Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems.

Transcription

1 S. R. Zinka School of Electronics Engineering Vellore Institute of Technology July 16, 2013

2 Outline 1 Vectors 2 Coordinate Systems 3 VC - Differential Elements 4 VC - Differential Operators 5 Important Theorems 6 Summary 7 Problems

3 Outline 1 Vectors 2 Coordinate Systems 3 VC - Differential Elements 4 VC - Differential Operators 5 Important Theorems 6 Summary 7 Problems

4 Norm (Absolute/Modulus/Magnitude) Definition Given a vector space V over a subfield F of the complex numbers, a norm a on V is a function : V R with the following properties: For all a F and all u, v V, 1 a v = a v (positive scalability). 2 u + v u + v (triangular inequality) 3 If v = 0 then v is the zero vector 0 (separates points) a Sometimes the vertical line, Unicode Ux007c ( ), is used (e.g., v ), but this latter notation is generally discouraged, because it is also used to denote the absolute value of scalars and the determinant of matrices.

5 Norm - A Few Examples Euclidean Norm On an n-dimensional Euclidean space R n, the intuitive notion of length of the vector x = (x 1, x 2,..., x n) is captured by the formula x 2 := x x x2 n. (1) On an n-dimensional complex space C n, the most common norm is z 2 := z z z n 2. (2) Taxicab Norm / Manhattan Norm The name relates to the distance a taxi has to drive in a rectangular street grid to get from the origin to the point x. It is defined as x 1 := n x i. (3) i=1

6 Norm - A Few Examples Maximum Norm Maximum norm is defined as x := max ( x 1, x 2,..., x n ). (4) p-norm p-norm is defined as x p := ( n ) 1/p x i p. (5) i=1

7 Norm - The Concept of Unit Circle x 2 x 1 x

8 Addition & Subtraction b b a a+b a a-b a a-b b b Addition Subtraction

9 Dot or Scalar Product Definition The dot product of two vectors, a = [a 1, a 2,..., a n] and b = [b1, b 2,..., b n] in a vector space of dimension n is defined as n a b = a i b i = a 1 b 1 + a 2 b a n b n = a b cos θ. (6) i=1 Properties a b = b a (commutative) ( ) a b + c = a b + a c (distributive over vector addition) ) ) a (r b + c = r ( a b + a c (bilinear)

10 Dot or Scalar Product - Physical Interpretation Projection of a in the direction of b, ab is given by a b = a b b (7) Corollary ( ) If a b = a c and a = 0, then we can write: a b c = 0 by the distributive law; the result above ( ) ( ) says this just means that a is perpendicular to b c, which still allows b c = 0, and therefore b = c.

11 Cross or Vector Product Definition The cross product a b is defined as a vector c that is perpendicular to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span. Properties a b = ( a ) b sin θ n (8) a b = b a (anti-commutative) ( ) a b + c = a b + a c (distributive over vector addition) ) ) a (r b + c = r ( a b + a c (bilinear)

12 Cross or Vector Product - Physical Interpretation

13 Cross or Vector Product - Why the Name Cross Product? a b = ˆx ŷ ẑ a x a y a z b x b y b z

14 Scalar Triple Product Definition The scalar triple product of three vectors is defined as the dot product of one of the vectors with the cross product of the other two, ( ) ) a b c = b ( c a) = c ( a b. (9) Properties ( ) a b c ( ) a b c ) = a ( c b = a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3

15 Scalar Triple Product - Physical Interpretation a α h c θ b base Corollary If the scalar triple product is equal to zero, then the three vectors a, b, and c are coplanar, since the parallelepiped defined by them would be flat and have no volume.

