Page 285 - Determinants and Their Applications in Mathematical Physics

270 6. Applications of Determinants in Mathematical Physics

Hence,

∂v
v x = − b r e r
r ∂e r

= b r e r θ r . (6.7.35)
r
Similarly,
3
v t = − b e r θ r . (6.7.36)
r
r
From (6.7.35) and (6.7.23),

= b r δ qr θ r + e r ∂θ r
∂v x
∂e q
r ∂e q

= b q θ q + b r e r ∂θ r . (6.7.37)
r ∂e q
Referring to (6.7.32),
2 2

+ b r δ pr ∂θ r + e r ∂ θ r
∂ v x
∂θ q
= b q
∂e p
∂e p ∂e q
r ∂e q ∂e p ∂e q
2

=(b p + b q ) ∂θ p + b r e r ∂ θ r . (6.7.38)
∂e q
r ∂e p ∂e q
To obtain a formula for v xxx , put y = v x in (6.7.22), apply (6.7.37) with
q → p and r → q, and then apply (6.7.38):
2
2
v xxx = ∂v x + ∂ v x
b e p
p b p b q e p e q
p ∂e p p,q ∂e p ∂e q

2
= b e p b p θ p + ∂θ q
p b q e q
p q ∂e p

2

+ b p b q e p e q (b p + b q ) ∂θ p + b r e r ∂ θ r
p,q ∂e q r ∂e p ∂e q
= Q + R + S + T (6.7.39)
where, from (6.7.36), (6.7.32), and (6.7.31),
3
Q =
b e p θ p
p
p
= −v t
2
R = ∂θ p
b b q e p e q
p
p,q ∂e q