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VECTOR CALCULUS

13

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13.8

Stokes Theorem

In this section, we will learn about:

The Stokes Theorem and

using it to evaluate integrals.

VECTOR CALCULUS

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STOKES VS. GREENS THEOREM

Stokes Theorem can be regarded as

a higher-dimensional version of Greens

Theorem.

Greens Theorem relates a double integral over

a plane region Dto a line integral around its planeboundary curve.

Stokes Theorem relates a surface integral over

a surface Sto a line integral around the boundarycurve ofS(a space curve).

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INTRODUCTION

The figure shows an oriented surface with

unit normal vectorn.

The orientation ofSinduces the positiveorientation of the

boundary curve C.

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INTRODUCTION

This means that:

If you walk in the positive direction around Cwith your head pointing in the direction ofn,

the surface will always be on your left.

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STOKES THEOREM

Let:

Sbe an oriented piecewise-smooth surfacebounded by a simple, closed, piecewise-smooth

boundary curve Cwith positive orientation.

F be a vector field whose components havecontinuous partial derivatives on an open region

in that contains S.

Then,

3

curlC

S

d d F r F S

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STOKES THEOREM

The theorem is named after the Irish

mathematical physicist Sir George Stokes

(18191903).

What we call Stokes Theorem was actually

discovered by the Scottish physicist Sir William

Thomson (18241907, known as Lord Kelvin).

Stokes learned of it in a letter from Thomson in 1850.

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STOKES THEOREM

Thus, Stokes Theorem says:

The line integral around the boundary curve ofSof the tangential component ofF is equal to the surfaceintegral of the normal component of the curl ofF.

and

curl curl

C C

S S

d ds

d d

F r F T

F S F n S

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STOKES THEOREM

The positively oriented boundary curve of

the oriented surface Sis often written as S.

So,the theorem can be expressed as:

curlS

S

d d

F S F r

Equation 1

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STOKES THEOREM, GREENS THEOREM, & FTC

There is an analogy among Stokes Theorem,

Greens Theorem, and the Fundamental

Theorem of Calculus (FTC).

As before, there is an integral involving derivatives

on the left side of Equation 1 (recall that curl F isa sort of derivative ofF).

The right side involves the values ofF only onthe boundaryofS.

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In fact, consider the special case

where the surface S:

Is flat.

Lies in the xy-plane with upward orientation.

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Then,

The unit normal is k.

The surface integral becomes a double integral.

Stokes Theorem becomes:

curl curlC

S S

d d dA F r F S F k

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STOKES THEOREM

Stokes Theorem is too difficult for us to

prove in its full generality.

Still, we can give a proof when:

Sis a graph.

F, S, and Care well behaved.

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STOKES TH.SPECIAL CASE

We assume that the equation ofS

is:

z= g(x, y), (x, y) D

where:

ghas continuous second-order partial derivatives.

Dis a simple plane region whose boundary curveC1 corresponds to C.

Proof

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If the orientation ofSis upward, the positive

orientation ofCcorresponds to the positive

orientation ofC1.

ProofSTOKES TH.SPECIAL CASE

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We are also given that:

F = Pi + Qj + Rk

where the partial derivatives of

P, Q, and Rare continuous.

Proof

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Sis a graph of a function.

Thus, we can apply Formula 10 in

Section 12.7 with F replaced by curl F.

Proof

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Suppose

x =x(t) y =y(t) atb

is a parametric representation ofC1.

Then, a parametric representation ofCis:

x =x(t) y =y(t) z =g(x(t), y(t)) a

t

b

Proof

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Four terms in that double integral cancel.

The remaining six can be arranged to

coincide with the right side of Equation 2.

Hence,

curlCS

d d F r F S

Proof

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STOKES THEOREM

We first compute:

2 2

curl 1 2yx y z

y x z

i j k

F k

Example 1

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STOKES THEOREM

There are many surfaces with

boundary C.

The most convenientchoice, though, isthe elliptical region Sin the plane y+ z= 2that is bounded by C.

If we orient Supward,Chas the inducedpositive orientation.

Example 1

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STOKES THEOREM

The projection DofSon the xy-plane

is the disk x2 + y2 1.

So, using Equation 10in Section 12.7 with

z =g(x, y) = 2y,we have the following

result.

Example 1

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STOKES THEOREM

2 1

0 0

12 32

00

21 2

2 30

1

2

curl 1 2

1 2 sin

2 sin2 3

sin

2 0

CS D

d d y dA

r r dr d

r rd

d

F r F S

Example 1

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STOKES THEOREM

Use Stokes Theorem to compute

where:

F(x, y, z) = xzi + yzj + xyk

Sis the part of

the spherex2 + y2 + z2 = 4that lies inside

the cylinder

x2 + y2 =1and above

the xy-plane.

curlS

d F SExample 2

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STOKES THEOREM

A vector equation ofCis:

r(t) = cos ti + sin tj + k 0 t 2

Therefore, r(t) =sin ti + cos tj

Also, we have:

3

Example 2

3 cos 3 sin cos sint t t t t F r i j k

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STOKES THEOREM

Thus, by Stokes Theorem,

2

0

2

0

2

0

curl

( ( )) '( )

3 cos sin 3 sin cos

3 0 0

CS

d d

t t dt

t t t t dt

dt

F S F r

F r r

Example 2

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STOKES THEOREM

In general, ifS1 and S2 are oriented surfaces

with the same oriented boundary curve C

and both satisfy the hypotheses of Stokes

Theorem, then

This fact is useful when it is difficult to integrate

over one surface but easy to integrate over the other.

1 2

curl curlC

S S

d d d F S F r F S

Equation 3

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CIRCULATION

Thus, is a measure of the tendency

of the fluid to move around C.

It iscalled the circulation ofv around C.

C

d v r

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CURL VECTOR

Now, let:

P0(x0, y0, z0) be a point in the fluid.

Sabe a small disk with radius aand centerP0.

Then, (curl F)(P) (curl F)(P0) for all pointsPon Sabecause curl F is continuous.

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CURL VECTOR

Thus, by Stokes Theorem, we get

the following approximation to the circulation

around the boundary circle Ca:

0 0

2

0 0

curl curl

curl

curl

a

a a

a

CS S

S

d d dS

P P dS

P P a

v r v S v n

v n

v n

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CURL VECTOR

The approximation becomes better as a 0.

Thus, we have:

0 0 201

curl limaCa

P P da

v n v r

Equation 4

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CURL & CIRCULATION

Equation 4 gives the relationship

between the curl and the circulation.

It shows that curl v n is a measure of

the rotating effect of the fluid about the axis n.

The curling effect is greatest about the axis

parallel to curl v.

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CURL & CIRCULATION

Imagine a tiny paddle wheel placed in

the fluid at a point P.

The paddle wheel

rotates fastest

when its axis is

parallel to curl v.

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CLOSED CURVES

Finally, we mention that Stokes Theorem

can be used to prove Theorem 4 in

Section 12.5:

If curl F = 0 on all of , then F is conservative.3

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CLOSED CURVES

From Theorems 3 and 4 in Section 12.3,

we know that F is conservative if

for every closed path C.

Given C, suppose we can find an orientablesurface Swhose boundary is C.

This can be done, but the proof requiresadvanced techniques.

0C

d F r

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CLOSED CURVES

Then, Stokes Theorem gives:

A curve that is not simple can be broken into

a number of simple curves. The integrals around these curves are all 0.

curl 0 0C

S S

d d d F r F S S

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CLOSED CURVES

Adding these integrals,

we obtain:

for any closed curve C.

0C

d F r

chap 13 econ

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