Vector Calculus - IIT Hyderabadsuku/vectorcalculus2016/Lecture12.pdf · Vector Calculus Dr. D. Sukumar February 1, 2016. Green’s Theorem Tangent form or Ciculation-Curl form ˘
Post on 01-Jun-2020
16 Views
Preview:
Transcript
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
top related