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: