Mathematical Methods for Physics and Engineering : A Comprehensive Guide

LINE, SURFACE AND VOLUME INTEGRALS

Substituting this into (11.26) and takingcout of both integrals because it is constant, we
find

c·

∫

S

dS×∇φ=c·

∮

C

φdr.

Sincecis an arbitrary constant vector we therefore obtain the stated result (11.24).

Equation (11.25) may be proved in a similar way, by lettinga=b×cin Stokes’

theorem, wherecis again a constant vector. We also note that by settingb=r

in (11.25) we find
∫

S

(dS×∇)×r=

∮

C

dr×r.

Expanding out the integrand on the LHS gives

(dS×∇)×r=dS−dS(∇·r)=dS− 3 dS=− 2 dS.

Therefore, as we found in subsection 11.5.2, the vector area of an open surfaceS

is given by

S=

∫

S

dS=

1

2

∮

C

r×dr.

11.9.2 Physical applications of Stokes’ theorem

Like the divergence theorem, Stokes’ theorem is useful in converting integral

equations into differential equations.

From Amp`ere’s law, derive Maxwell’s equation in the case where the currents are steady,

i.e.∇×B−μ 0 J= 0.

Ampere’s rule for a distributed current with current density` Jis
∮

C

B·dr=μ 0

∫

S

J·dS,

for any circuitCbounding a surfaceS. Using Stokes’ theorem, the LHS can be transformed
into

∫

S(∇×B)·dS; hence ∫

S

(∇×B−μ 0 J)·dS=0

foranysurfaceS. This can only be so if∇×B−μ 0 J= 0 , which is the required relation.
Similarly, from Faraday’s law of electromagnetic induction we can derive Maxwell’s
equation∇×E=−∂B/∂t.

In subsection 11.8.3 we discussed the flow of an incompressible fluid in the

presence of several sources and sinks. Let us now considervortexflow in an

incompressible fluid with a velocity field

v=

1

ρ

eˆφ,

in cylindrical polar coordinatesρ, φ, z. For this velocity field∇×vequals zero