Page 281 - Determinants and Their Applications in Mathematical Physics

266 6. Applications of Determinants in Mathematical Physics

1
c. D t (φ ij )= φ i0 φ j2 − φ i1 φ j1 + φ i2 φ j0 − (φ i+3,j + φ i,j+3 ).
2
2
Proof. Put f r = −g r = b in identity (D).
r
2
2
2
2

(b − b )A = δ rs (b − b )e r +2(b r − b s ) A A rj
ij
is
i j r s
r s

= 0+2 (b r − b s )A A rj
is
r s

=2 A is b r A rj − 2 A rj b s A is
s r r s
=2(ψ 0i ψ 1j − ψ 0j ψ 1i ).
It follows that if
2
F ij =2ψ 0i ψ 1j − b A ,
ij
i
then
F ji = F ij .
Furthermore, if G ij is any function with the property
G ji = −G ij ,
then

G ij F ij =0. (6.7.17)
i j
The proof is trivial and is obtained by interchanging the dummy suffixes.
The proof of (a) can now be obtained by expanding the quadruple series

S = (b b − b b )b s A A rs
pq
i j
j i
p r p r
p,q,r,s
in two different ways and equating the results.

S = b A pq b b s A rs − b A pq b b s A rs
i
j
i
j
p r p r
p,q r,s p,q r,s
= φ i0 φ j1 − φ j0 φ i1 ,
which is identical to the left side of (a). Also, referring to (6.7.17) with
i, j → p, r,

S = (b b − b b ) A pq b s A rs
i j
j i
p r p r
p,r q s

= i j j i
(b b − b b )ψ 0p ψ 1r
p r p r
p,r
1 2
= (b b − b b )(F pr + b A )
pr
i j
j i
2 p r p r p
p,r