Tensors for Physics

8.2 Surface Integrals, Stokes 125

It is understood that the closed curveC=∂Ais the rim, or the contour line, of the
surfaceA.
The standard Stokes law follows from (8.35) when the function fis identified
with the Cartesian componentvμof a vector fieldv.Dueto

ελνμ∇νvμ=(∇×v)λ,

theStokes lawcan be cast into the form
∫

A

(∇×v)·ds=

∮

∂A

v·dr. (8.36)

The line integral on the right hand side of (8.36) is referred to as the circulation of
the vector fieldv.
A remark on parity is in order. The nabla-operator∇and the line elementdr
occurring in (8.35) and (8.36) are polar vectors. Parity is conserved, i.e. both sides
of the equation in the generalized Stokes law have the same parity behavior since the
surface elementdsis an axial vector.
Furthermore, when the integration in (8.35) and (8.36) is extended over a closed
surface, there is no contour line and thus these integrals are equal to zero. Notice,
however, that the generalized Stokes law applies to simply connected surfaces which
have no holes. On the other hand, the circulation, i.e. the line integral, as it occurs on
the right hand side of (8.36), can be non-zero, when the integration is around a hole
in a surface, even when∇×v, occurring on the left hand side of (8.36), is zero.
Aproofof (8.35), which includes the proof for the conventional Stokes law (8.36),
is presented next. The surface is parameterized byr=r(p,q). The area over which
the integration is extended is assumed to be given by a rectangle in thep–q-parameter
plane, see Fig.8.11.
According to (8.23), the surface element can be written as

dsλ=ελαβ

∂rα

∂p

∂rβ

∂q

dpdq.

Fig. 8.11Area and rim
curve inp–q-plane for the
proof of the Stokes law