16 Vector Triple Product Definition The vector triple product is defined as the cross product of one vector with the cross product of the other two, ( ) ) a b c = b ( a c) c ( a b. (10)

17 Vectors - Independency & Orthogonality

18 Outline 1 Vectors 2 Coordinate Systems 3 VC - Differential Elements 4 VC - Differential Operators 5 Important Theorems 6 Summary 7 Problems

19 Remember Complex Numbers? Cartesian Polar Euler s formula is our jewel and one of the most remarkable, almost astounding, formulas in all of mathematics - Richard Feynman

20 Typical 2D Coordinate Systems Cartesian x = ρ cos φ y = ρ sin φ Polar ρ = x 2 + y 2 ( φ = tan 1 y ) x

21 2D Coordinate Transformations [ Aρ [ Ax A φ ] A y ] = = [ cos φ sin φ sin φ cos φ [ cos φ sin φ sin φ cos φ ] [ Ax ] [ Aρ A y ] A φ ]

22 Typical 3D Coordinate Systems (RHS) Z Z (x,y,z) (ρ,φ,z) z X O x y z Y X ρ O φ Y Cartesian x = ρ cos φ y = ρ sin φ z = z Cylendrical ρ = x 2 + y 2 ( φ = tan 1 y ) x z = z

23 Typical 3D Coordinate Systems (RHS) Spherical x = r sin θ cos φ y = r sin θ sin φ z = r cos θ r = x 2 + y 2 + z 2 θ = ( ) cos 1 z x2 + y 2 + z 2 ( φ = tan 1 y ) x

24 Cross Product of Standard Basis Vectors Z O (ρ,φ,z) z X ρ φ Y ˆx ŷ = ẑ ŷ ẑ = ˆx ẑ ˆx = ŷ ˆx ˆx = ˆ0 ˆρ ˆφ = ẑ ˆφ ẑ = ˆρ ẑ ˆρ = ˆφ ˆρ ˆρ = ˆ0 ˆr ˆθ = ˆφ ˆθ ˆφ = ˆr ˆφ ˆr = ˆθ ˆr ˆr = ˆ0 and so on...

25 Dot Product of Standard Basis Vectors Z O (ρ,φ,z) z X ρ φ Y ˆx ˆx = ŷ ŷ = ẑ ẑ = 1 ˆx ŷ = ŷ ẑ = ˆx ẑ = 0 ˆρ ˆρ = ˆφ ˆφ = ẑ ẑ = 1 ˆρ ˆφ = ˆφ ẑ = ẑ ˆρ = 0 ˆr ˆr = ˆθ ˆθ = ˆφ ˆφ = 1 ˆr ˆθ = ˆθ ˆφ = ˆφ ˆr = 0 and so on...

26 3D Coordinate Transformations Cartesian Cylindrical Z O (ρ,φ,z) z X ρ φ Y A ρ A φ A z A x A y A z = = cos φ sin φ 0 sin φ cos φ cos φ sin φ 0 sin φ cos φ A x A y A z A ρ A φ A z

27 3D Coordinate Transformations Cartesian Spherical A r A θ A φ A x A y A z = = sin θ cos φ sin θ sin φ cos θ cos θ cos φ cos θ sin φ sin θ sin φ cos φ 0 sin θ cos φ cos θ cos φ sin φ sin θ sin φ cos θ sin φ cos φ cos θ sin θ 0 A x A y A z A r A θ A φ

28 3D Coordinate Transformations Cylindrical Spherical Z O (ρ,φ,z) z X ρ φ Y A r A θ A φ A ρ A φ A z = = sin θ 0 cos θ cos θ 0 sin θ sin θ cos θ cos θ sin θ 0 A ρ A φ A z A r A θ A φ

29 Would you like to see a few more coordinate systems?

30 Parabolic Coordinate System

31 Curvilinear Coordinate System b 2 b 1 b 2 e 2 b 1 e 1

32 Outline 1 Vectors 2 Coordinate Systems 3 VC - Differential Elements 4 VC - Differential Operators 5 Important Theorems 6 Summary 7 Problems

