Vector Calculus. Dr. D. Sukumar

Transcription

1 Vector Calculus Dr. D. Sukumar

2 Space co-ordinates Change of variable

3 Cartesian co-ordinates < x <

4 Cartesian co-ordinates < x < < y <

5 Cartesian co-ordinates < x < < y < < z <

6 Cylindrical

7 Cylindrical

8 Cylindrical

9 Cylindrical 0 r <

10 Cylindrical 0 r <

11 Cylindrical 0 r <

12 Cylindrical 0 r <

13 Cylindrical 0 r < 0 θ 2π

14 Cylindrical 0 r < 0 θ 2π

15 Cylindrical 0 r < 0 θ 2π

16 Cylindrical 0 r < 0 θ 2π

17 Cylindrical 0 r < 0 θ 2π < z <

18 Spherical

19 Spherical

20 Spherical

21 Spherical 0 ρ<

22 Spherical 0 ρ<

23 Spherical 0 ρ<

24 Spherical 0 ρ<

25 Spherical 0 ρ< 0 θ 2π

26 Spherical 0 ρ< 0 θ 2π

27 Spherical 0 ρ< 0 θ 2π

28 Spherical 0 ρ< 0 θ 2π

29 Spherical 0 ρ< 0 θ 2π 0 φ π

30 Multivariable function A function f is given on a domain D f : D R 3 R

31 Multivariable function A function f is given on a domain D D:= Set form f : D R 3 R

32 Multivariable function A function f is given on a domain D f : D R 3 R D:= Set form {(x, y, z) a x b, c y d, e z f }

33 Multivariable function A function f is given on a domain D f : D R 3 R D:= Set form {(x, y, z) a x b, c y d, e z f } D:= Description by words

34 Multivariable function A function f is given on a domain D f : D R 3 R D:= Set form {(x, y, z) a x b, c y d, e z f } D:= Description by words Tetrahedron cut from the first octant by the plane 6x + 3y + 2z = 6

35 Multivariable function A function f is given on a domain D f : D R 3 R D:= Set form {(x, y, z) a x b, c y d, e z f } D:= Description by words Tetrahedron cut from the first octant by the plane 6x + 3y + 2z = 6 D:= Picture

36 Triple Integral(Rectangular co-ordinates) Describe the regions and setup the integral 1. Write six different iterated triple integrals for the volume of the rectangular solids in the first octant bounded by the co-ordinates planes and the planes x = 1,y = 2,z = 3 2. Region in the first octant enclosed by the cylinder x 2 + z 2 = 4 and the plane y = Region bounded by paraboloids z = 8 x 2 y 2 and z = x 2 + y 2

37 Triple integral (Cylindrical co-ordinates) 1. Solid enclosed by the cylinder x 2 + y 2 = 4 bounded above by the paraboloid z = x 2 + y 2 2. Cylinder whose base is the circle r = 3cosθ and whose top lies in the plane z = 5 x 3. Find the average value of the function f (r, θ, z) = r over the region bounded by the cylinder r = 1 between the planes z = 1 and z = 1.

38 Triple integral (Spherical co-ordinates) 1. Region bounded below by z = 0, and above by the sphere x 2 + y 2 + z 2 = 4 side by x 2 + y 2 = Find the average value of the function f (ρ, φ, θ) over the solid ball ρ 1.

39 Cylindrical to Rectangular

40 Cylindrical to Rectangular P(r, θ, z)

41 Cylindrical to Rectangular P(r, θ, z)

42 Cylindrical to Rectangular P(r, θ, z) A(r, θ, 0)

43 Cylindrical to Rectangular B r cos θ O θ r P(r, θ, z) r sin θ A(r, θ, 0)

44 Cylindrical V = dzr drdθ

45 Cylindrical V = dzr drdθ x = r cos θ y = r sin θ z = z

46 Cylindrical V = dzr drdθ x = r cos θ y = r sin θ z = z

47 Spherical to Rectangular O

48 Spherical to Rectangular P(ρ, φ, θ) O

49 Spherical to Rectangular B ρ cos φ φ P(ρ, φ, θ) O

50 Spherical to Rectangular B ρ cos φ φ P(ρ, φ, θ) O ρ sin φ

51 Spherical to Rectangular B ρ cos φ φ P(ρ, φ, θ) O θ ρ sin φ A

52 Spherical to Rectangular B ρ cos φ φ P(ρ, φ, θ) O θ C ρ sin φ ρ sin φ cos θ D ρ sin φ sin θ A

53 Spherical V = ρ 2 sin φdρdφdθ x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ

54 Spherical V = ρ 2 sin φdρdφdθ x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ

55 Change of variable In evaluating the integration b f (x)dx we do a change of a variable t = 2x ˆ b a f (x) dx

56 Change of variable In evaluating the integration b f (x)dx we do a change of a variable t = 2x ˆ b ( t f dx 2) a t = 2x x = t 2

57 Change of variable In evaluating the integration b f (x)dx we do a change of a variable t = 2x ˆ 2b ( t f dx 2) 2a t = 2x x = t 2 x = a t = 2a, x = b t = 2b

58 Change of variable In evaluating the integration b f (x)dx we do a change of a variable t = 2x ˆ 2b ( t ) dt f 2 2 2a t = 2x x = t 2 x = a t = 2a, x = b t = 2b x = t 2 dx = dt 2

59 Change of variable Let G be the region in u, v plane. It is transformed into another region R in the xy-plane as a one-to-one map by the equation. x = g(u, v), y = h(u, v). We call, R is the image of G, G is the pre-image of R. Any function f defined on R can be thought of as a function f (g(u, v), h(u, v)) defined on G as well.

