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# Spherical Polar

• Vectors:

But Note Well: is now a function of ! Similarly:

with a function of the angles that define the direction of . Specifically:

• Unit vectors (relative to Cartesian :

The fact that complicates all of the spherical coordinate vector differential forms, although we indicate above a different, more direct way of evaluating them than applying derivatives to the unit vectors themselves before completing the tensor construction of the vector differential operators.
• Direct Length

• Directed Area

And again, there are many other possible 's, for example, for the bounding surface for hemispherical volume where one piece of it would be a circular surface with an normal like (for example) . This is precisely the surface needed for certain problems you will tackle this semester.
• Volume Element

• Divergence; The divergence is constructed by the same argument that proves the divergence theorem in a general curvilinear coordinate system, or alternatively picks up pieces from , etc, hence its complexity:

Note that this follows from:

with , , , and , , . Take the contribution from :

because does not depend on , similarly for the other two pieces.

• Curl The curl is evaluated in exactly the same way from the expression above, but it ends up being much more complex:

• Laplacian The Laplacian follows by applying the divergence rule to the gradient rule and simplifying:

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Robert G. Brown 2017-07-11