33 Infinitesimal Differential Elements - Cartesian - dl dl = dxˆx + dyŷ + dzẑ

34 Infinitesimal Differential Elements - Cartesian - ds ds = ±dxdyẑ (or) ± dydzˆx (or) ± dzdxŷ

35 Infinitesimal Differential Elements - Cartesian - dv dv = dxdydz

36 Infinitesimal Differential Elements - Cylindrical - dl dl = dρ ˆρ + ρdφ ˆφ + dzẑ

37 Infinitesimal Differential Elements - Cylindrical - ds ds = ±ρdφdρẑ

38 Infinitesimal Differential Elements - Cylindrical - ds ds = ±ρdφdz ˆρ

39 Infinitesimal Differential Elements - Cylindrical - dv dv = ρdρdφdz

40 Infinitesimal Differential Elements - Spherical - dl dl = drˆr + rdθ ˆθ + r sin θdφ ˆφ

41 Infinitesimal Differential Elements - Spherical - ds ds = ±r 2 sin θdθdφˆr

42 Infinitesimal Differential Elements - Spherical - dv dv = r 2 sin θdrdθdφ

43 Outline 1 Vectors 2 Coordinate Systems 3 VC - Differential Elements 4 VC - Differential Operators 5 Important Theorems 6 Summary 7 Problems

44 Divergence Definition The divergence of a vector field F at a point P is defined as the limit of the net flow of F across the smooth boundary of a three dimensional region V divided by the volume of V as V shrinks to P. Formally, ( ) div F F ˆn (P) = F = lim V {P} S(V) V ds = Properties ) (k 1 A + k2 B = k 1 A + k2 B (linearity) ) (w A = w A + A w ( ) ) ) A B = B ( A A ( B lim V {P} S(V) F ds V. (11)

45 Divergence - Physical Interpretation n S n V n n F x F = x + F y y + F z z

46 Curl Definition If ˆn is any unit vector, the curl of F is defined to be the limiting value of a closed line integral in a plane orthogonal to ˆn as the path used in the integral becomes infinitesimally close to the point, divided by the area enclosed. ( ) curl F (P) = F = lim A 0 C Properties ) (k 1 A + k2 B = k 1 A + k2 B (linearity) ) (w A = w A A w ( ) [ ) )] [( ) ( ) ] A B = A ( B B ( A A B B A F dl ˆn. (12) A

47 Curl - Physical Interpretation ( Fz F = y F ) ( y Fx ˆx + z z F ) ( z Fy ŷ + x x F ) x ẑ y

48 Gradient Definition In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is that rate of increase, Properties (k 1 v + k 2 w) = k 1 v + k 2 w (Linearity) (vw) = v w + w v (Product Rule) grad (w) = w = w w ˆx + x y ŷ + w ẑ. (13) z

49 Gradient - Physical Interpretation w = w w ˆx + x y ŷ + w z ẑ

50 Solenoidal and Lamellar Fields Definition In vector calculus a solenoidal vector field (also known as an incompressible vector field) is a vector field v with divergence zero at all points in the field: Definition v = 0. (14) A vector field is said to be lamellar or irrotational if its curl is zero. That is, if v = 0. (15)

51 Curvilinear Coordinate Systems - Divergence, Curl, and Gradient v = v = [ q 1 (h 2 h 3 v 1) + q 2 (h 3 h 1 v 2) + 1 h 1 h 2 h 3 h 1 1 qˆ 1 h 1 qˆ 2 h 1 qˆ 3 h 1 h 2 h 3 q 1 q 2 q 3 h 1 v 1 h 2 v 2 h 3 v 3 ( ) 1 w w = ˆq i h i i q i where when (q 1, q 2, q 3) = (x, y, z) = (h 1, h 2, h 3) = (1, 1, 1), when (q 1, q 2, q 3) = (ρ, φ, z) = (h 1, h 2, h 3) = (1, ρ, 1), and when (q 1, q 2, q 3) = (r, θ, φ) = (h 1, h 2, h 3) = (1, r, r sin θ). ] (h 1 h 2 v 3) q 3

52 Second Order Derivatives - DCG Chart 2 w = w = ( w) A = ( A ) 2 A

53 Scalar Laplacian - Curvilinear Coordinate System 2 w = 1 [ ( h2 h 3 h 1 h 2 h 3 q 1 h 1 ) w + ( h3 h 1 q 1 q 2 h 2 ) w + ( h1 h 2 q 2 q 3 h 3 )] w q 3

54 Outline 1 Vectors 2 Coordinate Systems 3 VC - Differential Elements 4 VC - Differential Operators 5 Important Theorems 6 Summary 7 Problems

