Mathematical Methods for Physics and Engineering : A Comprehensive Guide

11.5 SURFACE INTEGRALS

dS

S

z

C

a

a

a

x

y

dA=dx dy

Figure 11.7 The surface of the hemispherex^2 +y^2 +z^2 =a^2 ,z≥0.

11.5.2 Vector areas of surfaces

The vector area of a surfaceSis defined as

S=

∫

S

dS,

where the surface integral may be evaluated as above.

Find the vector area of the surface of the hemispherex^2 +y^2 +z^2 =a^2 withz≥ 0.

As in the previous example,dS=a^2 sinθdθdφˆrin spherical polar coordinates. Therefore
the vector area is given by

S=

∫∫

S

a^2 sinθˆrdθ dφ.

Now, sinceˆrvaries over the surfaceS, it also must be integrated. This is most easily
achieved by writingˆrin terms of the constant Cartesian basis vectors. OnSwe have

ˆr=sinθcosφi+sinθsinφj+cosθk,

so the expression for the vector area becomes

S=i

(

a^2

∫ 2 π

0

cosφdφ

∫π/ 2

0

sin^2 θdθ

)

+j

(

a^2

∫ 2 π

0

sinφdφ

∫π/ 2

0

sin^2 θdθ

)

+k

(

a^2

∫ 2 π

0

dφ

∫π/ 2

0

sinθcosθdθ

)

= 0 + 0 +πa^2 k=πa^2 k.

Note that the magnitude ofSis the projected area, of the hemisphere onto thexy-plane,
and not the surface area of the hemisphere.