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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