Page 161 - Vector Analysis

§5.4 Exercises 157

Problem 2. Let ψ(¨, t) : Ω Ñ Ω(t) be a diffeomorphism as defined in Theorem 5.1, and
J = det(∇ψ) and A = (∇ψ)´1. Complete the proof of the Piola identity, identities (2.7),
(5.3) and (5.4) by the following argument:

1. Let u(¨, t) : Ω(t) Ñ Rn be a smooth vector field. Show that

żż
divu dx = JAij(u ˝ ψ),ij dy ;

Ω(t) Ω

thus by the divergence theorem,

żż ż

u ¨ n dSx = JAij(u ˝ ψ)iNj dSy ´ (JAji ),j (u ˝ ψ)i dy . (5.11)

B Ω(t) BΩ Ω

2. Using (5.11),

ż @ u(¨, t) : Ω(t) Ñ Rn vanishing on B Ω(t).

(JAji ),j (u ˝ ψ)i dy = 0

Ω

As a consequence, the Piola identity is valid.

3. By the Piola identity, (5.11) implies that

żż

u ¨ n dSx = JAij(u ˝ ψ)iNj dSy @ u(¨, t) : Ω(t) Ñ Rn smooth.

B Ω(t) BΩ

Therefore, identities (5.3) and (5.4) are also valid.

4. Using identity (2.7) (which is obtained independent of the Piola identity) to show that

J,k = JAji ψ,ijk .