Determinants and Their Applications in Mathematical Physics

270 6. Applications of Determinants in Mathematical Physics

Hence,

vx=−

∑

r

brer

∂v

∂er

=

∑

r

brerθr. (6.7.35)

Similarly,

vt=−

∑

r

b

3

rerθr. (6.7.36)

From (6.7.35) and (6.7.23),

∂vx

∂eq

=

∑

r

br

(

δqrθr+er

∂θr

∂eq

)

=bqθq+

∑

r

brer

∂θr

∂eq

. (6.7.37)

Referring to (6.7.32),

∂

2

vx

∂ep∂eq

=bq

∂θq

∂ep

+

∑

r

br

(

δpr

∂θr

∂eq

+er

∂

2

θr

∂ep∂eq

)

=(bp+bq)

∂θp

∂eq

+

∑

r

brer

∂

2

θr

∂ep∂eq

. (6.7.38)

To obtain a formula forvxxx, puty=vxin (6.7.22), apply (6.7.37) with

q→pandr→q, and then apply (6.7.38):

vxxx=

∑

p

b

2

pep

∂vx

∂ep

+

∑

p,q

bpbqepeq

∂

2

vx

∂ep∂eq

=

∑

p

b

2

p

ep

[

bpθp+

∑

q

bqeq

∂θq

∂ep

]

+

∑

p,q

bpbqepeq

[

(bp+bq)

∂θp

∂eq

+

∑

r

brer

∂

2

θr

∂ep∂eq

]

=Q+R+S+T (6.7.39)

where, from (6.7.36), (6.7.32), and (6.7.31),

Q=

∑

p

b

3

p

epθp

=−vt

R=

∑

p,q

b

2

p

bqepeq

∂θp

∂eq