in cylindrical coordinates and that 
satisfies Laplace's equation. Why is this Answers
the gradient of f, namely
which mean
Suppose however, we are given f as a function of r and 
, that is, in polar coordinates, (or g in spherical coordinates, as a function of 
, , and 
).
For example, suppose 
How do we find the gradient of f or g?
One way to find the gradient of such a function is to convert r or 
or 
into rectangular coordinates using the appropriate formulae for them, and perform the partial differentiation on the resulting expressions.
Thus we can write
and find, by ordinary partial differentiating
It is a bit more convenient sometimes, to be able to express the gradient directly in polar coordinates or spherical coordinates, like it is expressed in rectangular coordinates as above.
We want here an expression involving partial derivatives with respect to r and multiplied by vectors pointing respectively in the r direction, and direction.
So we want to know: what vectors should these partial derivatives be multiplied by in order to form the gradient?
When we find the answer, the actual partial derivative with respect to each polar variable will be the dot product of a unit vector in a polar direction with the gradient.
We therefore digress to discuss what thes unit vectors are so that you can recognize them.
The r direction is the direction tilted by an angle counterclockwise from the x axis. A unit vector in that direction, call it ur, can be written in any of the three following forms
The unit vector in the direction lies in the direction 90o beyond the r direction, counterclockwisely, and is therefore given by
The presence of a magnetic moment m creates a magnetic field which is the gradient of some scalar field. To gain a better intuitive feel about the relationship between scalar fields and their gradient vector fields, see Appendix A.3.6. Because the divergence of the magnetic field is zero, by definition, the divergence of the gradient of the scalar field is also zero, or ?2?m = 0. The operator ?2 is called the Laplacian and ?2?m = 0 is Laplace