Page 155 - Vector Analysis

Chapter 5
Additional Topics

5.1 Reynolds’ Transport Theorem

Let Ω1 and Ω2 be two Lipschitz domains of Rn with outward-pointing unit normal N and
$
’ Ω1 Ñ Ω2

&

n, respectively, and the map ψ : B Ω1 Ñ B Ω2 be a diffeomorphism; that is, ψ is

’ y ÞÑ x = ψ(y)
%

one-to-one and onto, and has smooth inverse. Let f P C 1(Ω2) X C (Ω2), and F = f ˝ ψ

which in turns belongs to C 1(Ω1) X C (Ω1). By the divergence theorem,

ż B f (x) dx = ż dSx .
B xi
Ω2 (f ni)(x)

B Ω2

On the other hand, by the chain rule we have that

BF B (f ˝ ψ) n [ B f ] B ψj
B yi B yi ψ B yi
ÿ

= = j=1 B xj ˝ ;

thus if A = (∇ψ)´1, n

Bf ˝ ψ = ÿ Aji B F . (5.1)
B xi B yj
j=1

Letting J = det(∇ψ) be the Jacobian of ψ, by the change of variable y = ψ(y) and the Piola

identity,

ż B f (x) dx = ż B f () det(∇ψ)(y)dy = n ż B (JAji F ) dy.
B xi ψ(y) B yj
Ω2 Ω1 B xi ÿ Ω1

j=1

The divergence theorem again implies that

ż B f (x) dx = nż JAji F NjdSy
B xi ÿ
Ω2
j=1 Ω1

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