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Section 13.7Stokes’s Theorem

Math 21a

April 28, 2008

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.Image: Flickr user Al Giordino

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Announcements

◮ Final Exam: 5/23 9:15 am (tentative)◮ Office hours Tuesday, Wednesday 2–4pm SC 323◮ Problem Sessions: Mon, 8:30; Thur, 7:30; SC 103b

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Outline

Last Time: Surface and Flux integrals

Stokes’s TheoremStatementProof

Worksheet

Next Time: The Divergence Theorem

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Surface integrals and flux integrals

Let r : D → S be a parametrized surface and f a function on S.◮ If f is a function on S, we define∫∫

S

f dS =

∫∫D

f(r(u, v)) |ru × rv| dA

◮ If F is a vector field on S, we defined∫∫S

F · dS =

∫∫S

F · n dS =

∫∫D

F · (ru × rv) dA

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Outline

Last Time: Surface and Flux integrals

Stokes’s TheoremStatementProof

Worksheet

Next Time: The Divergence Theorem

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Theorem (Green’s Theorem)Let D be a piecewise-smooth surface abounded by a simple, closed,piecewise-smooth curve C with positive orientation. If P and Q havecontinuous partial derivatives on an open region that contains D, then∫

C

(P dx + Q dy) =

∫∫D

(∂Q∂x

− ∂P∂y

)dA

or ∫C

F · dr =

∫∫D

dF dA

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Theorem (Stokes’s Theorem)Let S be an oriented piecewise-smooth surface that is bounded by a simple,closed, piecewise-smooth boundary curve C with positive orientation. Let Fbe a vector field whose components have continuous partial derivatives onan open region in R3 that contains S. Then∫

C

F · dr =

∫∫S

curlF · dS

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Proof (one case)

Proof.Assume S is the graph of a function g with domain D, where theboundary curve C1 of D is mapped to C. Then you can apply Green’sTheorem.

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Outline

Last Time: Surface and Flux integrals

Stokes’s TheoremStatementProof

Worksheet

Next Time: The Divergence Theorem

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Example (Worksheet Problem 1)Let S is the hemisphere x2 + y2 + z2 = 4 with z ≥ 0 and P be theparaboloid z = x2 + y2 − 4 both oriented upwards. Suppose that F isa vector field defined on R3 with continuous partial derivatives.Explain why ∫∫

S

curlF · dS =

∫∫P

curlF · dS.

SolutionBoth of these are equal to the line integral of F around their respectiveboundaries, which in this case are the same curve.

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Example (Worksheet Problem 1)Let S is the hemisphere x2 + y2 + z2 = 4 with z ≥ 0 and P be theparaboloid z = x2 + y2 − 4 both oriented upwards. Suppose that F isa vector field defined on R3 with continuous partial derivatives.Explain why ∫∫

S

curlF · dS =

∫∫P

curlF · dS.

SolutionBoth of these are equal to the line integral of F around their respectiveboundaries, which in this case are the same curve.

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Example (Worksheet Problem 2)

Use Stokes’s Theorem to evaluate∫∫

S

curlF · dS, where

F = x2eyz i+ y2exz j+ z2exyj

and S is the hemisphere x2 + y2 + z2 = 4 with z ≥ 0 orientedupwards.

SolutionWe integrate

∫CF · dr around the boundary curve x2 + y2 = 4. After

parametrization we have

I = 8∫ 2π

0(cos(t) sin2(t) − sin(t) cos2(t)) dt = 0.

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Example (Worksheet Problem 2)

Use Stokes’s Theorem to evaluate∫∫

S

curlF · dS, where

F = x2eyz i+ y2exz j+ z2exyj

and S is the hemisphere x2 + y2 + z2 = 4 with z ≥ 0 orientedupwards.

SolutionWe integrate

∫CF · dr around the boundary curve x2 + y2 = 4. After

parametrization we have

I = 8∫ 2π

0(cos(t) sin2(t) − sin(t) cos2(t)) dt = 0.

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Example (Worksheet Problem 3)

Use Stokes’s Theorem to evaluate∫∫

S

curlF · dS, where

F = (x + y2) i+ (y + z2) j+ (z + x2)j

and C is the triangle with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1)oriented in counterclockwise direction when viewed from above.

SolutionThe answer is 3.

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Example (Worksheet Problem 3)

Use Stokes’s Theorem to evaluate∫∫

S

curlF · dS, where

F = (x + y2) i+ (y + z2) j+ (z + x2)j

and C is the triangle with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1)oriented in counterclockwise direction when viewed from above.

SolutionThe answer is 3.

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Outline

Last Time: Surface and Flux integrals

Stokes’s TheoremStatementProof

Worksheet

Next Time: The Divergence Theorem