Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear
coordinates that are natural for describing positions on a sphere or spheroid.
Define to be the azimuthal angle in the
-plane
from the x-axis with
(denoted when referred to as the longitude), to be the polar angle (also known as the zenith angle and colatitude,
with where is the latitude) from the positive z-axis
with , and to be distance
(radius) from a point to the origin. This is the convention commonly used in mathematics.
In this work, following the mathematics convention, the symbols for the radial, azimuth, and zenith angle coordinates are taken as , , and , respectively.
Note that this definition provides a logical extension of the usual polar coordinates notation, with remaining the
angle in the -plane and becoming the
angle out of that plane. The sole exception to this convention in this work is
in spherical harmonics, where
the convention used in the physics literature is retained (resulting, it is hoped,
in a bit less confusion than a foolish rigorous consistency might engender).
Unfortunately, the convention in which the symbols and are reversed
is also frequently used, especially in physics. The symbol is sometimes
also used in place of , and and instead of . The following table summarizes a number of conventions
used by various authors; be very careful when consulting the literature.
(radial, azimuthal, polar) | reference | | this work, Zwillinger (1985, pp. 297-298) | | Beyer
(1987, p. 212) | | Korn and Korn (1968, p. 60) | | Misner et al.
(1973, p. 205) | (Rr, Pphi,
Ttheta) | SetCoordinates[Spherical[r, Ttheta, Pphi]] in
the Mathematica
package VectorAnalysis`) | | Arfken (1985,
p. 102) | | Moon and Spencer (1988, p. 24) |
The spherical coordinates are
related to the Cartesian coordinates by
where , ,
and , and the inverse tangent must be suitably defined to take the correct
quadrant of into account.
In terms of Cartesian coordinates,
The scale factors are
so the metric coefficients are
The line element is
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(13)
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the area element
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(14)
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and the volume element
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(15)
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The Jacobian is
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(16)
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The position vector is
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(17)
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so the unit vectors are
Derivatives of the unit vectors
are
The gradient is
|
(33)
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and its components are
(Misner et al. 1973, p. 213, who however use the notation convention ).
The Christoffel symbols of the second kind in the definition of Misner et al. (1973, p. 209)
are given by
(Misner et al. 1973, p. 213, who however use the notation convention ). The Christoffel symbols of the second kind in the definition of
Arfken (1985) are given by
(Walton 1967; Moon and Spencer 1988, p. 25a; both of whom however use the notation convention ).
The divergence is
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(49)
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or, in vector notation,
The covariant derivatives
are given by
|
(52)
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so
The commutation coefficients
are given by
|
(62)
|
|
(63)
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so ,
where .
|
(64)
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so ,
.
|
(65)
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so .
|
(66)
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so
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(67)
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Summarizing,
Time derivatives of the position
vector are
The speed is therefore given by
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(74)
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The acceleration is
Plugging these in gives
but
so
Time derivatives of the unit vectors are
The curl is
|
(89)
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The Laplacian is
The vector Laplacian in spherical
coordinates is given by
|
(93)
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To express partial derivatives with respect to Cartesian axes in terms of partial
derivatives of the spherical coordinates,
Upon inversion, the result is
|
(97)
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The Cartesian partial derivatives
in spherical coordinates are therefore
(Gasiorowicz 1974, pp. 167-168; Arfken 1985, p. 108).
The Helmholtz differential
equation is separable in spherical coordinates.
Arfken, G. "Spherical Polar Coordinates." §2.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL:
Academic Press, pp. 102-111, 1985.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton,
FL: CRC Press, 1987.
Gasiorowicz, S. Quantum Physics. New York: Wiley, 1974.
Korn, G. A. and Korn, T. M. Mathematical Handbook for Scientists and Engineers. New
York: McGraw-Hill, 1968.
Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravitation. San Francisco, CA: W. H. Freeman,
1973.
Moon, P. and Spencer, D. E. "Spherical Coordinates ."
Table 1.05 in Field Theory Handbook, Including Coordinate Systems, Differential
Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 24-27,
1988.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill,
p. 658, 1953.
Walton, J. J. "Tensor Calculations on Computer: Appendix." Comm.
ACM 10, 183-186, 1967.
Zwillinger, D. (Ed.). "Spherical Coordinates in Space." §4.9.3 in CRC Standard Mathematical Tables and Formulae. Boca Raton,
FL: CRC Press, pp. 297-298, 1995.
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