Vector Calculus

Dr. D. Sukumar

February 1, 2016

Green’s TheoremTangent form or Ciculation-Curl form

‰cMdx + Ndy =

¨R

(∂N

∂x− ∂M

∂y

)dA

‰CF · dr =

¨R

(∇× F ) · k dA

Stoke’s Theorem ‰CF · dr =

¨S∇× F · n dσ

Green’s TheoremTangent form or Ciculation-Curl form

‰cMdx + Ndy =

¨R

(∂N

∂x− ∂M

∂y

)dA

‰CF · dr =

¨R

(∇× F ) · k dA

Stoke’s Theorem ‰CF · dr =

¨S∇× F · n dσ

Green’s Theorem(Normal form or Flux-Divergence form)

˛CMdy − Ndx =

¨R

(∂M

∂x+∂N

∂y

)dA

˛CF · n ds =

¨R∇ · F dA

I C is a simple, closed, smooth curve

I R is the region enclosed by C

I dA is area element

I ds is length element¨SF · n dσ =

˚D∇ · F dV .

Green’s Theorem(Normal form or Flux-Divergence form)

˛CMdy − Ndx =

¨R

(∂M

∂x+∂N

∂y

)dA

˛CF · n ds =

¨R∇ · F dA

I C is a simple, closed, smooth curve

I R is the region enclosed by C

I dA is area element

I ds is length element¨SF · n dσ =

˚D∇ · F dV .

˛C

F · n ds =¨

R

∇ · F dA

¨SF · n dσ =

˚D∇ · F dV

I S is a simple, closed, oriented surface.

I D is solid regin bounded by S

I dσ surface area element

I dV is volume element

˛C

F · n ds =¨

R

∇ · F dA

¨SF · n dσ =

˚D∇ · F dV

I S is a simple, closed, oriented surface.

I D is solid regin bounded by S

I dσ surface area element

I dV is volume element

The Divergence TheoremGauss

The flux of a vector field F = M i + Nj + Pk across a closedoriented surface S in the direction of the surface’s outward unitnormal field n equals the integral of ∇ · F (divergence of F ) overthe region D enclosed by the surface:

¨SF · n dσ =

˚D∇ · F dV .

F = yi + xyi − zkD : The region inside the solid cylinder x2 + y2 ≤ 4 between theplane z = 0 and the parabolaid z = x2 + y2

∇ · F = 0 + x − 1 = x − 1˚

D

∇ · F dV =

ˆ 2

0

ˆ √4−x2

−√4−x2

ˆ x2+y2

0

(x − 1)dzdydx

=

ˆ 2

0

ˆ √4−x2

−√4−x2

(x − 1)(x2 + y2)dydx

=

ˆ 2

0

(x − 1)[x2y +y3

3]√4−x2

−√4−x2

=

ˆ 2

0

(x − 1)(2x2√

4− x2 +2

3(4− x)2

√4− x2)dx

=1

3

ˆ 2

0

(x − 1)√

4− x2[6x2 + 2(16− 8x + 8x2)]dx

=1

3

ˆ 2

0

(x − 1)√

4− x2[8x2 − 8x + 16]dx

= −16π

F = yi + xyi − zkD : The region inside the solid cylinder x2 + y2 ≤ 4 between theplane z = 0 and the parabolaid z = x2 + y2

∇ · F = 0 + x − 1 = x − 1˚

D

∇ · F dV =

ˆ 2

0

ˆ √4−x2

−√4−x2

ˆ x2+y2

0

(x − 1)dzdydx

=

ˆ 2

0

ˆ √4−x2

−√4−x2

(x − 1)(x2 + y2)dydx

=

ˆ 2

0

(x − 1)[x2y +y3

3]√4−x2

−√4−x2

=

ˆ 2

0

(x − 1)(2x2√

4− x2 +2

3(4− x)2

√4− x2)dx

=1

3

ˆ 2

0

(x − 1)√

4− x2[6x2 + 2(16− 8x + 8x2)]dx

=1

3

ˆ 2

0

(x − 1)√

4− x2[8x2 − 8x + 16]dx

= −16π

ExerciseDivergence theorem

Use divergence theorem to calculate outward flux

1. F = (y − x)i + (z − y)j + (y − x)kD :The cube bounded by the planes x ± 1, y ± 1 and z ± 1.−16

2. F = x2i− 2xy j + 3xzkD :The region cut from the first octant by the spherex2 + y2 + z2 = 4 3π

I F is conservative, F is irrotational=⇒ Ciruculation= 0

I F is incompressible, ∇.F is 0 =⇒ Flux= 0

Fundamental Theorem of Calculus

ˆ[a,b]

df

dxdx = f (b)− f (a)

Let F = f (x)i

ˆ[a,b]

df

dxdx = f (b)− f (a)

Fundamental Theorem of Calculus

ˆ[a,b]

df

dxdx = f (b)− f (a)

Let F = f (x)i

ˆ[a,b]

df

dxdx = f (b)− f (a)

