Spherical Coordinates.pdf

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Geometry > Coordinate Geometry >

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Spherical Coordinates

Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear coordinatesthat are natural for describing

positions on a sphereor spheroid. Define to be the azimuthal angle in the -planefrom thex-axiswith (denoted when referred to as the longitude),

to be the polar angle(also known as the zenith angleand colatitude, with where is the latitude) from the positive z-axiswith , 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 anglecoordinates are taken as , , and , respectively. Note that

this definition provides a logical extension of the usual polar coordinatesnotation, with remaining the anglein the -planeand becoming the angleout 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 Mathematicapackage VectorAnalysis`)

Arfken (1985, p. 102)

Moon and Spencer (1988, p. 24)

The spherical coordinates are related to the Cartesian coordinates by

(1)

(2)

(3)

where , , and , and the inverse tangentmust be suitably defined to take the correct quadrant of into account.

In terms of Cartesian coordinates,

(4)

(5)

(6)

The scale factorsare

(7)

(8)

(9)

so the metriccoefficientsare

(10)

(11)

(12)

The line elementis

(13)

the areaelement

S

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ical Coordinates -- from Wolfram MathWorld http://mathworld.wolfram.com/SphericalCoordin

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(14)

and the volume element

(15)

The Jacobianis

(16)

The position vectoris

(17)

so the unit vectorsare

(18)

(19)

(20)

(21)

(22)

(23)

Derivatives of the unit vectorsare

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

The gradientis

(33)

and its components are

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(Misner et al. 1973, p. 213, who however use the notation convention ).


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The Christoffel symbols of the second kind in the definition of Misner et al. (1973, p. 209) are given by

(43)

(44)

(45)

(Misner et al. 1973, p. 213, who however use the notation convention ). The Christoffel symbols of the second kindin the definition of Arfken (1985) are

given by

(46)

(47)

(48)

(Walton 1967; Moon and Spencer 1988, p. 25a; both of whom however use the notation convention ).

The divergenceis

(49)

or, in vectornotation,

(50)

(51)

The covariant derivativesare given by

(52)

so

(53)

(54)

(55)

(56)

(57)

(58)

(59)

(60)

(61)

The commutation coefficientsare given by

(62)

(63)

so , where .


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(64)

so , .

(65)

so .

(66)

so

(67)

Summarizing,

(68)

(69)

(70)

Time derivatives of the position vectorare

(71)

(72)

(73)

The speedis therefore given by

(74)

The accelerationis

(75)

(76)

(77)

(78)

(79)

(80)

Plugging these in gives

(81)

but

(82)

(83)

so

(84)

(85)


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Time derivativesof the unit vectorsare

(86)

(87)

(88)

The curlis

(89)

The Laplacianis

(90)

(91)

(92)

The vector Laplacianin spherical coordinates is given by

(93)

To express partial derivativeswith respect to Cartesian axes in terms of partial derivativesof the spherical coordinates,

(94)

(95)

(96)

Upon inversion, the result is

(97)

The Cartesian partial derivativesin spherical coordinates are therefore

(98)

(99)

(100)

(Gasiorowicz 1974, pp. 167-168; Arfken 1985, p. 108).

The Helmholtz differential equationis separable in spherical coordinates.

SEE ALSO:Azimuth, Colatitude, Great Circle, Helmholtz Differential Equation--Spherical Coordinates, Latitude, Longitude, Oblate Spheroidal Coordinates, Polar

Angle, Prolate Spheroidal Coordinates, Zenith Angle

REFERENCES:

Arfken, G. "Spherical Polar Coordinates." 2.5 inMathematical 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. ACM10, 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.

CITE THIS AS:

Weisstein, Eric W."Spherical Coordinates." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SphericalCoordinates.html


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