Vector Calculus. Dr. D. Sukumar. January 31, 2014

Transcription

1 Vector Calculus Dr. D. Sukumar January 31, 2014

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3 Green s Theorem Tangent form or Ciculation-Curl form c Mdx +Ndy = R ( N x M ) da y

4 Green s Theorem Tangent form or Ciculation-Curl form c Mdx +Ndy = C F dr = R R ( N x M ) da y ( F) k da C is a simple, closed, smooth curve in counterclockwise direction R is the region enclosed by C da is area element dr is tangential length

5 Stokes Theorem The circulation of F = Mi+Nj+Pk around the boundary C of an oriented surface S in the direction counterclockwise with respect to the surface s unit normal vector n equals the integral of F n over S F dr = F ndσ C S C is a simple,closed, smooth curve considered counterclockwise direction

6 Stokes Theorem The circulation of F = Mi+Nj+Pk around the boundary C of an oriented surface S in the direction counterclockwise with respect to the surface s unit normal vector n equals the integral of F n over S F dr = F ndσ C S C is a simple,closed, smooth curve considered counterclockwise direction S is a surface (oriented) with boundary C

7 Stokes Theorem The circulation of F = Mi+Nj+Pk around the boundary C of an oriented surface S in the direction counterclockwise with respect to the surface s unit normal vector n equals the integral of F n over S F dr = F ndσ C S C is a simple,closed, smooth curve considered counterclockwise direction S is a surface (oriented) with boundary C dσ is surface area element

8 Stokes Theorem The circulation of F = Mi+Nj+Pk around the boundary C of an oriented surface S in the direction counterclockwise with respect to the surface s unit normal vector n equals the integral of F n over S F dr = F ndσ C S C is a simple,closed, smooth curve considered counterclockwise direction S is a surface (oriented) with boundary C dσ is surface area element dr is tangential length

9 Use Stoke s Theorem to calculate the circulation of the Field F = x 2 i +2xj +z 2 k around the curve C: The ellipse 4x 2 +y 2 = 4 in the xy plane counter clockwise when viewed from above.

10 Use Stoke s Theorem to calculate the circulation of the Field F = x 2 i +2xj +z 2 k around the curve C: The ellipse 4x 2 +y 2 = 4 in the xy plane counter clockwise when viewed from above. The surface S is 4x 2 +y 2 = 4 ie f = 4x 2 +y 2 = 4

11 Use Stoke s Theorem to calculate the circulation of the Field F = x 2 i +2xj +z 2 k around the curve C: The ellipse 4x 2 +y 2 = 4 in the xy plane counter clockwise when viewed from above. The surface S is 4x 2 +y 2 = 4 ie f = 4x 2 +y 2 = 4 i j k F = x y z x 2 2x z 2 ( F) k = 2

12 Use Stoke s Theorem to calculate the circulation of the Field F = x 2 i +2xj +z 2 k around the curve C: The ellipse 4x 2 +y 2 = 4 in the xy plane counter clockwise when viewed from above. The surface S is 4x 2 +y 2 = 4 ie f = 4x 2 +y 2 = 4 i j k F = x y z x 2 2x z 2 ( F) k = 2 Circulation F dr c

13 Use Stoke s Theorem to calculate the circulation of the Field F = x 2 i +2xj +z 2 k around the curve C: The ellipse 4x 2 +y 2 = 4 in the xy plane counter clockwise when viewed from above. The surface S is 4x 2 +y 2 = 4 ie f = 4x 2 +y 2 = 4 i j k F = x y z x 2 2x z 2 ( F) k = 2 Circulation F dr = c ˆ 2 ˆ 4 y y ˆ 2 2dxdy = 2 4 y 2 dy 0

14 Use Stoke s Theorem to calculate the circulation of the Field F = x 2 i +2xj +z 2 k around the curve C: The ellipse 4x 2 +y 2 = 4 in the xy plane counter clockwise when viewed from above. The surface S is 4x 2 +y 2 = 4 ie f = 4x 2 +y 2 = 4 i j k F = x y z x 2 2x z 2 ( F) k = 2 Circulation F dr = c ˆ 2 ˆ 4 y y ˆ 2 2dxdy = 2 ( y =2 4 y sin 1y 2 0 ) y 2 dy