= f (b)i · i + f (a)i · −i

Fundamental Theorem of Calculus

ˆ[a,b]

df

dxdx = f (b)− f (a)

Let F = f (x)i

ˆ[a,b]

df

dxdx = f (b)− f (a)

= f (b)i · i + f (a)i · −i= F (b) · n + F (a) · n

Fundamental Theorem of Calculus

ˆ[a,b]

df

dxdx = f (b)− f (a)

Let F = f (x)i

ˆ[a,b]

df

dxdx = f (b)− f (a)

= f (b)i · i + f (a)i · −i= F (b) · n + F (a) · n= total outward flux of F across the boundary

Fundamental Theorem of Calculus

ˆ[a,b]

df

dxdx = f (b)− f (a)

Let F = f (x)i

ˆ[a,b]

df

dxdx = f (b)− f (a)

= f (b)i · i + f (a)i · −i= F (b) · n + F (a) · n= total outward flux of F across the boundary

=

ˆ[a,b]∇ · Fdx

Integral of the differential operator acting on a field over a regionequal the sum of (or integral of ) field components appropriate tothe operator on the boundary of the region

Scalar integration

1. Integration

I Area Between curvesI Volume by cross sectionI Surface area of revolution by

I DiskI WasherI Shell

2. Double integral

I Cartesian co-ordinatesI Polar co-ordinates

3. Triple integrals

I RectangularI CylindricalI Spherical

4. Change of variable

I Jacobian

Scalar integration

1. IntegrationI Area Between curves

I Volume by cross sectionI Surface area of revolution by

I DiskI WasherI Shell

2. Double integral

I Cartesian co-ordinatesI Polar co-ordinates

3. Triple integrals

I RectangularI CylindricalI Spherical

4. Change of variable

I Jacobian

Scalar integration

1. IntegrationI Area Between curvesI Volume by cross section

I Surface area of revolution by

I DiskI WasherI Shell

2. Double integral

I Cartesian co-ordinatesI Polar co-ordinates

3. Triple integrals

I RectangularI CylindricalI Spherical

4. Change of variable

I Jacobian

Scalar integration

1. IntegrationI Area Between curvesI Volume by cross sectionI Surface area of revolution by

I DiskI WasherI Shell

2. Double integral

I Cartesian co-ordinatesI Polar co-ordinates

3. Triple integrals

I RectangularI CylindricalI Spherical

4. Change of variable

I Jacobian

Scalar integration

1. IntegrationI Area Between curvesI Volume by cross sectionI Surface area of revolution by

I Disk

I WasherI Shell

2. Double integral

I Cartesian co-ordinatesI Polar co-ordinates

3. Triple integrals

I RectangularI CylindricalI Spherical

4. Change of variable

I Jacobian

Scalar integration

1. IntegrationI Area Between curvesI Volume by cross sectionI Surface area of revolution by

I DiskI Washer

I Shell

2. Double integral

I Cartesian co-ordinatesI Polar co-ordinates

3. Triple integrals

I RectangularI CylindricalI Spherical

4. Change of variable

I Jacobian

Scalar integration

1. IntegrationI Area Between curvesI Volume by cross sectionI Surface area of revolution by

I DiskI WasherI Shell

2. Double integral

I Cartesian co-ordinatesI Polar co-ordinates

3. Triple integrals

I RectangularI CylindricalI Spherical

4. Change of variable

I Jacobian

Scalar integration

1. IntegrationI Area Between curvesI Volume by cross sectionI Surface area of revolution by

I DiskI WasherI Shell

2. Double integral

I Cartesian co-ordinatesI Polar co-ordinates

3. Triple integrals

I RectangularI CylindricalI Spherical

4. Change of variable

I Jacobian

Scalar integration

1. IntegrationI Area Between curvesI Volume by cross sectionI Surface area of revolution by

I DiskI WasherI Shell

2. Double integralI Cartesian co-ordinates

I Polar co-ordinates

3. Triple integrals

I RectangularI CylindricalI Spherical

4. Change of variable

I Jacobian

Scalar integration

1. IntegrationI Area Between curvesI Volume by cross sectionI Surface area of revolution by

I DiskI WasherI Shell

2. Double integralI Cartesian co-ordinatesI Polar co-ordinates

3. Triple integrals

I RectangularI CylindricalI Spherical

4. Change of variable

I Jacobian

Scalar integration

1. IntegrationI Area Between curvesI Volume by cross sectionI Surface area of revolution by

I DiskI WasherI Shell

2. Double integralI Cartesian co-ordinatesI Polar co-ordinates

3. Triple integrals

I RectangularI CylindricalI Spherical

4. Change of variable

I Jacobian

Scalar integration

1. IntegrationI Area Between curvesI Volume by cross sectionI Surface area of revolution by

I DiskI WasherI Shell

2. Double integralI Cartesian co-ordinatesI Polar co-ordinates

3. Triple integralsI Rectangular

I CylindricalI Spherical

4. Change of variable

I Jacobian

Scalar integration

1. IntegrationI Area Between curvesI Volume by cross sectionI Surface area of revolution by

I DiskI WasherI Shell

2. Double integralI Cartesian co-ordinatesI Polar co-ordinates

