Page 54 - Vector Analysis

50 CHAPTER 2. Differentiation of Functions of Several Variables

Theorem 2.48 (Piola’s identity). Let ψ : Ω Ď Rn Ñ ψ(Ω) Ď Rn be a C 2-diffeomorphism,
and [aij]nˆn be the adjoint matrix of ∇ψ. Then

Einstein’s summationn

ÿ Bconvention (2.6)
a , ” a = 0.ji j
B xj=1 ji
j

In other words, each column of the adjoint matrix of the Jacobian matrix of ψ is divergence-
free (see Definition 4.74).

Proof. Let J = det(∇ψ) and A = (∇ψ)´1. Then aji = JAji . Moreover, since A∇ψ = In,

n
ř Arj ψ,rs = δjs; thus
[ n ] n [Ajr,k ψ,rs Ajr ψ,rsk ]
r=1 Ajrψ,rs ,k
ÿ ÿ

0 = = +

r=1 r=1

which, after multiplying the equality above by Asi and then summing over s, implies that

n (2.7)

Aij,k = ´ ÿ Arj ψ,rskAsi .

r,s=1

As a consequence, by Theorem 2.36 we conclude that

n B (JAij ) = n n[ ´ ] = 0. ˝
B xj ÿ ÿ JArsψ,srjAij JArj ψ,rsj Ais
ÿ
j=1 r,s=1
j=1

2.5.3 The Chain Rule

Theorem 2.49. Let U Ď Rn and V Ď Rm be open sets, f : U Ñ Rm and g : V Ñ Rℓ be
vector-valued functions, and f (U) Ď V. If f is differentiable at x0 P U and g is differentiable
at f (x0), then the map F = g ˝ f defined by

()
F (x) = g f (x) @ x P U

is differentiable at x0, and

( )( )
(DF )(x0)(h) = (Dg) f (x0) (Df )(x0)(h)

or in component,

[] = m B gi ( (x0 ) B fk (x0) .
(DF )(x0) ij B yk f ) B xj
ÿ

k=1