Page 94 - Vector Analysis

90 CHAPTER 3. Multiple Integrals


[D(g ˝ ψ)]
and M =  . By the property of determinants and the chain rule, we find that
[Dψ2]
[Dψ3]

...

[Dψn]

 n ( Bg ˝ ) B ψj n ( Bg ˝ ) B ψj ¨¨¨ n ( Bg ˝ ) B ψj 
det (  ψ ψ B x2 ψ B xn  )
ř ř ¨¨¨ ř
...
j=1 B yj B x1 j=1 B yj j=1 B yj

B ψ2 B ψ2 B ψ2

det(M) = B x1 B x2 B xn

... ... ...

B ψn B ψn ¨¨¨ B ψn
B x1 B x2 B xn

 ( Bg ˝ ) B ψ1 ( Bg ˝ ) B ψ1 ¨¨¨ ( B g ˝ ψ) B ψ1 
det (  ψ ψ  )
B y1 B x1 B y1 B x2 B y1 B xn

B ψ2 B ψ2 ¨ ¨ ¨ B ψ2

= B x1 B x2 B xn

... ... . . . ...

B ψn B ψn ¨¨¨ B ψn
B ψ1  B xn
B x1 B ψ1 B x2

(  B x1 B ψ1 ¨¨¨ B xn  ) = (f ˝ ψ)J .
B x2
= ( Bg ) ... ... . . . ...
˝ ψ det
B y1 B ψn B ψn
B ψn ¨ ¨ ¨
B x1 B x2 B xn

On the other hand, letting A = (Dψ)´1, then

Adj(M)j1 = (´1)1+j det () = Adj([Dψ])j1 = JAj1 .
M(p1, pj)

Computing the determinant by expanding along the first row, we obtain that

det(M) = n M1j Adj(M)j1 = n B (g ˝ ψ) JAj1 ;
B xj
ÿ ÿ

j=1 j=1

thus we conclude the identity

(f ˝ ψ)J = n B (g ˝ψ ) JA1j .
B xj
ÿ

j=1