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Chapter 2. Vector Analysis

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Cheng; 3/2/2009; 2-1 Chapter 2. Vector Analysis 2.1 Overview At a given position and time a scalar function → a magnitude, a vector function → a magnitude and a direction Function conversion between different coordinates Physical laws should be independent of the coordinates. Coordinate system is chosen by convenience Three main topics (1) Vector algebra : addition, subtraction, multiplication (2) Orthogonal coordinate system : Cartesian, Cylindrical, Spherical (3) Vector calculus : differentiation, integration(gradient, divergence, curl) 2.2 Vector Addition and Subtraction A vector has a magnitude and a direction AAa= A ↑ ↑ magnitude, A A unit vector, A Graphical representation Two vectors are equal if they have the same magnitude and direction, even though they may be displaced in space. • Vector addition, CA= + B Two vectors, AB and , form a plane Parallelogram rule : is the diagonal of the parallelogram C Head-To-Tail rule : The head of A touches the tail of B . C is drawn from the tail of A to the head of B . Cheng; 3/2/2009; 2-2 Note C =+=+A B B A Commutative law ABF++=++()() ABF Associative law • Vector subtraction ˆ AB−=+− Aej B, where −=−BaB()B 2.3 Vector Multiplication Multiplication of a vector by a positive scalar kA= bg kA a A A. Scalar or Dot Product AB•≡ ABcosθAB Note that •≤ (1) A B AB (2) •≤ •≥ A B 00 or A B •=× (3) A B A Projection of B onto A =B × Projection of A onto B (4) A •=B 0 when A and B are perpendicular to each other Note that A •=A A2 → AAA=• A •=•B B A : Commutative law ABF•+=•+•() ABAF : Distributive law Cheng; 3/2/2009; 2-3 Example Law of Cosine Use vectors to prove CAB222=+−2 ABcos α CBA=− CCCBABA2 =•⇒( −) •( −) ⇒•+•−•−•BB AAAB BA ⇒+−AB222cos AB α B. Vector or Cross Product AB×≡ aABnABsin θ ↑ unit vector normal to A and B (the right hand rule) AB× = Area of the parallelogram formed by A and B Note that AB×=−× BA NOT Commutative ABF××≠××()() ABF NOT Associative ABF×+=×+×() ABAF Distributive law C. Product of Three Vectors Scalar triple product ABC•×=•ejej AaBCn sin α ↑ ↑ Area of parallelogram Height of the parallelepiped → Volume of the parallelepiped Vector triple product ABCBACCAB××=( ) ( •−) ( •) back-cab rule Cheng; 3/2/2009; 2-4 2.4 Orthogonal Coordinate Systems A point in space is represented by three surfaces, u1 = const., u2 = const. and u3 = const. If they are mutually perpendicular, they form an orthogonal coordinate system. y Unit vectors along uuu,, → Base vectors, aaaˆˆˆ, , 123 uu123 u A vector in (uuu123,,) coordinate system → A =+Aauu11ˆˆˆ Aa u 2 u 2 + Aa u 3 u 3 y Differential change dui → Differential length change dliii= h du ↑ metric coefficient hi Vector differential length dl=+ aˆˆuu111( h du) a 2( h 2 du 2) + a ˆ u 3( h 3 du 3) Differential volume dv= ( h11 du) X( h 2 du 2) X( h 3 du 3) Vector differential area ds= ds aˆn , aˆn : surface normal Note ds11 aˆˆuu= h 232 h du du 31 a ds22 aˆˆuu= h 131 h du du 32 a ds33 aˆˆuu= h 121 h du du 23 a A. Cartesian Coordinate System bgbguuu123,,= xyz ,, A point bgxyz111,, : The intersection of three planes xx= 1 , yy= 1 , zz= 1 Base vectors : aaa xyz, , Properties of the base vectors aaxy×= a z, aayz×= a x, aazx×= a y aaxy•=•=•= aa yz aa xz0 aaxx•=•=•= aa yy aa zz1 The position vector to the point Pxbg111,, y z → → OP=++ x111 axyz y a z a A vector A in Cartesian coord. AAaAaAa=++xx yy zz Cheng; 3/2/2009; 2-5 y Vector differential length → dl=++ dx axyz dy a dz a Differential area dsx = dydz dsy = dxdz dsz = dxdy Differential volume dv= dxdydz y The dot product A•= Bdidi Aaxx + Aa yy + Aa zz • Ba xx + Ba yy + Ba zz ⇒ AB x x + AB y y + AB z z The cross product A×= Bdidi Aaxx + Aa yy + Aa zz × Ba xx + Ba yy + Ba zz aaa