15 Use Stoke s Theorem to calculate the circulation of the Field F = x 2 i +2xj +z 2 k around the curve C: The ellipse 4x 2 +y 2 = 4 in the xy plane counter clockwise when viewed from above. The surface S is 4x 2 +y 2 = 4 ie f = 4x 2 +y 2 = 4 i j k F = x y z x 2 2x z 2 ( F) k = 2 Circulation F dr = c ( y =2 ˆ 2 ˆ 4 y y ˆ 2 2dxdy = 2 4 y sin 1y 2 ( π ) =4 2 0 = 2π 0 ) y 2 dy

16 Exercise Stoke s Theorem Use Stoke s theorem to calculate the flux of the curl of the field F across the surface S in the direction of the outward unit normal n. 1. F = 2zi +3xj +5yk S : z +x 2 +y 2 = 4 2. F = 2zi +3xj +5yk S : r(r, θ) = (r cos θ)i +(r sin θ)j +(4 r 2 )k 0 r 2, 0 θ 2π 3. F = x 2 yi +2y 3 zj +3zk S : r(r, θ) = (r cos θ)i +(r sin θ)j +rk 0 r 1, 0 θ 2π 12π π 4

17 Green s Theorem (Normal form or Flux-Divergence form) C Mdy Ndx = R ( M x + N ) da y

18 Green s Theorem (Normal form or Flux-Divergence form) C ( M Mdy Ndx = R x + N ) y F nds = F da C S da C is a simple, closed, smooth curve R is the region enclosed by C da is area element ds is length element

19 C F nds = S F da

20 C F nds = S F da F ndσ = S D F dv S is a simple, closed, oriented surface. D is solid regin bounded by S dσ surface area element dv is volume element

21 The Divergence Theorem The flux of a vector field F = Mi+Nj+Pk across a closed oriented surface S in the direction of the surface s outward unit normal field n equals the integral of F (divergence of F) over the region D enclosed by the surface: F ndσ = F dv. S D

22 F = yi +xyi zk D : The region inside the solid cylinder x 2 +y 2 4 between the plane z = 0 and the parabolaid z = x 2 +y 2 D F = 0+x 1 = x 1 F dv

23 F = yi +xyi zk D : The region inside the solid cylinder x 2 +y 2 4 between the plane z = 0 and the parabolaid z = x 2 +y 2 D F = 0+x 1 = x 1 F dv = = = = ˆ 2 ˆ 4 x 2 ˆ x2 +y 2 = x 2 ˆ 2 ˆ 4 x 2 0 ˆ 2 0 ˆ 2 0 ˆ 2 0 (x 1)dzdydx 4 x 2 (x 1)(x 2 +y 2 )dydx (x 1)[x 2 y + y3 3 ] 4 x 2 4 x 2 (x 1)(2x 2 4 x (4 x)2 4 x 2 )dx 0 ˆ 2 (x 1) 4 x 2 [6x 2 +2(16 8x +8x 2 )]dx = 1 (x 1) 4 x 3 2 [8x 2 8x +16]dx 0 = 16π

24 Exercise Divergence theorem Use divergence theorem to calculate outward flux 1. F = (y x)i+(z y)j+(y x)k D :The cube bounded by the planes x ±1, y ±1 and z ± F = x 2 i 2xyj+3xzk D :The region cut from the first octant by the sphere x 2 +y 2 +z 2 = 4 3π

25 F is conservative, F is irrotational= Ciruculation= 0 F is incompressible,.f is 0 = Flux= 0

26 Fundamental Theorem of Calculus ˆ [a,b] df dx = f(b) f(a) dx

27 Fundamental Theorem of Calculus Let F = f(x)i ˆ [a,b] ˆ [a,b] df dx = f(b) f(a) dx df dx = f(b) f(a) dx = f(b)i i +f(a)i i

28 Fundamental Theorem of Calculus Let F = f(x)i ˆ [a,b] ˆ [a,b] df dx = f(b) f(a) dx df dx = f(b) f(a) dx = f(b)i i +f(a)i i = F(b) n+f(a) n

29 Fundamental Theorem of Calculus ˆ [a,b] Let F = f(x)i ˆ df dx = f(b) f(a) dx [a,b] df dx = f(b) f(a) dx = f(b)i i +f(a)i i = F(b) n+f(a) n = total outward flux of F across the boundary

30 Fundamental Theorem of Calculus ˆ [a,b] Let F = f(x)i ˆ df dx = f(b) f(a) dx [a,b] df dx = f(b) f(a) dx = f(b)i i +f(a)i i = F(b) n+f(a) n = total outward flux of F across the boundary ˆ = Fdx [a,b]

31 Integral of the differential operator acting on a field over a region equal the sum of (or integral of ) field components appropriate to the operator on the boundary of the region