Quick proof of the cyclic property of the cross
products of unit basis vectors in spherical polar
coordinates
10 December 2012 at 13:29
Public
In Cartesian coordinates there is a useful cyclic property of
the unit basis vectors which is convenient to know when calculating cross products of
vectors. For the purposes of some things I am doing at the moment I want to make
absolutely sure that the same cyclic property also holds for the unit basis vectors in
spherical polar coordinates. Somewhat surprisingly, I have not been able to find a proper
statement/proof of this cyclic property for spherical polar coordinates anywhere online or
in any textbook (though I'm sure there must be one somewhere!). For peace of mind I've
therefore decided to quickly work out a proof for myself here.
In Cartesian coordinates, the cross product of two vectors
is conveniently calculated as a determinant
When working out cross products of the unit basis vectors themselves in Cartesian
coordinates, there is a useful cyclic property which helps you remember the results:
Going clockwise around the cycle we have
and going anticlockwise around the cycle we have
Now let
be the unit basis vectors in spherical polar coordinates. What I want to prove for myself is
that the same cyclic property holds for these, i.e.,
so that going clockwise around the cycle we have
and going anticlockwise around the cycle we have
Probably the most straightforward way to do this is to express the unit basis vectors in
spherical polar coordinates in terms of the unit basis vectors in Cartesian coordinates,
and then work out the cross products of those expressions. We have
(Note that these are the 'normalized' columns of the Jacobian matrix
i.e. dividing each column of this matrix by its norm gives you each of the above unit
vectors. If you take the tanspose of this Jacobian matrix AFTER normalizing its columns,
the columns of the resulting transpose matrix would give the unit vectors for Cartesian
coordinates expressed in terms of the unit vectors for spherical polar coordinates).
Using the above expressions for the unit vectors in spherical polar coordinates in terms
of the unit vectors in Cartesian coordinates, we can now set about confirming the cross
products around the cycle:
This confirms the cyclic property in the clockwise direction. Going anticlockwise around
the cycle involves interchanging rows in the above determinants, which would have the
effect of changing the signs of the determinants to negative, so this confirms the
anticlockwise cyclic property. QED