Multivariate Calculus - Bruce E. Shapiro

202 LECTURE 25. STOKES’ THEOREM
Stokes’ theorem as we have presented here is really a special case in 3 dimensions.
A more general form would have S ⊂ R n and ∂S ⊂ R n−1 , and would be written
using ”generalized” surface and volume integrals as
∫ ∫
ω = dω (25.6)
∂S
where dω represents the “exterior derivative” of a generalized vector field ω called
a “differential form.” We have already met several special cases to equation 25.6:
1. In one dimension f : (a, b) ⊂ R ↦→ R, S is a line segment and ∂S is the set
of endpoints {a, b}. The derivative is f ′ , so that
∫ b
which is the fundamental theorem of calculus.
a
S
f ′ (t)dt = f(b) − f(a) (25.7)
2. In two dimensions we can write the vector field as F = (M, N, 0), S is an
area A ⊂ R 2 , ∂S is the boundary C of A, and n = (0, 0, 1) so that
(∇ × F)) · n = N x − M y (25.8)
Equation 25.5 then becomes
∮ ∮
F · dr = (Mdx + Ndy) = (N x − M y )dA (25.9)
which is Green’s theorem.
C
C
3. In three dimensions S = V , ∂S = ∂V , ω = F · n and dω = ∇ · F, giving
F · n dS = ∇ · F dV (25.10)
∂V
which is the divergence theorem (also called Gauss’ theorem).
Example 25.1 Find ∮ C
F · dr for
F = ( yz 2 − y, xz 2 + x, 2xyz )
where C is a cirlce of radius 3 centered at the origin in the xy-plane, using Stoke’s
theorem.
Solution. Letting S denote the disk that is enclosed by the circle C will can the
formula
∮
F · T ds = (∇ × F) · n dS
C
S
Revised December 6, 2006. Math 250, Fall 2006
A