Page 283 - 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
A that the variables x and t occur only in the exponential functions e r ,
1 ≤ r ≤ n. It is therefore possible to express the derivatives A x , v x , A t ,
and v t in terms of partial derivatives of A and v with respect to the e r .
The basic formulas are as follows.
If

y = y(e 1 ,e 2 ,...,e n ),
then

y x = ∂y ∂e r
∂e r ∂x
r
∂y
= − b r e r , (6.7.21)
r ∂e r

y xx = − b s e s ∂y x
s ∂e s
∂ ∂y
= b s e s b r e r
s r ∂e s ∂e r
2
∂y ∂ y
= b r b s e s δ rs + e r
r,s ∂e r ∂e r ∂e s
2
2 ∂y ∂ y
= + . (6.7.22)
b e r
r b r b s e r e s
r ∂e r r,s ∂e r ∂e s
Further derivatives of this nature are not required. The double-sum rela-
tions (A)–(D) in Section 3.4 are applied again but this time f is interpreted

as a partial derivative with respect to an e r .
The basic partial derivatives are as follows:
= δ rs , ∂e r (6.7.23)
∂e r
∂e s
∂a rs
= δ rs
∂e m
∂e m
= δ rs δ rm . (6.7.24)
Hence, applying (A) and (B),
∂
(log A)= ∂a rs A rs
∂e m
r,s ∂e m

= δ rs δ rm A rs
r,s
= A mm (6.7.25)