Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(lu) #1

LINE, SURFACE AND VOLUME INTEGRALS


where|∇f|and∂f/∂zare evaluated on the surfaceS. We can therefore express


any surface integral overSas a double integral over the regionRin thexy-plane.


Evaluate the surface integralI=


Sa·dS,wherea=xiandSis the surface of the
hemispherex^2 +y^2 +z^2 =a^2 withz≥ 0.

The surface of the hemisphere is shown in figure 11.7. In this casedSmay be easily
expressed in spherical polar coordinates asdS=a^2 sinθdθdφ, and the unit normal to the
surface at any point is simplyrˆ. On the surface of the hemisphere we havex=asinθcosφ
and so


a·dS=x(i·ˆr)dS=(asinθcosφ)(sinθcosφ)(a^2 sinθdθdφ).

Therefore, inserting the correct limits onθandφ, we have


I=



S

a·dS=a^3

∫π/ 2

0

dθsin^3 θ

∫ 2 π

0

dφcos^2 φ=

2 πa^3
3

.


We could, however, follow the general prescription above and project the hemisphereS
onto the regionRin thexy-plane that is a circle of radiusacentred at the origin. Writing
the equation of the surface of the hemisphere asf(x, y)=x^2 +y^2 +z^2 −a^2 =0andusing
(11.10), we have


I=



S

a·dS=


S

x(i·ˆr)dS=


R

x(i·ˆr)

|∇f|dA
∂f/∂z

.


Now∇f=2xi+2yj+2zk=2r,soonthesurfaceSwe have|∇f|=2|r|=2a.OnSwe


also have∂f/∂z=2z=2



a^2 −x^2 −y^2 andi·ˆr=x/a. Therefore, the integral becomes

I=


∫∫


R

x^2

a^2 −x^2 −y^2

dx dy.

Although this integral may be evaluated directly, it is quicker to transform to plane polar
coordinates:


I=

∫∫


R′

ρ^2 cos^2 φ

a^2 −ρ^2

ρdρdφ

=


∫ 2 π

0

cos^2 φdφ

∫a

0

ρ^3 dρ

a^2 −ρ^2

.


Making the substitutionρ=asinu, we finally obtain


I=


∫ 2 π

0

cos^2 φdφ

∫π/ 2

0

a^3 sin^3 udu=

2 πa^3
3

.


In the above discussion we assumed that any line parallel to thez-axis intersects

Sonly once. If this is not the case, we must split up the surface into smaller


surfacesS 1 ,S 2 etc. that are of this type. The surface integral overSis then the


sum of the surface integrals overS 1 ,S 2 and so on. This is always necessary for


closed surfaces.


Sometimes we may need to project a surfaceS(or some part of it) onto the

zx-oryz-plane, rather than thexy-plane; for such cases, the above analysis is


easily modified.

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