14.8 Stokes' Theorem
Stokes' Theorem can be regarded as a three-dimensional version of Green's Theorem,
which relates a surface integral over a surface to a line integral around its boundary
curve.
Let S be an oriented surface with unit normal vector n and with the boundary curve
C . The orientation on S induces the positive orientation of the boundary curve C : if
you walk in the positive direction around C with your head pointing in the direction
of n, then the surface will always be on your left.
. Let S be an oriented piecewise-smooth surface that is bounded
by a simple, closed, piecewise-smooth boundary curve C with positive orientation. Let
F be a vector eld whose components have continuous partial derivatives on an open
region in R3 that contains S . Then
Stokes' Theorem
ˆ
¨
F · dr =
C
curl F · dS
S
Remark. In the special case where the surface S is at and lies in the xy -plane with
upward orientation, the unit normal is k. For the vector eld F = P (x, y) i+Q(x, y) j,
Stokes' Theorem becomes Green's Theorem.
ˆ
˛
F · dr =
P dx + Q dy =
∂S
¨
∂S
¨ curl F · k dA =
S
S
∂Q ∂P
−
∂x
∂y
dA
´
Example 1. Use Stokes' Theorem to evaluate C F · dr if F = h 3x8 , 4xy 3 , y 2 x i and
C is in the plane z = y consisting of 4 line segments from (0, 0, 0) to (1, 0, 0), from
(1, 0, 0) to (1, 3, 3), from (1, 3, 3) to (0, 3, 3) and from (0, 3, 3) to (0, 0, 0).
Example 2. Use Stokes' Theorem to evaluate
˜
S
curl F · dS if F = h xyz, x, exy cos z i
and S is the hemisphere x2 + y 2 + z 2 = 1, z ≥ 0, oriented upward.