The Gradient in Cylindrical Coordinates

The gradient in rectangular coordinates is given by

To express the gradient in cylindrical coordinates, consider as a function of and note that and .

By the chain rule,

Since and .

and .

Since

Recall that

Hence,

and

Substitution of these partial derivatives in the expressions for and yields

Substituting these expressions, and those for i and j in terms of the cylindrical coordinates, into we find that

Thus,

Finally, since the gradient in cylindrical coordinates is expressed as

.

Note that if f is a function of only the variables and , the above constitutes a derivation of the expression for the gradient in polar coordinates:


Copyright 2001 Rensselaer Polytechnic Institute. All Rights Reserved.