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Vector Calculus What Is Field? Electromagnetic Fields Lecture 1 Vector Calculus What is field? Mathematics: a map Rn →Rm which assigns each point a quantity (scalar or vector). Physics: property of space – action (observed as a force acting on objects) In this course: “Field” means property or space in which this property is observed or function which describes this property. Electromagnetic Fields, Lecture 1, slide 2 Vector calculus Vector calculus (or vector analysis) is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3 dimensional Euclidean space R3 Euclidean vector is a geometric object that has both a magnitude (or length) and direction. Vectors are fundamental in the physical sciences. Electromagnetic Fields, Lecture 1, slide 3 Why we use vectors Under changes of coordinate systems they behave the same way points do. Thus, vector notation of most of the equations used to describe physical fields' properties is independent on the coordinate system being used. VECTORS ARE CONVENIENT! [Physics] General covariance: the essential idea is that coordinates do not exist a priori in nature, but are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws. Electromagnetic Fields, Lecture 1, slide 4 Coordinate systems A coordinate system is a system which uses a set of numbers, or coordinates, to uniquely determine the position of a point or other geometric element. Using CS we can transform problems of geometry into problems concerning numbers (and then solve these problems by calculation). y 2.65 P(3.33,2.65) 3.33 x Vectors allow nice and general presentation of ideas and general behaviour of field, but we, engineers, do need numbers to quantitatively calculate, predict and design... Electromagnetic Fields, Lecture 1, slide 5 Cartesian coordinate system Three perpendicular planes are chosen and the three coordinates of a point are the signed distances to each of the planes x – the signed distance from the yz plane y – the signed distance from the xz plane z – the signed distance from the xy plane Electromagnetic Fields, Lecture 1, slide 6 Cylindrical coordinate system The origin, the plane and the reference direction on this plane are chosen. r – the radius is the Euclidean distance from the origin φ – the azimuth (or azimuthal angle) is the signed angle measured from the azimuth reference direction to the orthogonal projection of the line segment OP on the reference plane. z – the height is the signed distance from the chosen plane Electromagnetic Fields, Lecture 1, slide 7 Spherical coordinate system The origin, the plane and the azimuth reference direction are chosen. The zenith direction direction is perpendicular to the plane. r – the radius is the Euclidean distance from the origin θ – the inclination (or polar angle) is the angle between the zenith direction and the line segment OP. φ – the azimuth (or azimuthal angle) is the signed angle measured from the azimuth reference direction to the orthogonal projection of the line segment OP on the reference plane. Electromagnetic Fields, Lecture 1, slide 8 Coordinate transformation Cartesian Cylindrical Spherical (x,y,z) (ρ,φ,z) (r,θ,φ) Cartesian 2 2 2 2 2 r= x y z (x,y,z) = x y =arctan y / x =arccos z/r z=z = Cylindrical 2 2 x=cos = z (ρ,φ,z) y=sin =arctan/ z z=z =arctan y / x Spherical x=r sincos =r sin (r,θ,φ) y=r sinsin = z=r cos z=r cos teta These are not all elements we must change! More follows.... Electromagnetic Fields, Lecture 1, slide 9 Unit vectors transformation Cartesian Cylindrical Spherical (x,y,z) (ρ,φ,z) (r,θ,φ) Cartesian x y x 1x y 1 yz1 z 1 = 1x 1 y 1 = r r 1 ,1 ,1 y z 2 x y z 1 =− 1x 1 y x z1x y z 1 y− 1 z 1 = 1 =1 r z z −y 1 x 1 1 = x y z Cylindrical 1 =cos1 −sin1 1 = 1 1 x r r r z 1 =sin1 cos1 z 1 ,1 ,1z y 1 = 1 − 1 r r z 1z=1z 1 =1 Spherical 1x=sincos1rcoscos1 −sin1 1 =sinsin1 coscos1 cos1 1r ,1 ,1 y r 1 =sin1rcos1 1z=cos1r−sin1 1 =1 Still, these are not all elements we must change! More follows.... 