Page 265 - Determinants and Their Applications in Mathematical Physics

250 6. Applications of Determinants in Mathematical Physics

ij
(A ) = − e c r u A rj e c s u A , (6.4.4)
is
r s

2 b r c r A rr + e (c r +c s )u A rs =2 c r , (6.4.5)
r r,s r

2 b r c r A A rj + e c r u A rj e c s u A =(c i + c j )A . (6.4.6)
ir
ij
is
r r s
Put

φ i = e c s u A . (6.4.7)
is
s
Then (6.4.4) and (6.4.6) become
(A ) = −φ i φ j , (6.4.8)
ij

2 b r c r A A rj + φ i φ j =(c i + c j )A . (6.4.9)
ir
ij
r
Eliminating the φ i φ j terms,

ij
(A ) +(c i + c j )A =2 b r c r A A ,
ir
ij
rj
r
ij

e (c i +c j )u A =2e (c i +c j )u b r c r A A . (6.4.10)
ir
rj
r
Differentiating (6.4.3),

(log A) = e (c i +c j )u A ij
i,j

=2 b r c r e c i u A ir e c j u A rj
r
i j
2
=2 b r c r φ . (6.4.11)
r
r
Replacing s by r in (6.4.7),

e c i u φ i = e (c i +c r )u A ,
ir
r

e c j u φ i =2 b r c r e c i u A ir e c j u A rj
r j

=2 b r c r φ r e c i u A ,
ir
r

φ + c i φ i =2 b r c r φ r A .
ir
i
r
Interchange i and r, multiply by b r c r A , sum over r, and refer to (6.4.9):
rj

b r c r A (φ + c r φ r )=2 b r c r A A rj
rj
ir
r b i c i φ i
r i r