55 Open and Closed Surfaces & &

56 Divergence Theorem Definition Suppose V is a subset of R n (in the case of n = 3, V represents a volume in 3D space) which is compact and has a piecewise smooth boundary S. If F is a continuously differentiable vector field defined on a neighborhood of V, then we have V ) ( ) ( F dv = F ˆn ds = F ds. (16) S S

57 Divergence Theorem - Physical Interpretation ) [F (y + y) F (y)] x z = ( F vol 1 vol 1 ) [F (y + 2 y) F (y + y)] x z = ( F vol 2 vol 2 ) Sum : [F (y + 2 y) F (y)] x z = ( F vol i vol i i

58 Stokes Theorem Definition The surface integral of the curl of a vector field over a surface S in Euclidean three-space is related to the the line integral of the vector field over its boundary as ) ( F ds = F dl. (17) S C

59 Stokes Theorem - Physical Interpretation ) = ( F ds1 1 1 ) = ( F ds2 2 2 Sum : i i ) = ( F dsi i i

60 Outline 1 Vectors 2 Coordinate Systems 3 VC - Differential Elements 4 VC - Differential Operators 5 Important Theorems 6 Summary 7 Problems

61 Important Vectorial Identities A B = B A = A B cos θ A B = A B B B B A B = B A = ( A B sin θ) n = A (B C) = B (C A) = C (A B) = ˆx ŷ ẑ A x A y A z B x B y B z A x A y A z B x B y B z C x C y C z A (B C) = B (A C) C (A B) (A B) (C D) = (A C) (B D) (B C) (A D) *** (A B) (C D) = (A B D) C (A B C) D ***

62 Coordinate Transformations (Point) x = ρ cos φ y = ρ sin φ ρ = x 2 + y 2 ( φ = tan 1 y ) x x = r sin θ cos φ y = r sin θ sin φ z = r cos θ r = x 2 + y 2 + z 2 θ = ( ) cos 1 z x2 + y 2 + z 2

63 Coordinate Transformations (Vector) A ρ A φ A z A x A y A z A r A θ A φ A x A y A z A r A θ A φ A ρ A φ A z = = = = = = cos φ sin φ 0 sin φ cos φ cos φ sin φ 0 sin φ cos φ A x A y A z A ρ A φ A z sin θ cos φ sin θ sin φ cos θ cos θ cos φ cos θ sin φ sin θ sin φ cos φ 0 sin θ cos φ cos θ cos φ sin φ sin θ sin φ cos θ sin φ cos φ cos θ sin θ 0 sin θ 0 cos θ cos θ 0 sin θ sin θ cos θ cos θ sin θ 0 A ρ A φ A z A r A θ A φ A x A y A z A r A θ A φ

64 Differential Elements Cartesian Coordinate System: Cylindrical Coordinate System: dl = dxˆx + dyŷ + dzẑ ds = ±dxdyẑ (or) ± dydzˆx (or) ± dzdxŷ dv = dxdydz Spherical Coordinate System: dl = dρ ˆρ + ρdφ ˆφ + dzẑ ds = ±ρdφdρẑ (or) ± ρdφdz ˆρ dv = ρdρdφdz dl = drˆr + rdθ ˆθ + r sin θdφ ˆφ ds = ±r 2 sin θdθdφˆr dv = r 2 sin θdrdθdφ

65 Divergence, Curl, and Gradient where, v = v = [ q 1 (h 2 h 3 v 1) + q 2 (h 3 h 1 v 2) + 1 h 1 h 2 h 3 h 1 1 qˆ 1 h 2 qˆ 2 h 3 qˆ 3 h 1 h 2 h 3 q 1 q 2 q 3 h 1 v 1 h 2 v 2 h 3 v 3 ( ) 1 w w = ˆq i h i i q i [ 2 1 w = h 1 h 2 h 3 q 1 ( h2 h 3 h 1 ) w + ( h3 h 1 q 1 q 2 h 2 ] (h 1 h 2 v 3) q 3 ) w + ( h1 h 2 q 2 q 3 h 3 )] w q 3 (q 1, q 2, q 3) (v 1, v 2, v 3) (h 1, h 2, h 3) ( ) Catersian (x, y, z) vx, v y, v z (1, 1, 1) ( ) Cylindrical (ρ, φ, z) vρ, v φ, v z (1, ρ, 1) ( ) Spherical (r, θ, φ) vr, v θ, v φ (1, r, r sin θ)

