Convert F from Cartesian coordinates to spherical polar coordinates.
Relative to the coordinate axes, in what direction is F pointing?
The coordinate conversion matrix A for Cartesian coordinates to spherical polar coordinates is
so that Fsph = A Fcart where
Show that the conversion matrix B for the
inverse conversion from spherical polar back to Cartesian coordinates,
such that
B = AT (the transpose of matrix A).
Convert from spherical polar to Cartesian coordinates the vector field
Find the divergence and curl of each of the following:
For the vector field defined in spherical polar coordinates by
find .
Consider the purely radial vector field F (r, q, f) = f (r) , where is the unit radial vector in the spherical polar coordinate system and f (r) is any function of r that is differentiable everywhere in (except possibly at the origin).
Of particular interest is the central force law
Show that the divergence of F vanishes
everywhere in
(except possibly at the origin) if and only if
[Two of the four fundamental forces of nature, electromagnetism and
gravity, both obey this inverse square law.]
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