Spherical Coordinates

A system of Curvilinear Coordinates which is 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 from the z-Axis with (Colatitude, equal to where is the Latitude), and to be distance (Radius) from a point to the Origin.

Unfortunately, the convention in which the symbols and are reversed is frequently used, especially in physics, leading to unnecessary confusion. The symbol is sometimes also used in place of . Arfken (1985) uses , whereas Beyer (1987) uses . Be very careful when consulting the literature.

In this work, the symbols for the azimuthal, polar, and radial 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 the Plane.

 (1) (2) (3)

where , , and . In terms of Cartesian Coordinates,
 (4) (5) (6)

The Scale Factors are
 (7) (8) (9)

so the Metric Coefficients are
 (10) (11) (12)

The Line Element is
 (13)

the Area element
 (14)

and the Volume Element
 (15)

The Jacobian is
 (16)

The Position Vector is

 (17)

so the Unit Vectors are
 (18) (19) (20)

Derivatives of the Unit Vectors are

 (21) (22) (23) (24) (25) (26) (27) (28) (29)

 (30)

so
 (31) (32) (33) (34) (35) (36)

Now, since the Connection Coefficients are given by ,

 (37) (38) (39)

The Divergence is
 (40)
or, in Vector notation,

 (41)

The Covariant Derivatives are given by

 (42)

so
 (43) (44) (45) (46) (47) (48) (49) (50) (51)

The Commutation Coefficients are given by

 (52)

 (53)

so , where .
 (54)

so , .
 (55)

so .
 (56)

so
 (57)

Summarizing,
 (58) (59) (60)

Time derivatives of the Position Vector are

 (61)

The Speed is therefore given by
 (62)

The Acceleration is
 (63) (64) (65)

Plugging these in gives
 (66)

but
 (67)

so
 (68)

Time Derivatives of the Unit Vectors are
 (69) (70) (71)

The Curl is
 (72)

The Laplacian is

 (73)

The vector Laplacian is

 (74)

To express Partial Derivatives with respect to Cartesian axes in terms of Partial Derivatives of the spherical coordinates,

 (75) (76)

Upon inversion, the result is

 (77)

The Cartesian Partial Derivatives in spherical coordinates are therefore
 (78) (79) (80)

(Gasiorowicz 1974, pp. 167-168).

The Helmholtz Differential Equation is separable in spherical coordinates.

See also Colatitude, Great Circle, Helmholtz Differential Equation--Spherical Coordinates, Latitude, Longitude, Oblate Spheroidal Coordinates, Prolate Spheroidal Coordinates

References

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, p. 212, 1987.

Gasiorowicz, S. Quantum Physics. New York: Wiley, 1974.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 658, 1953.