Mathematical Methods for Physics and Engineering : A Comprehensive Guide

LINE, SURFACE AND VOLUME INTEGRALS

11.9 Stokes’ theorem and related theorems

Stokes’ theorem is the ‘curl analogue’ of the divergence theorem and relates the

integral of the curl of a vector field over an open surfaceSto the line integral of

the vector field around the perimeterCbounding the surface.

Following the same lines as for the derivation of the divergence theorem, we

can divide the surfaceSinto many small areasSiwith boundariesCiand unit

normalsnˆi. Using (11.17), we have for each small area

(∇×a)·ˆniSi≈

∮

Ci

a·dr.

Summing overiwe find that on the RHS all parts of all interior boundaries

that are not part ofCare included twice, being traversed in opposite directions

on each occasion and thus contributing nothing. Only contributions from line

elements that are also parts ofCsurvive. If eachSiis allowed to tend to zero

then we obtain Stokes’ theorem,

∫

S

(∇×a)·dS=

∮

C

a·dr. (11.23)

We note that Stokes’ theorem holds for both simply and multiply connected open

surfaces, provided that they are two-sided. Stokes’ theorem may also be extended

to tensor fields (see chapter 26).

Just as the divergence theorem (11.18) can be used to relate volume and surface

integrals for certain types of integrand, Stokes’ theorem can be used in evaluating

surface integrals of the form

∮

S(∇×a)·dSas line integrals or vice versa.

Given the vector fielda=yi−xj+zk, verify Stokes’ theorem for the hemispherical

surfacex^2 +y^2 +z^2 =a^2 ,z≥ 0.

Let us first evaluate the surface integral
∫

S

(∇×a)·dS

over the hemisphere. It is easily shown that∇×a=− 2 k, and the surface element is
dS=a^2 sinθdθdφrˆin spherical polar coordinates. Therefore

∫

S

(∇×a)·dS=

∫ 2 π

0

dφ

∫π/ 2

0

dθ

(

− 2 a^2 sinθ

)

ˆr·k

=− 2 a^2

∫ 2 π

0

dφ

∫π/ 2

0

sinθ

(z

a

)

dθ

=− 2 a^2

∫ 2 π

0

dφ

∫π/ 2

0

sinθcosθdθ=− 2 πa^2.

We now evaluate the line integral around the perimeter curveCof the surface, which