1
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