Vector Calculus. Dr. D. Sukumar. February 1, 2016

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1 Vector Calculus Dr. D. Sukumar February 1, 2016

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

3 Green s Theorem Tangent form or Ciculation-Curl form Stoke s Theorem c Mdx + Ndy = C C F dr = R R S ( N x M ) da y ( F ) k da F dr = F n dσ

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

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

6 C F n ds = R F da

7 C F n ds = R F da F n dσ = 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

8 The Divergence Theorem Gauss 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 n dσ = F dv. S D

9 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 = = = = = 1 3 = 1 3 ˆ 2 ˆ 4 x ˆ 2 x 2 +y 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 + y 3 3 ] 4 x 2 4 x 2 (x 1)(2x 2 4 x (4 x)2 4 x 2 )dx 0 ˆ 2 0 = 16π (x 1) 4 x 2 [6x 2 + 2(16 8x + 8x 2 )]dx (x 1) 4 x 2 [8x 2 8x + 16]dx

10 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 = = = = = 1 3 = 1 3 ˆ 2 ˆ 4 x ˆ 2 x 2 +y 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 + y 3 3 ] 4 x 2 4 x 2 (x 1)(2x 2 4 x (4 x)2 4 x 2 )dx 0 ˆ 2 0 = 16π (x 1) 4 x 2 [6x 2 + 2(16 8x + 8x 2 )]dx (x 1) 4 x 2 [8x 2 8x + 16]dx

11 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

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

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

14 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

15 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

16 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

17 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 = F dx [a,b]

18 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

19 Scalar integration 1. Integration

20 Scalar integration 1. Integration Area Between curves

21 Scalar integration 1. Integration Area Between curves Volume by cross section

22 Scalar integration 1. Integration Area Between curves Volume by cross section Surface area of revolution by

23 Scalar integration 1. Integration Area Between curves Volume by cross section Surface area of revolution by Disk

24 Scalar integration 1. Integration Area Between curves Volume by cross section Surface area of revolution by Disk Washer

25 Scalar integration 1. Integration Area Between curves Volume by cross section Surface area of revolution by Disk Washer Shell

26 Scalar integration 1. Integration Area Between curves Volume by cross section Surface area of revolution by Disk Washer Shell 2. Double integral

27 Scalar integration 1. Integration Area Between curves Volume by cross section Surface area of revolution by Disk Washer Shell 2. Double integral Cartesian co-ordinates

28 Scalar integration 1. Integration Area Between curves Volume by cross section Surface area of revolution by Disk Washer Shell 2. Double integral Cartesian co-ordinates Polar co-ordinates

29 Scalar integration 1. Integration Area Between curves Volume by cross section Surface area of revolution by Disk Washer Shell 2. Double integral Cartesian co-ordinates Polar co-ordinates 3. Triple integrals

30 Scalar integration 1. Integration Area Between curves Volume by cross section Surface area of revolution by Disk Washer Shell 2. Double integral Cartesian co-ordinates Polar co-ordinates 3. Triple integrals Rectangular

31 Scalar integration 1. Integration Area Between curves Volume by cross section Surface area of revolution by Disk Washer Shell 2. Double integral Cartesian co-ordinates Polar co-ordinates 3. Triple integrals Rectangular Cylindrical

32 Scalar integration 1. Integration Area Between curves Volume by cross section Surface area of revolution by Disk Washer Shell 2. Double integral Cartesian co-ordinates Polar co-ordinates 3. Triple integrals Rectangular Cylindrical Spherical

33 Scalar integration 1. Integration Area Between curves Volume by cross section Surface area of revolution by Disk Washer Shell 2. Double integral Cartesian co-ordinates Polar co-ordinates 3. Triple integrals Rectangular Cylindrical Spherical 4. Change of variable

34 Scalar integration 1. Integration Area Between curves Volume by cross section Surface area of revolution by Disk Washer Shell 2. Double integral Cartesian co-ordinates Polar co-ordinates 3. Triple integrals Rectangular Cylindrical Spherical 4. Change of variable Jacobian

35 Vector integration 5. Line integral

36 Vector integration 5. Line integral 6. Vector fields

37 Vector integration 5. Line integral 6. Vector fields Gradient

38 Vector integration 5. Line integral 6. Vector fields Gradient Divergent Flux density

39 Vector integration 5. Line integral 6. Vector fields Gradient Divergent Flux density Curl Circulation density

40 Vector integration 5. Line integral 6. Vector fields Gradient Divergent Flux density Curl Circulation density 7. Green s theorem

41 Vector integration 5. Line integral 6. Vector fields Gradient Divergent Flux density Curl Circulation density 7. Green s theorem Normal form

42 Vector integration 5. Line integral 6. Vector fields Gradient Divergent Flux density Curl Circulation density 7. Green s theorem Normal form Tangent form

43 Vector integration 5. Line integral 6. Vector fields Gradient Divergent Flux density Curl Circulation density 7. Green s theorem Normal form Tangent form 8. Surface integral

44 Vector integration 5. Line integral 6. Vector fields Gradient Divergent Flux density Curl Circulation density 7. Green s theorem Normal form Tangent form 8. Surface integral Equation form

45 Vector integration 5. Line integral 6. Vector fields Gradient Divergent Flux density Curl Circulation density 7. Green s theorem Normal form Tangent form 8. Surface integral Equation form Parametric form

46 Vector integration 5. Line integral 6. Vector fields Gradient Divergent Flux density Curl Circulation density 7. Green s theorem Normal form Tangent form 8. Surface integral Equation form Parametric form 9. Stoke s theorem

47 Vector integration 5. Line integral 6. Vector fields Gradient Divergent Flux density Curl Circulation density 7. Green s theorem Normal form Tangent form 8. Surface integral Equation form Parametric form 9. Stoke s theorem 10. Gauss divergence theorem

48 Test No particular Model. Only exact answer will carry full marks.

49 Best wishes

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