Page 53 - Vector Analysis

§2.5 Properties of Differentiable Functions 49

As a consequence,

lim ››(gf )(x) ´ (gf )(x0) ´ A(x ´ x0)››Rm
xÑx0 }x ´ x0}Rn

ď ˇˇg(x0)ˇˇ lim ››f (x) ´ f (x0) ´ (Df )(x0)(x ´ x0)››Rm
}x ´ x0}Rn
xÑx0 ]
[
ˇˇg(x) ´ g(x0) ´ (Dg)(x0)(x ´ x0)ˇˇ
+ lim }x ´ x0}Rn }f (x)}Rm

xÑx0 [ ]
(x0)››Rm
+ lim ˇˇ(Dg)(x0)(x ´ x0)ˇˇ ››f (x) ´ f = 0
}x ´ x0}Rn
xÑx0

which implies that gf is differentiable at x0 with derivative D(gf )(x0) given by (2.3). ˝

‚ The differentiation of the Jacobian

Before going into the next section, we study the differentiation of a special determinant, the
Jacobian.

Example 2.46. Suppose that ψ : Ω Ď Rn Ñ ψ(Ω) Ď Rn is a given diffeomorphism
(thus det(∇ψ) ‰ 0). Let M = ∇ψ, and J = det(M). By Corollary 1.72, the adjoint
matrix of M is JM´1. Letting δ be a (first order) partial differential operator which satisfies
δ(f g) = f δg + (δf )g, by Theorem 1.73 we find that

n Einstein’s summation

δJ = tr(JM´1δM) = ÿ JAijδψ,ij ”convention JAji δψ,ij , (2.5)

i,j=1

where Aji = aji with M´1 = [aji]nˆn, and f,j ” Bf .

B xj

Remark 2.47. From now on we sometimes write the row index of a matrix as a super-script

for the following reason: if ψ : Ω Ď Rn Ñ Rm is a differentiable vector-valued function, then

∇ψ is usually expressed by 

B ψ1 B ψ1 ¨¨¨ B ψ1
B x2
∇ψ =  B x1 B ψ2 ¨¨¨ B xn  ;
B ψ2 B x2 ... B ψ2
B x1 ¨¨¨ B xn
...
... ...
B ψm
B ψm B x2 B ψm

B x1 B xn

thus the (i, j) element of ∇ψ is B ψi , and the row index i appears “above” the column index

B xj

j.