Page 117 - Vector Analysis

§4.3 The Surface Integrals 113

Theorem 4.39. Let Σ Ď R3 be a regular surface, tV1, ψ1u and tV2, ψ2u be two local C 1-
parameterizations of Σ at a point p P Σ, and U = ψ1(V1) X ψ2(V2) Ď Σ. Then for (i, j) =
(1, 2) and (2, 1), the transition function ψj´1 ˝ ψi : ψi´1(U ) Ñ ψj´1(U ) is of class C 1.

Proof. We first note that ψj´1 ˝ ψi is continuous on ψi´1(U ). Moreover, by the chain rule we

find that B (ψj´1 ˝ ψi) is the unique 2-vector satisfying

Bu

[ B ψi ] [ B ψj´1 ] [ ][ B (ψj´1 ˝ ψi) (u, ]
v) Bu v) (Dψj )(ψj´1 v) Bu v)
Bu (u, = (ψj ˝ ˝ ψi)(u, = ˝ ψi)(u, .

Similarly, B (ψj´1 ˝ ψi) is the unique 2-vector satisfying

Bv

[ B ψi (u, ] = [B (ψj ˝ ψj´1 ˝ ψi)(u, ] = [ ˝ ψi)(u, v ][ B (ψj´1 ˝ ψi) (u, ] .
v) v) (Dψj )(ψj´1 ) v)
Bv Bv Bv

Therefore, we obtain that

[] = [ ˝ (ψj´1 ˝ ][[ B (ψj´1 ˝ ψi) ]...[ B (ψj´1 ˝ ψi) ]] . (4.7)
Dψi (Dψj ) ψi) Bu
Bv

Since [] has full rank, [ ]T[Dψj ] is an invertible 2 ˆ 2 matrix (for if ATAx = 0 then
Dψj Dψj

}Ax}R2 n = xTATAx = 0 which implies x = 0 since A has full rank); thus (4.7) implies that

[[ B (ψj´1 ˝ ψi) ]...[ B (ψj´1 ˝ ψi) ]] = (([ ]T [ ) ˝ (ψj´1 ˝ )´1[ ) ˝ (ψj´1 ˝ ψi)]T[Dψi] ;
Bu Bv Dψj Dψj ] ψi) (Dψj

thus the partial derivatives of ψj´1 ˝ ψi exist and are continuous. Theorem 2.30 then implies

that ψj´1 ˝ ψi is of class C 1. ˝

Similar to how the directional derivative is defined, we intend to define the differentia-
bility of f through the differentiability of the function f ˝ ψ : V Ñ Rn, where tV, ψu is a

local parametrization of Σ (at some point). Again, we need to talk about if this definition

depends on the choice of local parameterizations. Nevertheless, if tV1, ψ1u and tV2, ψ2u
are two C 1-local parametrization of Σ at p, and f ˝ ψ1 is differentiable at ψ1´1(p), then
the chain rule and Theorem 4.39 imply that f ˝ ψ2 is also differentiable at ψ2´1(p) since
f ˝ ψ2 = (f ˝ ψ1) ˝ (ψ1´1 ˝ ψ2). This induces the following

Definition 4.40. Let Σ Ď R3 be a C 1-regular surface. A scalar function f : Σ Ñ R is
said to be differentiable at p P Σ if for every parametrization tV, ψu of Σ at p, the function