60 so What is the relation between integral f over R and integral f (g(u, v), h(u, v)) over G. If 1. f,g,h have continuous partial derivatives and 2. J(u, v) is zero only at isolated points then f (x, y)dxdy = f (g(u, v), h(u, v)) J(u, v) dudv R G

61 Jacobian Jacobian determinant or Jacobian of the co-ordinate transformation x = g(u, v), y = h(u, v). is J(x, y; u, v) = ( x u x u x v y v ) = (x, y) (u, v) = J(u, v)

62 Example Solve the system u = x y; v = 2x + y for x and y in-terms of u and v.

63 Example Solve the system u = x y; v = 2x + y for x and y in-terms of u and v. Find the Jacobian.

64 Example Solve the system u = x y; v = 2x + y for x and y in-terms of u and v. Find the Jacobian. Let R be the triangle region with vertices (0,0), (1,1) and (1, -2) in the xy-plane. Find the image of R and sketch the transformed region in uv-plane.

65 Example Solve the system u = x y; v = 2x + y for x and y in-terms of u and v. Find the Jacobian. Let R be the triangle region with vertices (0,0), (1,1) and (1, -2) in the xy-plane. Find the image of R and sketch the transformed region in uv-plane. Evaluate R (2x 2 xy y 2 )dxdy enclosed by the triangular region.

66 Example Solve the system u = x y; v = 2x + y for x and y in-terms of u and v. Find the Jacobian. Let R be the triangle region with vertices (0,0), (1,1) and (1, -2) in the xy-plane. Find the image of R and sketch the transformed region in uv-plane. Evaluate R (2x 2 xy y 2 )dxdy enclosed by the triangular region. Evaluate the same integral in the region bounded by the lines R : y = 2x + 4, y = 2x + 7, y = x 2, y = x + 1

67 u = x y; v = 2x + y x = u+v ; y = 2u+v 3 3

68 u = x y; v = 2x + y x = u+v ; y = 2u+v 3 3 ( x J(u, v) = (x,y) = (u,v) u y y u v x v y v ) = (0,3) ( ) = = 1 3 x G u + v = 3 (3,0) u

69 u = x y; v = 2x + y x = u+v ; y = 2u+v 3 3 ( x J(u, v) = (x,y) = (u,v) u y y u v x v y v ) = (0,3) ( ) = = 1 3 x G u + v = 3 (3,0) u

70 u = x y; v = 2x + y x = u+v ; y = 2u+v 3 3 ( x J(u, v) = (x,y) = (u,v) u y (1,1) y u v x v y v ) = ( ) = = 1 3 R x (1,-2) u

71 u = x y; v = 2x + y x = u+v ; y = 2u+v 3 3 ( x J(u, v) = (x,y) = (u,v) u y (1,1) y u v x v y v ) = (0,3) ( ) = = 1 3 R x G u + v = 3 (1,-2) (3,0) u

72 u = x y; v = 2x + y x = u+v ; y = 2u+v 3 3 ( x J(u, v) = (x,y) = (u,v) y (1,1) y u u v x v y v ) = (0,3) ( ) = = 1 3 R x G u + v = 3 (1,-2) (3,0) u ˆ 3 ˆ v (2x 2 xy y 2 ) 1 3 dudv

73 Example(cont.) R (2x 2 xy y 2 )dxdy u = x y; v = 2x + y x = u+v ; y = 2u+v 3 3

74 Example(cont.) R (2x 2 xy y 2 )dxdy u = x y; v = 2x + y x = u+v 3 ; y = 2u+v 3 R : y = 2x + 4, y = 2x + 7, y = x + 1, y = x 2 G : v = 4, v = 7, u = 1, u = 2

75 Example(cont.) R (2x 2 xy y 2 )dxdy u = x y; v = 2x + y x = u+v ; y = 2u+v 3 3 R : y = 2x + 4, y = 2x + 7, y = x + 1, y = x 2 G : v = 4, v = 7, u = 1, u = 2 dxdy = J(x, y; u, v)dudv = 1 3 dudv

76 Example(cont.) R (2x 2 xy y 2 )dxdy u = x y; v = 2x + y x = u+v ; y = 2u+v 3 3 R : y = 2x + 4, y = 2x + 7, y = x + 1, y = x 2 G : v = 4, v = 7, u = 1, u = 2 dxdy = J(x, y; u, v)dudv = 1dudv 3 ( ) 2 ( ) ( ) ( u + v u + v 2u + v 2u + v G ) dudv

77 Example(cont.) R (2x 2 xy y 2 )dxdy u = x y; v = 2x + y x = u+v ; y = 2u+v 3 3 R : y = 2x + 4, y = 2x + 7, y = x + 1, y = x 2 G : v = 4, v = 7, u = 1, u = 2 dxdy = J(x, y; u, v)dudv = 1dudv 3 ˆ 7 ( ) 2 ( ) ( ) ( u + v u + v 2u + v 2u + v ˆ ) dud 33 4

78 Evaluate 1. Find the Jacobian (x,y,z) (u,v,w) 1.1 x = u cos v, y = u sin v, z = w 1.2 x = 2u 1, y = 3v 4, z = 1 2 (w 4) 2. Evaluate R (3x xy + 8y 2 )dxdy on the region R : y = x + 1, y = x + 3, y = x, y = x using the substitution u = 3x + 2y, v = x + 4y 64/5