Page 214 - Determinants and Their Applications in Mathematical Physics

5.4 The Cofactors of the Matsuno Determinant 199
∂
2 2 2 2
= E rr (c + c )E + (c + c )E ij
ij
i j i j
r i,j r ∂x r i,j

∂ 2 2
= E rr + (c + c )E ij
i j
r ∂x r i,j

∂ 1
2
= E vv + 2 c E rr + E rst,rst
3
r
v ∂x v r r,s,t
1
2
=2 c E rv,rv + E rstv,rstv ,
3
r
r,v r,s,t,v
which is equivalent to (5.4.36).
Since
2
2
2
2
(c − c )(c r + c s ) − 2c r c s (c r − c s )=(c + c )(c r − c s ),
r s r s
it follows immedately that
Q − 2R = P. (5.4.39)
A second relation between Q and R is found as follows. Let

U = c r E ,
rr
r
1
V = E rs,rs . (5.4.40)
2
r,s
It follows from (5.4.17) that

V = c r E rs − c r E rr
r,s r

= c r E rr − c s E .
rs
r r,s
Hence

c r E rs = U + V,
r,s

c s E rs = U − V. (5.4.41)
r,s
To obtain a formula for R, multiply (5.4.10) by c i c j , sum over i and j, and
apply the third equation of (5.4.4):

R = c i c j E E rj − c i c j E E rj
ir
is
i,j,r,s i,j,r
   
∂

=  c i E is   c j E rj  + c i c j E ij
r ∂x r
i,s j,r i,j