Determinants and Their Applications in Mathematical Physics

268 6. Applications of Determinants in Mathematical Physics

6.7.3 The First Form of Solution, Second Proof

Second Proof of Theorem 6.13.It can be seen from the definition of

Athat the variablesxandtoccur only in the exponential functionser,

1 ≤r≤n. It is therefore possible to express the derivativesAx,vx,At,

andvtin terms of partial derivatives ofAandvwith respect to theer.

The basic formulas are as follows.

If

y=y(e 1 ,e 2 ,...,en),

then

yx=

∑

r

∂y

∂er

∂er

∂x

=−

∑

r

brer

∂y

∂er

, (6.7.21)

yxx=−

∑

s

bses

∂yx

∂es

=

∑

s

bses

∑

r

br

∂

∂es

(

er

∂y

∂er

)

=

∑

r,s

brbses

[

δrs

∂y

∂er

+er

∂

2

y

∂er∂es

]

=

∑

r

b

2

rer

∂y

∂er

+

∑

r,s

brbseres

∂

2

y

∂er∂es

. (6.7.22)

Further derivatives of this nature are not required. The double-sum rela-

tions (A)–(D) in Section 3.4 are applied again but this timef

′

is interpreted

as a partial derivative with respect to aner.

The basic partial derivatives are as follows:

∂er

∂es

=δrs, (6.7.23)

∂ars

∂em

=δrs

∂er

∂em

=δrsδrm. (6.7.24)

Hence, applying (A) and (B),

∂

∂em

(logA)=

∑

r,s

∂ars

∂em

A

rs

=

∑

r,s

δrsδrmA

rs

=A

mm

(6.7.25)