# The Jacobian for Polar and Spherical Coordinates

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# Spherical Coordinates

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
(both in meaning and in order listed) is also frequently used, especially in physics.
This is especially confusing since the identical notation typically
to a physicist. The symbol is sometimes
also used in place of , instead of
, and and instead of . The following table summarizes a number of conventions
used by various authors. Extreme care is therefore needed when consulting the literature.

 order notation reference (radial, azimuthal, polar) this work (radial, azimuthal, polar) Apostol (1969, p. 95), Anton (1984, p. 859), Beyer (1987, p. 212) (radial, polar, azimuthal) ISO 31-11, Misner et al. (1973, p. 205) (radial, polar, azimuthal) Arfken (1985, p. 102) (radial, polar, azimuthal) Moon and Spencer (1988, p. 24) (radial, polar, azimuthal) Korn and Korn (1968, p. 60), Bronshtein et al. (2004, pp. 209-210) (radial, polar, azimuthal) Zwillinger (1996, pp. 297-299)

The spherical coordinates are
related to the Cartesian coordinates by

 (1) (2) (3)

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

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)

 (17)

so the unit vectors are

 (18) (19) (20) (21) (22) (23)

Derivatives of the unit vectors are

 (24) (25) (26) (27) (28) (29) (30) (31) (32)

 (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 ).

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 kind in 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 divergence is

 (49)

or, in vector notation,

 (50) (51)

The covariant derivatives are given by

 (52)

so

 (53) (54) (55) (56) (57) (58) (59) (60) (61)

The commutation coefficients are given
by

 (62)
 (63)

so ,
where .

 (64)

so ,
.

 (65)

so .

 (66)

so

 (67)

Summarizing,

 (68) (69) (70)

Time derivatives of the radius vector are

 (71) (72) (73)

The speed is therefore given by

 (74)

The acceleration is

 (75) (76) (77) (78) (79) (80)

Plugging these in gives

 (81)

but

 (82) (83)

so

 (84) (85)

Time derivatives of the unit
vectors are

 (86) (87) (88)

The curl is

 (89)

The Laplacian is

 (90) (91) (92)

The vector Laplacian in spherical coordinates
is given by

 (93)

To express partial derivatives with respect to Cartesian axes in terms of partial derivatives
of the spherical coordinates,

 (94) (95) (96)

Upon inversion, the result is

 (97)

The Cartesian partial derivatives in spherical
coordinates are therefore

 (98) (99) (100)

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

The Helmholtz differential equation
is separable in spherical coordinates.

Differential Equation–Spherical Coordinates , Latitude ,
Longitude , Oblate
Spheroidal Coordinates , Polar Angle , Polar
Coordinates , Polar Plot , Polar
Vector , Prolate Spheroidal Coordinates ,
Zenith Angle

REFERENCES:

Anton, H. Calculus
with Analytic Geometry, 2nd ed.
New York: Wiley, 1984.

Apostol, T. M. Calculus, 2nd ed., Vol. 2: Multi-Variable Calculus and Linear Algebra, with Applications
to Differential Equations and Probability.
Waltham, MA: Blaisdell, 1969.

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.

Bronshtein, I. N.; Semendyayev, K. A.; Musiol, G.; and Muehlig, H. Handbook
of Mathematics, 4th ed.
New York: Springer-Verlag, 2004.

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.

Referenced on Wolfram|Alpha: Spherical Coordinates

CITE THIS AS:

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

# Wolfram Web Resources

 Mathematica » The #1 tool for creating Demonstrations and anything technical. Wolfram|Alpha » Explore anything with the first computational knowledge engine. Wolfram Demonstrations Project » Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Computerbasedmath.org » Join the initiative for modernizing math education. Online Integral Calculator » Solve integrals with Wolfram|Alpha. Step-by-step Solutions » Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own. Wolfram Problem Generator » Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Wolfram Education Portal » Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. Wolfram Language » Knowledge-based programming for everyone.
THINGS TO TRY:

parametric equations
spherical coordinates
of vector (1, 2, 3)
spherical coordinates
for 2,1,-2

 Damped Spherical Spring Pendulum Erik Mahieu Surface Morphing Yu-Sung Chang Zebra-Painted Egg Michael Schreiber Cone Placed on a Sphere Sándor Kabai
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
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
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
(both in meaning and in order listed) is also frequently used, especially in physics.
This is especially confusing since the identical notation typically
to a physicist. The symbol is sometimes
also used in place of , instead of
, and and instead of . The following table summarizes a number of conventions
used by various authors. Extreme care is therefore needed when consulting the literature.

 order notation reference (radial, azimuthal, polar) this work (radial, azimuthal, polar) Apostol (1969, p. 95), Anton (1984, p. 859), Beyer (1987, p. 212) (radial, polar, azimuthal) ISO 31-11, