Begin2.DVI

The coordinate curves are

The straight-lines,
r (θ 0 , z ), 0 ≤z≤h

and the circles,
r (θ, z 0 ), 0 ≤θ≤ 2 π,

where θ 0 and z 0 are constants. The tangent vectors to the coordinate curves are

given by

∂
r

∂θ =−asin θ

ˆe 1 +acos θeˆ 2 and ∂
r

∂z =

ˆe 3

Consequently, we have E=a^2 , F = 0,and G= 1 .The element of surface area is then

dS =

√

EG −F^2 dθ dz =a dθ dz. The surface area of the cylinder of height his therefore

S=

∫h

0

∫ 2 π

0

a dθ dz = 2πah.

Volume Integrals

The summation of scalar and vector fields over a region of space can be expressed

by volume integrals having the form

∫∫∫

V

f(x, y, z )dV and

∫∫∫

V

F
(x, y, z )dV,

where dV =dx dy dz is an element of volume and V is the region over which the

integrations are to extend.

The integral of the scalar field is an ordinary triple integral. The triple integral

of the vector function F
=F
(x, y, z )can be expressed as

∫∫∫

V

F dV
 =eˆ 1 ∫∫∫

V

F 1 (x, y, z )dV +ˆe 2

∫∫∫

V

F 2 (x, y, z )dV +ˆe 3

∫∫∫

V

F 3 (x, y, z )dV, (7 .106)

where each component is a scalar triple integral.

Whenever appropriate, the above integrals are sometimes expressed

(i) in cylindrical coordinates (r, θ, z ), where x=rcos θ, y =rsin θ, z =zand the element

of volume is represented dV =r dr dθ dz

(ii) in spherical coordinates (ρ, θ, φ )where x=ρsin θcos φ,y=ρsin θsin φ,z=ρcos θand

the element of volume is dV =ρ^2 sin θ dρ dφ dθ.

(iii) in curvilinear coordinates (u, v, w )where x=x(u, v, w ), y =y(u, v, w ), z =z(u, v, w )

and the element of volume is given by^8 dV =

∣∣

∣∣∂
r

∂u ·

(

∂
r

∂v ×

∂
r

∂w

)∣∣

∣∣du dv dw where

r =x(u, v, w )ˆe 1 +y(u, v, w )ˆe 2 +z(u, v, w )ˆe 3

(^8) See pages 143 and 156 for details.