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Showing posts from September, 2012

Laurent Series as Fourier Expansion

During our math methods lecture yesterday, this pair of formulas came up:

Which was very cool because it has the feel of a Fourier series.  It's a sum that approximates a function... The coefficients are line integrals over the contour as opposed to integrals over the conjugate variable.  This is the last in a string of similarities between complex analysis and Fourier analysis that have been building in class lately.  So, it was very cool to find a set of class notes from Dr. Jeffrey Rauch of UMich that directly addresses this:

Thanks Dr. Rauch!

An Intuitive Way to the Spherical Gradient and Laplacian

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 element in the r direction is easy to understand.  A small displacement along the r direction is represented as dr.  The theta and phi displacements might not be as obvious.  The graphic to the left illustrates what's going on.  With small displacement along the theta direction you're moving along a circle with radius r.  The distance you've moved is equal to the length of the arc which is equal to the radius…