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=∫
Aελνμ∇νf(r)dsλ=∫p 2p 1dp∫q 2q 1dq(
∇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 2q 1dq(
f(p 2 ,q)∂rμ(p 2 ,q)
∂q−f(p 1 ,q)∂rμ(p 1 ,q)
∂q)
−
∫p 2p 1dp(
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≡∮
∂Afdrμaround the closed rim∂Aof the surfaceA, thusSμ=Iμ. This completes the proof
of the generalized Stokes law (8.35).