Mathematical Methods for Physics and Engineering : A Comprehensive Guide

LINE, SURFACE AND VOLUME INTEGRALS

R

y

C

x

dx

dy

dr

nˆds

Figure 11.11 A closed curveCin thexy-plane bounding a regionR.Vectors

tangent and normal to the curve at a given point are also shown.

The surface integral overS 1 is easily evaluated. Remembering that the normal to the
surface points outward from the volume, a surface element onS 1 is simplydS=−kdx dy.
OnS 1 we also havea=(y−x)i+x^2 k,sothat

I=−

∫

S 1

a·dS=

∫∫

R

x^2 dx dy,

whereRis the circular region in thexy-plane given byx^2 +y^2 ≤a^2. Transforming to plane
polar coordinates we have

I=

∫∫

R′

ρ^2 cos^2 φ ρ dρ dφ=

∫ 2 π

0

cos^2 φdφ

∫a

0

ρ^3 dρ=

πa^4

4

.

It is also interesting to consider the two-dimensional version of the divergence

theorem. As an example, let us consider a two-dimensional planar regionRin

thexy-plane bounded by some closed curveC(see figure 11.11). At any point

on the curve the vectordr=dxi+dyjis a tangent to the curve and the vector

nˆds=dyi−dxjis a normal pointing out of the regionR. If the vector fieldais

continuous and differentiable inRthen the two-dimensional divergence theorem

in Cartesian coordinates gives
∫∫

R

(

∂ax

∂x

+

∂ay

∂y

)

dx dy=

∮

a·nˆds=

∮

C

(axdy−aydx).

LettingP=−ayandQ=ax, we recover Green’s theorem in a plane, which was

discussed in section 11.3.

11.8.1 Green’s theorems

Consider two scalar functionsφandψthat are continuous and differentiable in

some volumeVbounded by a surfaceS. Applying the divergence theorem to the