Page 143 - Vector Analysis

§4.6 The Divergence Theorem 139

where [gαβ] is the inverse matrix of the metric tensor [gαβ] associated with tV, ψu, and
! B ψ )n are tangent vectors to Σ.

B yβ β=1

The surface divergence operator divΣ is defined as the formal adjoint of ´∇Σ; that is, if

u P TΣ, then

żż @ f P Cc1(Σ; R) .

´ u ¨ ∇Σf dS = f divΣu dS

ΣΣ

In a local parametrization (V, ψ),

(divΣu) ˝ ψ = 1 n´1 B [?ggαβ ( ˝ ψ) ¨ B ψ )] ,
?g (u
ÿ B yα B yβ

α,β=1

where g = det(g) is the first fundamental form associated with tV, ψu.

Remark 4.81. Suppose that f : O Ď R3 Ñ R for some open set containing Σ. Then the
surface gradient of f at p P Σ is the projection of the gradient vector (∇f )(p) onto the
tangent plane TpΣ. In other words, let N : Σ Ñ R3 be a continuous unit normal vector field
on Σ, then

[]
(∇Σf )(p) = (∇f )(p) ´ (∇f )(p) ¨ N(p) N(p) (or simply ∇Σf = ∇f ´ (∇f ¨ N)N) .

Definition 4.82 (Surfaces with Boundary). An oriented C k-surface Σ Ď R3 is said to have
C ℓ-boundary B Σ if there exists a collection of pairs tVm, ψmuKm=1, called a collection of local
parametrization of Σ, if

1. Vm Ď R2 is open and ψm : Vm Ñ R3 is one-to-one map of class C k for all m P
t1, ¨ ¨ ¨ , Ku;

2. ψm(Vm) X Σ ‰ H for all m P t1, ¨ ¨ ¨ , Ku and Σ Ď ŤK ψm(Vm);

m=1

3. ψm : Vm Ñ ψm(Vm) is a C k-diffeomorphism if ψm(Vm) Ď Σ;

4. ψm : Vm+ ” Vm X ty2 ą 0u Ñ ψm(Vm) X Σ is a C k-diffeomorphism if Um X B Σ ‰ H;

5. ψm : Vm X ty2 = 0u Ñ Um X B Σ is of class C ℓ if Um X B Σ ‰ H.

Now we are in the position of stating the divergence theorem on surfaces with boundary.