of change the with NB Proof: ∂x ∂x 2 f 4 f V T p i ′ x πr ( ( det( = ) ij...k ′ i ′ x x = en clrfnto and function scalar a being ) : r um aibe nteintegral). the in variables dummy are 2 = ) I dr T V ij sivratunder invariant is M ij . f = = : ( = x Z R 5 4 ′ V and ) πa R ip transformation V ρ ,ie h aoin1or Jacobian=1 the i.e. 1, = ) 5 0 f x R Hence . ( i T ip x x o n rotation any for 1 = k j R x dV jq k π δ · · · ij or T with = − = R R ij π/ ij Z x kr Z = . V i bu oriaeae,te ofimn htthe that confirming then axes, coordinate about 2 V x x T ′ V α j 15 pq...r f f = 4 ) u aigtetae 3 trace, the taking But . ( ( πa en oi peewith sphere solid a being x x = x dV 6– 76 – ′ ′ i ) ) x rotation 5 = e V T x x δ i ′ i ij...k i ′ i ′ ij = Z x x apdto mapped = . ssm oue ie rotation a Given volume. some is V j ′ j ′ R V ρ · · · · · · f 0 so , R (3 smpe to mapped is ( x ip x x R − ) k ′ k ′ x x T nwihcase which in , dV dV dV p i ′ 1) ij , x j ′ ′ ′ 15 sisotropic is x ∂x ∂x 4 · · · = ′ p i ′ πa = x dV = 5 x n o naineunder invariance for ons k ′ hr fcntn density constant of phere δ dV i ′ V h euti partic- is result The . ij e α R i ′ ups htunder that Suppose . = = with ip | r 5 2 T ⇒ T ≤ | acntn)and constant) (a Ma ii ij...k x = T i ′ n ij a 2 = R δ sisotropic. is h result The . ≤ V ij = R , x 3 ij αδ i x x i ij j dV and , (NB R ij = , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. etakPo.R oz o ro reading. proof for Josza R. Prof. thank We Acknowledgements .Spoethat Suppose 1. .Spoethat Suppose 2. .Spoethat Suppose 3. oainthrough rotation a T oainthrough rotation oainthrough rotation a R Hence R xsi iia a,w deduce we way, similar a in axes osdrartto through rotation a consider above, Hence ( ihohridxpstosw write we positions index other with R te ne oiin n te xso rotation, of axes other and positions index other hence − 2 12 12 12 1)( = R R R 33 12 21 R − T T T T R T T 2 1)( ijk p ij 23 22 133 13 33 T = p − T = (+1)(+1) = (+1)(+1) = = = 213 = 1) αδ βǫ T T R T R T T ij i ij (+1)( = ijk 111 1 23 22 ijk . p srn stoi.Consider isotropic. 1 rank is π srn stoi.Consider isotropic. 2 rank is R . T ihsmlrrslsfor results similar with , n similarly and 0 = 3 2 π bu the about srn stoi.Uigtertto by rotation the Using isotropic. 3 rank is q π/ = R bu the about 3 lcws bu the about clockwise 2 − T r − T T 22 23 T 1)(+1) pqr yapplying By . 2 T . Then . ( R ( ( = R π/ 23 x ij ij 1 = ) x si 2 above: (2) in as 2 − T = ) xstoyields too axis = 7– 77 – 3 213 1)(+1)(+1) T xs Then axis. T   R 11 T 1   = ijk 2 = − p T = 0 1 0 0 − R − 0 0 1 0 1 0 1 0 0 31 unless 0 = T 0 0 1 3 T T − 2 q π/ T 213 22 = .Te,rqiigiorp fa of isotropy requiring Then, 0. = 0 1 π pq oain bu the about rotations 2 T ln ihsmlrrslsfor results similar with Along . = T   oain bu te xsand axes other about rotations T 133   x = 32 1 T 3 T , T = 3 , and 33 R ijk 0. = axis: ,adtherefore and 0, = ,j k j, i, R 21 T = 1 sttlyantisymmetric, totally is 123 R T p T 111 α, 33 p T r l itnt Now distinct. all are T T = = 13 a and say 13 11 = R R R = ( = = 11 1 1 p p T R R R R 1 − 2 π 1 1 1 q = q p p 1)(+1) T R x R R R si (1) in as ij 1 3 − 1 T 3 1 r ∀ r q q and T T i i T T T pqr 1 0. = pqr pq pq 6= T and 13 x = = = = j 2 . .