Vector (Nabla) Operations in Curvilinear Coordinates

Vector calculus is an essential ingredient of modern scientific communication. First proposed by Josiah Willard Gibbs, vector analysis is compact, elegant, and simple. Fundamental components of vector analysis are the dot and cross product, as well as gradient, divergence, and curl. This post is intended as a quick and handy reminder of these various operations in different coordinates.

Dot and cross products are invariant under coordinate transformation, so they have the same form for all coordinates

and

The remaining operations take specific forms under different coordinates. For Cylindrical coordinates

while for Spherical coordinates

Voila!

P.S. most people refer to the "inverted triangle" operator as the "del" operator; however, this symbol is called "nabla" just like epsilone, eta, gamma etc... and I find no reason for calling it otherwise! Therefore, I usually use terms such as "nabla phi" for the gradient of a scalar, "nabla dot A" and "nabla cross A" for the divergence and curl of a vector field, respectively.

Cite as:
Saad, T. "Vector (Nabla) Operations in Curvilinear Coordinates." Weblog entry from Please Make A Note. http://pleasemakeanote.blogspot.com/2008/05/nabla-operations-gradient-curl-and-dot.html