Multivariate Calculus - Bruce E. Shapiro

Lecture 25
Stokes’ Theorem
We have already seen the circulation, whose definition we recall here. We will use
this definition to give a geometric definition of the curl of a vector field, as we did
in the previous section for the divergence.
Definition 25.1 Let D ⊂ R 3 be an open set, C ⊂ D a path, and F a vector field
on D. Then the circulation of the vector field is
∮
circ F = F · dr (25.1)
The circulation density about a vector n is
circ F
circ n F = lim
A→0 A
= lim 1
A→0 A
C
∮
C
F · dr (25.2)
We define the curl of a vector field at a point as vector field having magnitude
equal to the maximum circulation at the point and direction normal to the plane of
circulation, so that
(∇ × F) · n = circ n F (25.3)
Theorem 25.1 Let F = (M, N, P ) be a differentiable vector field. Then
curl F = ∇ × F = (P y − N z , M z − P x , N x − M y ) (25.4)
Theorem 25.2 Stokes’ Theorem. Let D ⊂ R 3 be a connected set, let S ⊂ D be
a surface with boundary ∂S and surface normal vector n. If F = (M, N, P ) is a
differentiable vector field on D then
∮
∂S
F · T ds = (∇ × F) · n dS (25.5)
S
where T is a unit tangent vector of ∂S. Here ds is the distance element along C
and dS is the surface element on S.
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