Page 135 - Vector Analysis

§4.5 Manifolds, Charts, Atlas and Differentiable Structure 131

where gi is the first fundamental form associated with the parametrization ␣φi(Ui), φ´1(.

Remark 4.71. Let C Ď R3 be a regular C 1-curve. The line integral of a scalar function
f : C Ñ R over C is the “surface integral” of f over C defined in (4.70). In other words,
dS = ds in the case that M is a one-dimensional manifold.

4.5.1 Some useful identities

Let Σ Ď Rn be the boundary of an open set Ω (thus an oriented surface), tV, ψu be a local

parametrization of Σ, and N : Σ Ñ Rn be the normal vector on Σ which is compatible with

the parametrization ψ; that is,

([ ... ψ,2 ... ¨¨¨ ... ψ,n´1 ... N ˝ ψ ]) ą 0.
det ψ,1

Define Ψ(y 1, yn) = ψ(y 1) + yn(N ˝ ψ)(y 1). Then Ψ : V ˆ (´ε, ε) Ñ T for some tubular
neighborhood T of Σ.

Φ(O+) Ψ Ω
O+ B Ω
y1 = (y1, ¨ ¨ ¨ , yn´1) P Rn´1 ψ = φ´1
ψ(y1) P B Ω
φ

Φ = Ψ´1

Figure 4.3: The map Ψ constructed from the local parametrization tV, ψu

Since (∇Ψ)ˇˇtyn=0u = [ ... ψ,2 ... ¨¨¨ ... ψ,n´1 ... N ˝ ψ ] Corollary 1.65 and 1.66 implies that
ψ,1 ,

det(∇Ψ)2ˇˇtyn=0u = [ det ((∇Ψ)T) ]ˇ = det ((∇Ψ)T∇Ψ)ˇˇ
 det(∇Ψ) ˇ  ˇtyn=0u

ˇtyn=0u

det ( g11 g12 ¨¨¨ g(n´1)1 0 ) = g .
g21 g22 ¨¨¨ g(n´... 1)2
= ... ... ... 0
g(n´1)1 g(n´1)2 ¨¨¨ g(n´1)(n´1) ...
0

0 0 ¨¨¨ 0 1

Defining J as the Jacobian of the map Ψ; that is, J = det(∇Ψ), then the identity above

implies that J = ?g on tyn = 0u .