Mathematical Methods for Physics and Engineering : A Comprehensive Guide

11.9 STOKES’ THEOREM AND RELATED THEOREMS

is the circlex^2 +y^2 =a^2 in thexy-plane. This is given by
∮

C

a·dr=

∮

C

(yi−xj+zk)·(dxi+dyj+dzk)

=

∮

C

(ydx−xdy).

Using plane polar coordinates, onCwe havex=acosφ,y=asinφso thatdx=
−asinφdφ,dy=acosφdφ, and the line integral becomes
∮

C

(ydx−xdy)=−a^2

∫ 2 π

0

(sin^2 φ+cos^2 φ)dφ=−a^2

∫ 2 π

0

dφ=− 2 πa^2.

Since the surface and line integrals have the same value, we have verified Stokes’ theorem
in this case.

The two-dimensional version of Stokes’ theorem also yields Green’s theorem in

a plane. Consider the regionRin thexy-plane shown in figure 11.11, in which a

vector fieldais defined. Sincea=axi+ayj, we have∇×a=(∂ay/∂x−∂ax/∂y)k,

and Stokes’ theorem becomes
∫∫

R

(

∂ay

∂x

−

∂ax

∂y

)

dx dy=

∮

C

(axdx+aydy).

LettingP=axandQ=aywe recover Green’s theorem in a plane, (11.4).

11.9.1 Related integral theorems

As for the divergence theorem, there exist two other integral theorems that are

closely related to Stokes’ theorem. Ifφis a scalar field andbis a vector field,

and bothφandbsatisfy our usual differentiability conditions on some two-sided

open surfaceSbounded by a closed perimeter curveC,then
∫

S

dS×∇φ=

∮

C

φdr, (11.24)

∫

S

(dS×∇)×b=

∮

C

dr×b. (11.25)

Use Stokes’ theorem to prove (11.24).

In Stokes’ theorem, (11.23), leta=φc,wherecis a constant vector. We then have
∫

S

[∇×(φc)]·dS=

∮

C

φc·dr. (11.26)

Expanding out the integrand on the LHS we have

∇×(φc)=∇φ×c+φ∇×c=∇φ×c,

sincecis constant, and the scalar triple product on the LHS of (11.26) can therefore be
written

[∇×(φc)]·dS=(∇φ×c)·dS=c·(dS×∇φ).