66 Important Differential Identities (vw) = v w + w v (A B) = (A ) B + (B ) A + A ( B) + B ( A) *** (wa) = w A + A w (A B) = B ( A) A ( B) (wa) = w A A w *** (A B) = [A ( B) B ( A)] [(A ) B (B ) A] *** A = ( A) 2 A r = r r *** 1 r = r (. ) r r 3 r 3 *** = 2 ( 1 r ) = 4πδ (r) ***

67 Important Integral Identities ) V ( F dv = S F ds (Divergence Theorem) ) S ( F ds = C F dl (Stokes Theorem)

68 Outline 1 Vectors 2 Coordinate Systems 3 VC - Differential Elements 4 VC - Differential Operators 5 Important Theorems 6 Summary 7 Problems

69 Divergence 1 Evaluate the G at P Cart (2, 3, 4), if G = xˆx + y2ŷ + z 3 ẑ; P Cyl (2, 110, 1), if G = 2ρz 2 sin 2 φ ˆρ + ρz 2 sin 2φ ˆφ + 2ρ 2 z sin 2 φẑ; and P Spherical (1.5, 30, 50 ), if G = 2r sin θ cos φˆr + r cos θ cos φ ˆθ r sin φ ˆφ. Ans: 43; 9.06; 1.28 [H1, D3.7, P73] 2 Given the electric flux density, D = 0.3r2ˆr nc/m 2 in free space: ) find E (= D ε, where ε F/m at point P (r = 2, θ = 25, φ = 90 ); ) find the total charge (ρ v = D within the sphere r = 3; find the total electric flux leaving the sphere r = 4. Ans: 135.5ˆr V/m; 305 nc; 965 nc [H1, D3.3, P61]

70 Curl ( ) 1 Calculate the value of the vector current density Je = H : in Cartesian coordinates at P Cart (2, 3, 4), if H = x 2 zŷ y 2 xẑ; in cylindrical coordinates at P Cyl (1.5, 90, 0.5), if H = 2 ρ (cos 0.2φ) ˆρ; and in spherical coordinates at P Spherical (2, 30, 20 ), if H = 1 ˆθ. sin θ Ans: 16ˆx + 9ŷ + 16ẑ A/m 2 ; ẑ A/m 2 ; ˆφ A/m 2 [H1, D8.7, P246]

71 Gradient 1 Given the potential field, V = 2x 2 y 5z, find the electric field intensity ( E = V) at a given point P Cart (x, y, z). Ans: 4xyˆx 2x 2 ŷ + 5ẑ V/m [H1, E4.3, P104] 2 Given the potential field, V = 100 z 2 +1 ρ cos φ V, find the electric field intensity ( E = V) at a given point P Cyl (3, 60, 2). Ans: 10 ˆρ ˆφ + 24ẑ V/m [H1, D4.8, P106]

72 Divergence Theorem 1 Evaluate both sides of the divergence theorem for the field, D = 2xyˆx + x2ŷand the rectangular parallelepiped formed by the planes x = 0 and 1, y = 0 and 2, and z = 0 and 3. Ans: 12 [H1, E3.5, P77] 2 Given the field, D = 6ρ sin φ 2 ˆρ + 1.5ρ cos φ 2 ˆφ, evaluate both sides of the divergence theorem for the region bounded by ρ = 2, φ = 0, φ = π, z = 0, and z = 5. Ans: 225; 225 [H1, D3.9, P78] V ) ( F dv = F ds S

73 Stokes Theorem 1 Evaluate both sides of the Stokes theorem for the field, H = 6r sin φˆr + 18r sin θ cos φ ˆφ, and the patch around the region, r = 4, 0 θ 0.1π, and 0 φ 0.3π. Ans: 22.2 [H1, E8.3, P248] 2 Evaluate both sides of the Stokes theorem for the field, H = 6xyˆx 3y2ŷ, and the path around the region, 2 x 5, 1 y 1, and z = 0. Let the positive direction of ds be ẑ. Ans: -126 [H1, D8.6, P251] ) ( F ds = F dl S C