Page 142 - Vector Analysis

138 CHAPTER 4. Vector Calculus

quarter disc D1 = ␣(x, 0, z) P R3 ˇ x2 + z2 ď a2, x, z ě 0( and D2 = ␣(x, b, z) P R3 ˇ x2 + z2 ď
ˇ ˇ

a2, x, z ě 0(. Therefore,

ż żażb

F ¨ N dS = (0, y2, z) ¨ (´1, 0, 0) dydz = 0 ,

R1 0 0

ż żażb

F ¨ N dS = (x, y2, 0) ¨ (0, 0, ´1) dydx = 0 ,

R2 00
?
ż ż a ż a2´x2

F ¨ N dS = (x, 0, z) ¨ (0, ´1, 0) dzdx = 0 ,

D1 0 0

and

?? a2´x2
żaż a2´x2 żaż πa2b2
ż F ¨ N dS =

(x, b2, z) ¨ (0, 1, 0) dzdx = b2 dzdx = 4 .

D1 0 0 00

Together with the result in Example 4.72, we find that

ż ¨ N dS = (ż ż ż ż ż ) ¨ N dS = πa2b2 + πa2b = πa2(b2 + 2b)
F + F 4 2 4
+ + +
BΩ Σ R1
R2 D1 D2

ż

= divF d(x, y, z) .

Ω

4.6.4 The divergence theorem on surfaces with boundary

This section is devoted to the divergence theorem on surfaces in R3 instead of domains of
Rn. To do so, we need to define what the divergence operator on a surface is, and this
requires that we first define the vector fields on which the surface divergence operator acts.

Definition 4.79. Let Σ Ď R3 be an open C 1-surface; that is, Σ is of class C 1 and Σ X B Σ =
H. A vector field u defined on Σ is called a tangent vector field on Σ, denoted by u P TΣ,
if u ¨ N = 0 on Σ, where N : Σ Ñ S2 is a unit normal vector field on Σ.

Having established (4.18), we find that the divergence operator div is the formal adjoint
of the operator ´∇. The following definition is motivated by this observation.

Definition 4.80 (The surface gradient and the surface divergence). Let Σ Ď Rn be a
regular C 1-surface. The surface gradient of a function f : Σ Ñ R, denoted by ∇Σf , is a
vector-valued function from Σ to TpΣ given, in a local parametrization tV, ψu, by

(∇Σf ) n´1 gαβ B (f ˝ ψ) Bψ
B yβ
ÿ

˝ ψ = B yα ,

α,β=1