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March 2nd, 2016, 04:35 PM   #1
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divergence,grad, curl etc

What is the most efficient way to derive the divergence, gradient, curl, and other vector calculus operations in spherical and cylindrical,coordinates?

So far, I have seen, for example, all of 2 ways to go about deriving divergence in spherical/cylindrical coordinates.

Method one comes out of a book I read called 'div, grad, curl and all that'...

the derivation of them was to start from scratch with each coordinate system and form a kind of cube, in the coordinate system of choice, and then take a limit of the (flux/volume), as volume goes to 0... similar method for curl.

This takes some time.

The other method is to start with the definition of the divergence in rectangular coordinates, and then painstakingly use the chain rule to transform all variables into cylindrical/spherical coordinates.

This takes a lot of time.

Is there a better way? What is the best way?

Last edited by skipjack; March 2nd, 2016 at 06:13 PM.
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March 4th, 2016, 07:54 PM   #2
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Hi Kinroh,

I am not sure but it seems to me that the best way to derive these functions in multiple coordinate systems was presented to me in my E&M theory course. In this course we used the textbook Engineering Electromagnetic Fields and Waves by Carl T. A. Johnk (first edition). I wore out this book and about 20 years removed from taking the course (about 15 years ago), I bought the second edition.

This book is by far the best textbook I have ever used in any subject (I only have a BSEE, not an advanced degree). If you are really interested in this subject, I would suggest purchasing a used copy of either edition at a reasonable price.

Sorry for being so wordy in the introduction, but pertinent to your question, I think Johnk gives pretty good solutions to your questions: He presents a generalized orthogonal coordinate system and then applies this concept to rectangular, circular cylindrical, and spherical coordinates.

I had grand delusions of presenting the derivations here but I soon recognized the enormity of the task, which probably gives you an idea that there really is no 'easy' method of solution. However, I was able to find a copy of the second edition of Johnk's book for viewing here.

I do not condone the availability of this book's content online, but I think it will help answer your questions and may even pique your interest in purchasing the book (and it sure made my reply a lot easier).

The generalized coordinate system and its application to the rectangular, circular cylindrical, and spherical coordinate systems is perhaps best shown on pages 9-10.

The gradient is dealt with on pages 63-65, the divergence on pages 67-70, and curl on pages 76-80.

Curiously, for curl, Johnk does not present the equations for curl for the rectangular coordinate system (at least on page 80 with those for the other 2 coordinate systems), but as you probably know (or can derive from Johnk's equation (2-50) on page 79),

$\displaystyle \text{curl} \ F = a_x \left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z} \right )+a_y \left(\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x} \right )+a_z \left(\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y} \right ) \qquad$ (check this though)

I hope this helps.
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curl, divergence, grad

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