Page 149 - Vector Analysis

§4.7 The Stokes Theorem 145

4.7.2 The Stokes theorem

The path we choose to circle around the point a does not have to be a circle. However, in

such a case the average of the angular velocity no longer makes sense (since u ¨ T might not

contribute to the motion in the angular direction), and we instead consider the limit of the

following quantity

lim 1 ¿ u ¨ T ds,
A
AÑ0 C

where A is the area enclosed by C. This limit is always curlu ¨ N because of the famous

Stokes’ theorem.

Theorem 4.86 (The Stokes theorem). Let u : Ω Ď R3 Ñ R3 be a smooth vector field, and
Σ be a C 1-surface with C 1-boundary B Σ in Ω. Then

żż
u ¨ T ds = curlu ¨ N dS ,

BΣ Σ

where N and T are compatible normal and tangent vector fields.

To prove the Stokes theorem, we first establish the following

Lemma 4.87. Let Ω Ď R3 be a bounded Lipschitz domain, and w : Ω Ñ Rn be a mooth
vector-valued function. If Σ Ď Ω is an oriented C 1-surface with normal N, then

curlw ¨ N = divΣ(w ˆ N) on Σ . (4.23)

Proof. Let O Ď Ω be a C 1-domain such that Σ Ď B O and N is the outward-pointing unit
normal on B O. In other words, Σ is part of the boundary of O. Since

(∇φ)i = B φ Ni + (∇BO φ)i on B O ,
B N

by the divergence theorem we conclude that for all φ P C 1(O),

ż żż ˝
(curlw ¨ N)φ dS = curlw ¨ ∇φ dx = (N ˆ w) ¨ ∇φ dS

BO O BO

żż
= (N ˆ w) ¨ ∇BOφ dS = divB O(w ˆ N)φ dS .

BO BO

Identity (4.23) is concluded since φ can be chosen arbitrarily on Σ.