3. Triple integralsI RectangularI Cylindrical

I Spherical

4. Change of variable

I Jacobian

Scalar integration

1. IntegrationI Area Between curvesI Volume by cross sectionI Surface area of revolution by

I DiskI WasherI Shell

2. Double integralI Cartesian co-ordinatesI Polar co-ordinates

3. Triple integralsI RectangularI CylindricalI Spherical

4. Change of variable

I Jacobian

Scalar integration

1. IntegrationI Area Between curvesI Volume by cross sectionI Surface area of revolution by

I DiskI WasherI Shell

2. Double integralI Cartesian co-ordinatesI Polar co-ordinates

3. Triple integralsI RectangularI CylindricalI Spherical

4. Change of variable

I Jacobian

Scalar integration

1. IntegrationI Area Between curvesI Volume by cross sectionI Surface area of revolution by

I DiskI WasherI Shell

2. Double integralI Cartesian co-ordinatesI Polar co-ordinates

3. Triple integralsI RectangularI CylindricalI Spherical

4. Change of variableI Jacobian

Vector integration

5. Line integral

6. Vector fields

I GradientI Divergent – Flux densityI Curl– Circulation density

7. Green’s theorem

I Normal formI Tangent form

8. Surface integral

I Equation formI Parametric form

9. Stoke’s theorem

10. Gauss divergence theorem

Vector integration

5. Line integral

6. Vector fields

I GradientI Divergent – Flux densityI Curl– Circulation density

7. Green’s theorem

I Normal formI Tangent form

8. Surface integral

I Equation formI Parametric form

9. Stoke’s theorem

10. Gauss divergence theorem

Vector integration

5. Line integral

6. Vector fieldsI Gradient

I Divergent – Flux densityI Curl– Circulation density

7. Green’s theorem

I Normal formI Tangent form

8. Surface integral

I Equation formI Parametric form

9. Stoke’s theorem

10. Gauss divergence theorem

Vector integration

5. Line integral

6. Vector fieldsI GradientI Divergent – Flux density

I Curl– Circulation density

7. Green’s theorem

I Normal formI Tangent form

8. Surface integral

I Equation formI Parametric form

9. Stoke’s theorem

10. Gauss divergence theorem

Vector integration

5. Line integral

6. Vector fieldsI GradientI Divergent – Flux densityI Curl– Circulation density

7. Green’s theorem

I Normal formI Tangent form

8. Surface integral

I Equation formI Parametric form

9. Stoke’s theorem

10. Gauss divergence theorem

Vector integration

5. Line integral

6. Vector fieldsI GradientI Divergent – Flux densityI Curl– Circulation density

7. Green’s theorem

I Normal formI Tangent form

8. Surface integral

I Equation formI Parametric form

9. Stoke’s theorem

10. Gauss divergence theorem

Vector integration

5. Line integral

6. Vector fieldsI GradientI Divergent – Flux densityI Curl– Circulation density

7. Green’s theoremI Normal form

I Tangent form

8. Surface integral

I Equation formI Parametric form

9. Stoke’s theorem

10. Gauss divergence theorem

Vector integration

5. Line integral

6. Vector fieldsI GradientI Divergent – Flux densityI Curl– Circulation density

7. Green’s theoremI Normal formI Tangent form

8. Surface integral

I Equation formI Parametric form

9. Stoke’s theorem

10. Gauss divergence theorem

Vector integration

5. Line integral

6. Vector fieldsI GradientI Divergent – Flux densityI Curl– Circulation density

7. Green’s theoremI Normal formI Tangent form

8. Surface integral

I Equation formI Parametric form

9. Stoke’s theorem

10. Gauss divergence theorem

Vector integration

5. Line integral

6. Vector fieldsI GradientI Divergent – Flux densityI Curl– Circulation density

7. Green’s theoremI Normal formI Tangent form

8. Surface integralI Equation form

I Parametric form

9. Stoke’s theorem

10. Gauss divergence theorem

Vector integration

5. Line integral

6. Vector fieldsI GradientI Divergent – Flux densityI Curl– Circulation density

7. Green’s theoremI Normal formI Tangent form

8. Surface integralI Equation formI Parametric form

9. Stoke’s theorem

10. Gauss divergence theorem

Vector integration

5. Line integral

6. Vector fieldsI GradientI Divergent – Flux densityI Curl– Circulation density

7. Green’s theoremI Normal formI Tangent form

8. Surface integralI Equation formI Parametric form

9. Stoke’s theorem

10. Gauss divergence theorem

Vector integration

5. Line integral

6. Vector fieldsI GradientI Divergent – Flux densityI Curl– Circulation density

7. Green’s theoremI Normal formI Tangent form

8. Surface integralI Equation formI Parametric form

9. Stoke’s theorem

10. Gauss divergence theorem

Test

I No particular Model.

I Only exact answer will carry full marks.

Best wishes