xyz AB×=AAAxyz ⇒−diAByz AB zy a x +−bg AB zx AB xz a y +−di AB xy AB yx a z BBBxyz Example A straight line L1 is given by 2xy+= 4 . (a) Find the unit normal to L1 starting from the origin (b) Fine the normal line to L1 passing through P(0,2) Solution: (a) A vector along L1 : −+24xyˆˆ (1) A vector from the origin to a point Q on L1 : xxˆˆ+−(42 x) y (2) From (1) and (2) (−+24xyxxˆˆˆ)i( +−(42 xy) ˆ) =0 → Q(1.6, 0.8) 1 The unit normal is ()2xyˆˆ+ 5 1 (b) The straight line parallel to the unit normal is yxc= + 2 1 This line should pass through P(0,2) → yx= + 2 2 Cheng; 3/2/2009; 2-6 B. Cylindrical coordinates bgbguuu123,,=φ r ,, z A point Prbg111,,φ z : a cylindrical surface with radius of r1 , a half-plane rotated by φ1 from x-axis, a plane cutting z-axis at z1 . Three base vectors : aaarz, φ , ↑ ↑ directions change with P aarz×=φ a , aaφ ×=zr a , aaazr×=φ y Differential length dl=+ dr arz rdφ aφ + dz a ↑ Differential areas dsr = rdφ dz dsφ = drdz dsz = rdrdφ Differential volume dv= rdrdφ dz Cheng; 3/2/2009; 2-7 y A vector in cylindrical coords. AAaAaAa=++rrφφ zz Transformation of A into Cartesian coord. AAaAaaAaaAaaxxrrxxzzxr=•=•+•+•⇒φφ Acosφφ − A φ sin ↑ ↑ ↑ =0 = cos φ = − sin φ Similarly using aary• =φsin , aaφ • y = cos φ AAyr=+sinφφ Aφ cos In matrix form MLAx PO MLcosφφ− sin 0POMLAr PO MAy P = Msinφφ cos 0PMAφ P NMAz QP NM 001QPNMAz QP y A point in cylindrical coord is transformed into Cartesian coord. x = r cos φ yr= sin φ zz= Conversion from Cartesian to cylindrical coords. rxy=+22 y φ = tan−1 x zz= → Exercise Convert a vector in Cartesian coords., OQ=++345 axyz a a , to cylindrical coords. → OQa=dididi345xyzrrxyz ++• a aaaa + 345 ++• a aaaaφφ + 345 xyzzz ++• a aaa ↑ ↑ ↑ 34cosφφ+ sin 34bg−+sinφφ cos 5 From Cartesian Coords. we know that 3 4 cos φ= and sin φ= 5 5 Therefore in cylindrical Coords. → OQ=+55 arz a Can you convert this vector back to Cartesian Coords.? Cheng; 3/2/2009; 2-8 C. Spherical Coordinates bgbguuu123,,=θφ R ,, A point PRbg111,,θφ : a sphere with radius R1 a cone with a half-angle of θ1 a half-plane rotated by φ1 from x-axis z φφ= 1 plane θθ= 1 cone θ1 aR P a φ a θ φ1 y x RR= 1 sphere Three base vectors aaaR ×=θφ, aaθφ×= a R , aaφθ×=R a A vector in spherical coord AAaAaAa=++RRθθ φφ y Vector differential length dl=+ dR aR Rdθθφ aθφ + Rsin d a ↑ ↑ Differential areas 2 dsR = Rsinθ dθφ d dsθ = Rsinθ dRdφ dsφ = RdRdθ Differential volume dv= R2 sinθ dRdθφ d Cheng; 3/2/2009; 2-9 • Conversion of a point xR= sinθφ cos yR= sinθφ sin zR= cos θ Conversion from Cartesian to spherical coords. Rxyz=++222 θ =+tan−122⎡⎤xyz / ⎣⎦ y φ = tan−1 x Integrals Containing Vector Functions Fdv Fxˆˆˆ++ Fy Fz dv ∫V ∫V ( xyz) Vdl V() xdxˆˆˆ++ ydy zdz ∫C ∫C → Fdli Fdli ∫C ∫C Adsi Adsi ∫S ∫S Sign convention for the surface integral Cheng; 3/2/2009; 2-10 Cheng; 3/2/2009; 2-11 2.5 Gradient of a Scalar Field A scalar field Vtbg,,, u123 u u Move from VV= 1 surface to VV=+1 dV surface PP12→ → dn PP13→ → dl ↑ ↑ same dV different distances The directional derivative dV : The space rate of change of V along l dl The gradient of a scalar function V is defined as dV grad V≡∇ V ≡ a : The maximum directional derivative of V dn n (an is perpendicular to V=const. plane) • The directional derivative dV dV dn dV dV =⇒cos α ⇒•⇒∇•aanlbg Va l : projection of gradV onto al direction dl dn dl dn dn → dV=∇bg V • dl , (1) In Cartesian coord. ∂ ∂ ∂ ∂ ∂ ∂ V V V F V V V I dV =++⇒dx dy dz axyz ++a adxadyadza•++di xyz ∂x ∂y ∂z HG ∂x ∂y ∂z KJ ↑ (2) = dl From (1) and (2) ∂V ∂V ∂V F ∂ ∂ ∂ I ∇=V a +a +a ⇒a +a + aV ∂x xyz∂y ∂z HG ∂x xyz∂y ∂z KJ ↑ ≡ ∇ , “del” operator In bguuu123,, coordinates 11∂ ∂ 1∂ ∇=a +a + a uu123 u hu11∂ hu 22∂ hu 3∂ 3 Cheng; 3/2/2009; 2-12 2.6 Divergence of a Vector Field Flux is defined as a quantity per unit area per unit time A vector field is represented by