Begin2.DVI

The ∇operator and integration

16.

∫∫∫

V

∇f dV =

∫∫

S

feˆndS Special case of divergence theorem

17.

∫∫∫

V

∇× A dV
 =

∫∫

S

ˆen×A dS
Special case of divergence theorem

18.

∫

C

©d
r ×A
=

∫∫

S

(ˆen×∇ )×A dS
Special case of Stokes theorem

19.

∫

C

©f d
r =

∫∫

S

dS
×∇ f Special case of Stokes theorem

Vector Operators in curvilinear coordinates

In this section the concept of curvilinear coordinates is introduced and the rep-

resentation of scalars and vectors in these new coordinates are studied.

If associated with each point (x, y, z)of a rectangular coordinate system there is

a set of variables (u, v, w )such that x, y, z can be expressed in terms of u, v, w by a

set of functional relationships or transformations equations, then (u, v, w )are called

the curvilinear coordinates^2 (x, y, z ).Such transformation equations are expressible

in the form

x=x(u, v, w ), y =y(u, v, w ), z =z(u, v, w ) (8 .68)

and the inverse transformation can be expressed as

u=u(x, y, z ), v =v(x, y, z ) w=w(x, y, z) (8 .69)

It is assumed that the transformation equations (8.68) and (8.69) are single valued

and continuous functions with continuous derivatives. It is also assumed that the

transformation equations (8.68) are such that the inverse transformation (8.69) ex-

ists, because this condition assures us that the correspondence between the variables

(x, y, z )and (u, v, w )is a one-to-one correspondence.

The position vector

r =xˆe 1 +yˆe 2 +zˆe 3 (8 .70)

(^2) Note how coordinates are defined and the order of their representation because there are no standard repre-
sentation of angles or directions. Depending upon how variables are defined and represented, sometime left-hand
coordinates are confused with right-handed coordinates.