126 8 Integration of Fields
Insertion of this expression intoSμ=
∫
Aελνμ∇νf(r)dsλ, which is the left hand
side of (8.35), leads to
Sμ=
∫
A
ελνμ∇νf(r)dsλ=
∫p 2
p 1
dp
∫q 2
q 1
dq
(
∇νf
∂rν
∂p
∂rμ
∂q
−∇νf
∂rμ
∂p
∂rν
∂q
)
.
Hereελνμελαβ=δναδμβ−δνβδμα, which corresponds to (4.10), has been used.
Thanks to the chain rule, one has
∇νf
∂rν
∂p
=
∂f
∂p
, ∇νf
∂rν
∂q
=
∂f
∂q
.
Due to
∂f
∂p
∂rμ
∂q
=
∂
∂p
(
f
∂rμ
∂q
)
−f
∂^2 rμ
∂p∂q
,
∂f
∂q
∂rμ
∂p
=
∂
∂q
(
f
∂rμ
∂p
)
−f
∂^2 rμ
∂q∂p
,
the integrand of the surface integral considered reduces to
∂
∂p
(
f
∂rμ
∂q
)
−
∂
∂q
(
f
∂rμ
∂p
)
.
The first term can immediately be integrated overp, the second one overq.This
then yields
Sμ=
∫q 2
q 1
dq
(
f(p 2 ,q)
∂rμ(p 2 ,q)
∂q
−f(p 1 ,q)
∂rμ(p 1 ,q)
∂q
)
−
∫p 2
p 1
dp
(
f(p,q 2 )
∂rμ(p,q 2 )
∂p
−f(p,q 1 )
∂rμ(p,q 1 )
∂p
)
. (8.37)
The two terms in the upper line of (8.37) are the line integralsIμII+IμIValong
the segments II and IV, those in the lower line areIμIII+IμIwhich pertain to the
segments III and I. The four terms in (8.37), viz.:Sμ=IμI+IμII+IμIII+IμIV
make up the line integral
Iμ≡
∮
∂A
fdrμ
around the closed rim∂Aof the surfaceA, thusSμ=Iμ. This completes the proof
of the generalized Stokes law (8.35).