flux lines Its direction → Direction of the flux line Its magnitude → Density of the flux lines • Divergence of A The net outward flux of A per unit volume AdS• zS divA ≡ lim : dS= dS an ΔV →0 ΔV Find divA at Pxbgooo,, y z For a small volume ΔxΔΔy z A•= dS A •+ dS A •+ dS A •+ dS A •+ dS A •+ dS A • dS zSz frontz backz rightz leftz topz botom (1) On the front surface 1 AdSA•=front •ΔΔΔΔ S front ⇒ Ax x(,,) o +2 xy o z o yz zfront ↑ ↑ ΔxA∂ x bgΔΔyzax , Axxo(,,) y o z o+ , Taylor series 2 ∂x bgxyzooo,, (2) On the back surface 1 A•= dSAback •ΔΔΔΔΔΔ S back ⇒ A back •−bg yza x ⇒−− A x(,,) x o2 x y o z o yz zback ↑ ΔxA∂ Ax(,,) y z − x xo o o 2 ∂x bgxyzooo,, Combining ∂A AdS•+ AdS •=x ΔΔΔxyz (1) zfrontz back ∂x bgxyzooo,, Cheng; 3/2/2009; 2-13 Similarly, combining contributions from the right and left surfaces ∂A AdS•+ AdS •=y ΔΔΔxyz (2) zrightz left ∂y bgxyzooo,, Similarly, combining contributions from the top and bottom surfaces ∂A AdS•+ AdS •=z ΔΔΔxyz (3) ztopz bottom ∂z bgxyzooo,, From (1), (2) and (3) F ∂A ∂Ay ∂A I AdS•=x + +z ΔΔΔxyz zS HG ∂x ∂y ∂z KJ bgxyzooo,, ∂A ∂Ay ∂A → divA =++x z ∂x ∂y ∂z We define ∇•AdivA ≡ In general 1 L ∂ ∂ ∂ O ∇•A =M bgbgbghhA231 +hh13 A 2 + hh12 A 3P hhh123N∂ u 1 ∂u2 ∂u3 Q 2.7 Divergence Theorem The volume integral of the divergence of a vector field equals the total outward flux of the vector through the surface ∇•AdV = A • dS zzVS Proof: Consider a differential volume ΔV j bounded by S j From the definition of divA ej∇•AVΔ j = AdS • j zs j Add all ΔV j ’s L N O L N O limM ∇•AVΔ j P =• lim M AdSAdSP ⇒• ∑ejj ∑ ΔΔV jj→0M P V →0M zs j P zS N j =1 Q N j =1 Q ↑ ↑ ↑ Integral at the external surface ∇•Adv An internal surface is shared by two adjacent volumes.
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  • Geometric Algebra: an Introduction with Applications in Euclidean and Conformal Geometry

    Geometric Algebra: an Introduction with Applications in Euclidean and Conformal Geometry

    San Jose State University SJSU ScholarWorks Master's Theses Master's Theses and Graduate Research Fall 2013 Geometric Algebra: An Introduction with Applications in Euclidean and Conformal Geometry Richard Alan Miller San Jose State University Follow this and additional works at: https://scholarworks.sjsu.edu/etd_theses Recommended Citation Miller, Richard Alan, "Geometric Algebra: An Introduction with Applications in Euclidean and Conformal Geometry" (2013). Master's Theses. 4396. DOI: https://doi.org/10.31979/etd.chd4-za3f https://scholarworks.sjsu.edu/etd_theses/4396 This Thesis is brought to you for free and open access by the Master's Theses and Graduate Research at SJSU ScholarWorks. It has been accepted for inclusion in Master's Theses by an authorized administrator of SJSU ScholarWorks. For more information, please contact [email protected]. GEOMETRIC ALGEBRA: AN INTRODUCTION WITH APPLICATIONS IN EUCLIDEAN AND CONFORMAL GEOMETRY A Thesis Presented to The Faculty of the Department of Mathematics and Statistics San Jos´eState University In Partial Fulfillment of the Requirements for the Degree Master of Science by Richard A. Miller December 2013 ⃝c 2013 Richard A. Miller ALL RIGHTS RESERVED The Designated Thesis Committee Approves the Thesis Titled GEOMETRIC ALGEBRA: AN INTRODUCTION WITH APPLICATIONS IN EUCLIDEAN AND CONFORMAL GEOMETRY by Richard A. Miller APPROVED FOR THE DEPARTMENT OF MATHEMATICS AND STATISTICS SAN JOSE´ STATE UNIVERSITY December 2013 Dr. Richard Pfiefer Department of Mathematics and Statistics Dr. Richard Kubelka Department of Mathematics and Statistics Dr. Wasin So Department of Mathematics and Statistics ABSTRACT GEOMETRIC ALGEBRA: AN INTRODUCTION WITH APPLICATIONS IN EUCLIDEAN AND CONFORMAL GEOMETRY by Richard A. Miller This thesis presents an introduction to geometric algebra for the uninitiated.