Page 215 - Determinants and Their Applications in Mathematical Physics

200 5. Further Determinant Theory
∂S

2
2
= U − V + , (5.4.42)
r ∂x r
where

S = c i c j E . (5.4.43)
ij
i,j
This function is identical to the left-hand side of (5.4.26). Let

T = (c i + c j )(c r + c s )E E . (5.4.44)
is
rj
i,j,r,s
Then, applying (5.4.6),

T = c i E is c r E rj + E rj c i c s E is
i,s j,r j,r i,s

+ E is c j c r E rj + c j E rj c s E is
i,s j,r j,r i,s
2
=(U + V ) +2S E rs +(U − V ) 2
r,s

2
2
=2(U + V )+2S E . (5.4.45)
rr
r
Eliminating V from (5.4.42),

∂
2

T +2R =4U +2 E rr + S. (5.4.46)
r ∂x r
To obtain a formula for Q, multiply (5.4.11) by (c i + c j ), sum over i and
j, and apply (5.4.13) with the modifications (i, j) ↔ (r, s) on the left and
(i, r) → (r, s) on the right:

Q = (c i + c j )(c r + c s )E E rj − 2 c r (c i + c j )E E rj
is
ir
i,j,r,s i,j,r

= T − 2 c r (c i + c j )E E rj
ir
r i,j

= T − 4 c r c s E E . (5.4.47)
rs
sr
r,s
Eliminating T from (5.4.46) and applying (5.4.26) and the fourth and sixth
lines of (5.4.4),

Q +2R =4 c r c s (E E ss − E E )
rr
rs
sr
r,s

∂ 2 1
+2 E rr + c E ss − E stu,stu
3
s
r ∂x r s s,t,u