Page 139 - Vector Analysis

§4.6 The Divergence Theorem 135

Definition 4.74. A vector field u : Ω Ď Rn Ñ Rn is called solenoidal or divergence-free if
divu = 0 in Ω.

4.6.3 The divergence theorem

Theorem 4.75 (The divergence theorem). Let Ω Ď Rn be a bounded Lipschitz domain, and
v P C 1(Ω) X C (Ω). Then

żż
divv dx = v ¨ N dS ,

Ω BΩ

where N is the outward-pointing unit normal of Ω.

Proof. To embrace the beauty of geometry (and the context that we have introduced), we
prove the case that Ω is a bounded open set of class C 3.

Let tUmuKm=1 be an open cover of B Ω such that for each m P t1, ¨ ¨ ¨ , Ku there exists
a C 3-parametrization ψm : Vm Ď Rn´1 Ñ Um which is compatible with the orientation N;

that is,

( ... ¨¨¨ ... ψm,n´1 ... N ˝ ) ą 0 on Vm .
det [ψm,1 ψm]

Define ϑm(y1, yn) = ψm(y1) + yn(N ˝ ψm)(y1) as in Section 4.5.1. Then there exists εm ą 0
such that ϑm : Vm ˆ (´εm, εm) Ñ Wm is a C 2-diffeomorphism for some open set in Rn such

that ϑm : Vm ˆ (´εm, 0) Ñ Ω X Wm while ϑm : Vm ˆ (0, εm) Ñ int(ΩA) X Wm.

K

Choose an open set W0 Ď Rn such that W0 Ď Ω and Ω Ď Ť Wm, and define ϑ0 as the

m=0

identity map. Let 0 ď ζm ď 1 in Cc8(Um) denote a partition-of-unity of Ω subordinate to
the open covering tWmumK=0; that is,

K

ÿ ζm = 1 and spt(ζm) Ď Um @ m .

m=0

Let Jm = det(∇ϑm), Am = (∇ϑm)´1, and gm denote the first fundamental form associated
with tVm, ψmu. Using (4.17), ?gm(N ˝ ϑm) = Jm(Am)Ten on Vm ˆ t0u for m P t1, ¨ ¨ ¨ , Ku.