It's that time of year again when physics students everywhere are deriving the spherical and cylindrical del, nabla, gradient, or Laplacian operators. Every derivation I saw prior to this week involved lots of algebra and the chain rule... even mine. Fortunately for me, a comment on my derivation, and a homework assignment from Rutgers [pdf] led me to a far simpler and more intuitive way of doing things. You just start from the differential displacement in a given coordinate system and go from there.
The differential displacement in spherical coordinates is:
The final task is to derive the Laplacian. The Laplacian is the divergence of the gradient. We simply substitute our gradient result in place of the vector A in our divergence experession to get the following three terms: