Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index Interactive Entries Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book Wolfram Web Resources » 13,668 entries Last updated: Thu Nov 29 2018 Created, developed, and nurtured by Eric Weisstein at Wolfram Research  Geometry > Coordinate Geometry > Interactive Entries > Interactive Demonstrations > Spherical Coordinates Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear coordinates In this work, following the mathematics convention, the symbols for the radial, azimuth , and zenith angle Unfortunately, the convention in which the symbols and are reversed
The spherical coordinates are
where , , In terms of Cartesian coordinates ,
The scale factors are
so the metric coefficients
The line element is
the area element
and the volume element
The Jacobian is
The radius vector is
so the unit vectors are
Derivatives of the unit vectors are
The gradient is
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
(Misner et al. 1973, p. 213, who however use the notation convention ). The Christoffel
(Walton 1967; Moon and Spencer 1988, p. 25a; both of whom however use the notation convention ). The divergence is
or, in vector notation,
The covariant derivatives are given by
so
The commutation coefficients are given
so ,
so ,
so .
so
Summarizing,
Time derivatives of the radius vector are
The speed is therefore given by
The acceleration is
Plugging these in gives
but
so
Time derivatives of the unit
The curl is
The Laplacian is
The vector Laplacian in spherical coordinates
To express partial derivatives with respect to Cartesian axes in terms of partial derivatives
Upon inversion, the result is
The Cartesian partial derivatives in spherical
(Gasiorowicz 1974, pp. 167168; Arfken 1985, p. 108). The Helmholtz differential equation SEE ALSO: Azimuth , Colatitude , Great Circle , Helmholtz REFERENCES: Anton, H. Calculus Apostol, T. M. Calculus, 2nd ed., Vol. 2: MultiVariable Calculus and Linear Algebra, with Applications Arfken, G. "Spherical Polar Coordinates." §2.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 102111, Beyer, W. H. CRC Bronshtein, I. N.; Semendyayev, K. A.; Musiol, G.; and Muehlig, H. Handbook Gasiorowicz, S. Quantum Korn, G. A. and Korn, T. M. Mathematical Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravitation. Moon, P. and Spencer, D. E. "Spherical Coordinates ." Morse, P. M. and Feshbach, H. Methods Walton, J. J. "Tensor Calculations on Computer: Appendix." Comm. Zwillinger, D. (Ed.). "Spherical Coordinates in Space." §4.9.3 in CRC Referenced on WolframAlpha: Spherical Coordinates CITE THIS AS: Weisstein, Eric W. "Spherical Coordinates." Wolfram Web Resources
 THINGS TO TRY: parametric equations spherical coordinates of vector (1, 2, 3) spherical coordinates for 2,1,2

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index Interactive Entries Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book Wolfram Web Resources » 13,668 entries Last updated: Thu Nov 29 2018 Created, developed, and nurtured by Eric Weisstein at Wolfram Research  Geometry > Coordinate Geometry > Interactive Entries > Interactive Demonstrations > Spherical Coordinates Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear coordinates In this work, following the mathematics convention, the symbols for the radial, azimuth , and zenith angle Unfortunately, the convention in which the symbols and are reversed
The spherical coordinates are
where , , In terms of Cartesian coordinates ,
The scale factors are
so the metric coefficients
The line element is
the area element
and the volume element
The Jacobian is
The radius vector is
so the unit vectors are
Derivatives of the unit vectors are
The gradient is
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
(Misner et al. 1973, p. 213, who however use the notation convention ). The Christoffel
(Walton 1967; Moon and Spencer 1988, p. 25a; both of whom however use the notation convention ). The divergence is
or, in vector notation,
The covariant derivatives are given by
so
The commutation coefficients are given
so ,
so ,
so .
so
Summarizing,
Time derivatives of the radius vector are
The speed is therefore given by
The acceleration is
Plugging these in gives
but
so
Time derivatives of the unit
The curl is
The Laplacian is
The vector Laplacian in spherical coordinates
To express partial derivatives with respect to Cartesian axes in terms of partial derivatives
Upon inversion, the result is
The Cartesian partial derivatives in spherical
(Gasiorowicz 1974, pp. 167168; Arfken 1985, p. 108). The Helmholtz differential equation SEE ALSO: Azimuth , Colatitude , Great Circle , Helmholtz REFERENCES: Anton, H. Calculus Apostol, T. M. Calculus, 2nd ed., Vol. 2: MultiVariable Calculus and Linear Algebra, with Applications Arfken, G. "Spherical Polar Coordinates." §2.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 102111, Beyer, W. H. CRC Bronshtein, I. N.; Semendyayev, K. A.; Musiol, G.; and Muehlig, H. Handbook Gasiorowicz, S. Quantum Korn, G. A. and Korn, T. M. Mathematical Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravitation. Moon, P. and Spencer, D. E. "Spherical Coordinates ." Morse, P. M. and Feshbach, H. Methods Walton, J. J. "Tensor Calculations on Computer: Appendix." Comm. Zwillinger, D. (Ed.). "Spherical Coordinates in Space." §4.9.3 in CRC Referenced on WolframAlpha: Spherical Coordinates CITE THIS AS: Weisstein, Eric W. "Spherical Coordinates." Wolfram Web Resources
 THINGS TO TRY: parametric equations spherical coordinates of vector (1, 2, 3) spherical coordinates for 2,1,2

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index Interactive Entries Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book Wolfram Web Resources » 13,668 entries Last updated: Thu Nov 29 2018 Created, developed, and nurtured by Eric Weisstein at Wolfram Research  Geometry > Coordinate Geometry > Interactive Entries > Interactive Demonstrations > Spherical Coordinates Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear coordinates In this work, following the mathematics convention, the symbols for the radial, azimuth , and zenith angle Unfortunately, the convention in which the symbols and are reversed
The spherical coordinates are
where , , In terms of Cartesian coordinates ,
The scale factors are
so the metric coefficients
The line element is
the area element
and the volume element
The Jacobian is
The radius vector is
so the unit vectors are
Derivatives of the unit vectors are
The gradient is
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
(Misner et al. 1973, p. 213, who however use the notation convention ). The Christoffel
(Walton 1967; Moon and Spencer 1988, p. 25a; both of whom however use the notation convention ). The divergence is
or, in vector notation,
The covariant derivatives are given by
so
The commutation coefficients are given
so ,
so ,
so .
so
Summarizing,
Time derivatives of the radius vector are
The speed is therefore given by
The acceleration is
Plugging these in gives
but
so
Time derivatives of the unit
The curl is
The Laplacian is
The vector Laplacian in spherical coordinates
To express partial derivatives with respect to Cartesian axes in terms of partial derivatives
Upon inversion, the result is
The Cartesian partial derivatives in spherical
(Gasiorowicz 1974, pp. 167168; Arfken 1985, p. 108). The Helmholtz differential equation SEE ALSO: Azimuth , Colatitude , Great Circle , Helmholtz REFERENCES: Anton, H. Calculus Apostol, T. M. Calculus, 2nd ed., Vol. 2: MultiVariable Calculus and Linear Algebra, with Applications Arfken, G. "Spherical Polar Coordinates." §2.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 102111, Beyer, W. H. CRC Bronshtein, I. N.; Semendyayev, K. A.; Musiol, G.; and Muehlig, H. Handbook Gasiorowicz, S. Quantum Korn, G. A. and Korn, T. M. Mathematical Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravitation. Moon, P. and Spencer, D. E. "Spherical Coordinates ." Morse, P. M. and Feshbach, H. Methods Walton, J. J. "Tensor Calculations on Computer: Appendix." Comm. Zwillinger, D. (Ed.). "Spherical Coordinates in Space." §4.9.3 in CRC Referenced on WolframAlpha: Spherical Coordinates CITE THIS AS: Weisstein, Eric W. "Spherical Coordinates." Wolfram Web Resources
 THINGS TO TRY: parametric equations spherical coordinates of vector (1, 2, 3) spherical coordinates for 2,1,2
