Spherical coordinates are an extension of polar coordinates to 3D.
The position of a given point in space is described by
three variables , defined as:
- : the radius or radial distance: distance to the origin.
- : the elevation, polar angle or colatitude:
angle to the positive -axis, or zenith, i.e. the “north pole”.
- : the azimuth, azimuthal angle or longitude:
angle from the positive -axis, typically in the counter-clockwise sense.
Note that this is the standard notation among physicists,
but mathematicians often switch the definitions of and ,
while still writing .
and the spherical system are related by:
Conversely, a point given in
can be converted to using these formulae,
where is the 2-argument arctangent,
which is needed to handle the signs correctly:
Spherical coordinates form
an orthogonal curvilinear system,
whose scale factors , and we need.
To get those, we calculate the unnormalized local basis:
By normalizing the local basis vectors
, and ,
we arrive at these expressions:
Thanks to these scale factors, we can easily convert calculus from the Cartesian system
using the standard formulae for orthogonal curvilinear coordinates.
For line integrals,
the tangent vector element for a curve is as follows:
For surface integrals,
the normal vector element for a surface is given by:
And for volume integrals,
the infinitesimal volume takes the following form:
The basic vector operations (gradient, divergence, curl and Laplacian) are given by:
Uncommon operations include:
the gradient of a divergence ,
the gradient of a vector ,
the advection of a vector with respect to ,
the Laplacian of a vector ,
and the divergence of a 2nd-order tensor :
- M.L. Boas,
Mathematical methods in the physical sciences, 2nd edition,
- B. Lautrup,
Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition,