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Chapter 16 – Vector Calculus16.8 Stokes Theorem

16.8 Stokes' Theorem

Objectives: Understand Stokes’

Theorem Use Stokes’ Theorem to

evaluate integrals

16.8 Stokes' Theorem 2

HistoryThe theorem is named after the

Irish mathematical physicist Sir George Stokes (1819–1903) who was known for his studies of fluid flow and light. He was a professor at Cambridge University.

16.8 Stokes' Theorem 3

History◦ What we call Stokes’ Theorem was

actually discovered by the Scottish physicist Sir William Thomson (1824–1907, known as Lord Kelvin).

◦ Stokes learned of it in a letter from Thomson in 1850 and had his students try to prove it on an exam at Cambridge University.

16.8 Stokes' Theorem 4

Stokes’ Theorem vs Greens’ TheoremStokes’ Theorem can be regarded as

a higher-dimensional version of Green’s Theorem.

◦ Green’s Theorem relates a double integral over a plane region D to a line integral around its plane boundary curve.

◦ Stokes’ Theorem relates a surface integral over a surface S to a line integral around the boundary curve of S (a space curve).

16.8 Stokes' Theorem 5

IntroductionThe figure shows an oriented

surface with unit normal vector n.

◦The orientation of S induces the positive orientation of the boundary curve C.

16.8 Stokes' Theorem 6

IntroductionThis means that If you walk in the positive

direction around C with your head pointing in the direction of n, the surface will always be on your left.

16.8 Stokes' Theorem 7

Stokes’ TheoremLet:

◦ S be an oriented piecewise-smooth surface bounded by a simple, closed, piecewise-smooth boundary curve C with positive orientation.

◦ F be a vector field whose components have

continuous partial derivatives on an open region in 3 that contains S.

Then,

curlC

S

d d F r F S

16.8 Stokes' Theorem 8

Stokes’ Theorem

Since

Stokes’ Theorem says that the line integral around the boundary curve of S of the tangential component of F is equal to the surface integral of the normal component of the curl of F.

and curl curl C C S S

d ds d dS F r F T F S F n

16.8 Stokes' Theorem 9

Stokes’ TheoremThe positively oriented boundary

curve of the oriented surface S is often written as ∂S.

So, the theorem can be expressed as:

curlS

S

d d

F S F r

16.8 Stokes' Theorem 10

Special CaseLet’s let surface S be flat, lie in

the xy-plane with upward orientation, the unit normal is k. Then the surface integral becomes

This is a special case of Green’s Theorem!

curl (curl )C S

d d dA F r F S F k

16.8 Stokes' Theorem 11

Example 1Use Stokes’ Theorem to evaluate . In

each case C is oriented counterclockwise as viewed from above.

CdF r

( , , ) ,

C is the boundary of the part of the plane

2 2 2 in the first octant.

x x zx y z e e e

x y z

F i j k

16.8 Stokes' Theorem 12

Example 2 – pg. 1151 # 9Use Stokes’ Theorem to evaluate .

In each case C is oriented counterclockwise as viewed from above.

CdF r

2 2

( , , ) 2 ,

C is the circle 16, 5.

xyx y z yz xz e

x y z

F i j k