1z=cos1r−sin1 Electromagnetic Fields, Lecture 1, slide 10 Representation of vectors Vectors are similar to points and thus their can be represented the same way the points are. y Z 2.65 P(3.33,2.65) v=[3.33,2.65] 3.33 x v=[0,0,5.3] A Electromagnetic Fields, Lecture 1, slide 11 Basic operations Scalar multiplication multiplication of a scalar (field) and a vector (field), yielding a vector field: v w = a v Vector addition (substraction) addition of two vectors (vector fields), yielding a vector (field): w = v + u v u w=v + u Electromagnetic Fields, Lecture 1, slide 12 Dot product Multiplication of two vectors (vector fields), yielding a scalar (field): s = v ∙ u = |v||u| cos θ v u Properties: the dot product is commutative: θ v ∙ u = u ∙ v s θ v| co the dot product is distributive over | vector addition: Using coordinates: w ∙ ( v + u ) = w ∙ u + w ∙ v v = [ v , v , v ], w = [ w , w , w ] i j k i j k v ∙ u = v w +v w +v w i i j j k k Electromagnetic Fields, Lecture 1, slide 13 Cross product Multiplication of two vectors (vector fields), yielding a vector (field): v×u w = v × u = |v||u| sin θ 1 n u Properties: the dot product is anticommutative: |v×u| θ v × u = - u × v v the dot product is distributive over Using coordinates: vector addition: v = [ v , v , v ], w = [ w , w , w ] i j k i j k i j k w × ( v + u ) = w u + w × v v × u = det vi v j vk [wi w j wk ] Electromagnetic Fields, Lecture 1, slide 14 Del – a special ,,vector” A convenient mathematical notation for three differential operators used in vector calculus: ∂ ∂ ∂ ∇= , , [ ∂ x ∂ y ∂ z ] Del operator represented by What is partial derivative? nabla symbol f(x,y,z)=2xy+sin y + y e- z ∂ f =2 y ∂ x ∂ f −z =cos ye ∂ y ∂ f −z =−y e ∂ z Electromagnetic Fields, Lecture 1, slide 15 Gradient The gradient of a scalar field f is a vector field pointing in the direction of fastest increase of f. ∂ f ∂ f ∂ f ∇ f x , y , z= , , [ ∂ x ∂ y ∂ z ] f x , y=x2 y2 ∇ f x , y=[ 2 x ,2 y ] Electromagnetic Fields, Lecture 1, slide 16 Gradient example Field of a dipole (+q,-q): 3D surface, equilines and gradient (direction only) (Example from http://www.gnuplot.info/demo/vector.html) Electromagnetic Fields, Lecture 1, slide 17 Line integral of a vector field b ur⋅d r= ur t⋅r ' t dt ∫L ∫a where r : [ a , b ] L is a bijective parametrization of L |dr| u u r ... i ... ur⋅d r≈ Liu i 3 ∫L ∑i r 2 1 Electromagnetic Fields, Lecture 1, slide 18 Gradient theorem ∇ f⋅d r=f e−f b ∫L e L We use this theorem when trying to integrate a vector field u along a line. If u is a gradient of a scalar field f, then the integral is path- b independent. Electromagnetic Fields, Lecture 1, slide 19 Flux of a vector field = u⋅ndS ∬S u n u n Electromagnetic Fields, Lecture 1, slide 20 Divergence The divergence of a vector field u is a scalar field that measures source or sink of u at a given point. ∯ u⋅n dS ∇⋅u=lim Sr r 0 ∣V r∣ u=[ux x , y , z,uy x , y , z,uz x , y , z] ∂u x , y , z ∂ u x , y , z ∂u x , y , z ∇⋅u= x y z ∂ x ∂ y ∂ z ∇⋅u=x2 y x 3 y2 u= , [ 3 2 ] Electromagnetic Fields, Lecture 1, slide 21 Divergence theorem Also knows as Gauss' theorem, Ostrogradsky's theorem or Gauss-Ostrogradsky theorem. ∇⋅u dV = u⋅n dS ∭V ∯∂V Volume integral of the divergence of a vector field u is equal to the outward flux of u through the volume boundary (surface). n S n V n n Electromagnetic Fields, Lecture 1, slide 22 Curl The curl of a vector field u is a vector field that measures rotation of u at a given point. n×u dS ∯Sr ∇×u=limr 0 ∣V r∣ u=[ux x , y , z,uy x , y , z,uz x , y , z] 1x 1 y 1z ∂ ∂ ∂ ∇×u= ∂ x ∂ y ∂ z ∣ ux uy uz ∣ y 3 x2 u= , ,0 [ 3 2 ] ∇×u=[ 0,0, x− y2 ] Electromagnetic Fields, Lecture 1, slide 23 Curl – interpretation y 3 x2 u= , ,0 [ 3 2 ] ∇×u=[ 0,0, x− y2 ] Electromagnetic Fields, Lecture 1, slide 24 Stokes' theorem Strictly speaking this is Kelvin-Stokes theorem (a special case of more general Stokes' theorem). ∇×u dS= u d r ∬S ∮∂ S Flux of the curl of a vector field u through a surface is equal to the line integral of u along the surface boundary (closed line). u S dS ∂S Electromagnetic Fields, Lecture 1, slide 25 Green's theorem Let's start with curl of a 2D vector, written in 3D as u=[ L, M, 0 ]: ∂ 0 ∂ M ∂ L ∂0 ∂ M ∂ L ∇ × u = − i − j − k ∂ y ∂ z ∂ z ∂ x ∂ x ∂ y Calculating a flux of this curl through any 2D surface dS=[dx×dy]: ∂ M ∂ L ∬ ∇×u d S=∬ ∇×u⋅k dS=∬ − dS s s S ∂ x ∂ y ∇×u dS= u d r According to the Stokes' theorem ∬S ∮∂ S ∂ M ∂ L ∬ − dS=∮ u⋅d r=∮ [ L, M ,0]⋅[dx ,dy , dz]=∮ LdxM dy S ∂ x ∂ y ∂ S ∂ S ∂ S Green's theorem: ∂ M ∂ L ∮ L dxM dy=∬ − dS ∂ S S ∂ x ∂ y Electromagnetic Fields, Lecture 1, slide 26 Second order derivatives In math we have several possibilities: for scalar field: ∇ ⋅ ∇ f and ∇×∇ f ≡0 for vector field: ∇ ∇⋅u and ∇ × ∇ × u and ∇⋅∇×u≡0 The most important second order differential operator is Laplacian: f =∇⋅∇ f =∇2 f which may be defined for vector field as: ∇ 2 u=∇ ∇⋅u−∇×∇×u Electromagnetic Fields, Lecture 1, slide 27 Identities Curl of the gradient of any scalar field is always zero vector: ∇×∇ f ≡0 This identity allows us express any curl free vector field by a scalar field.
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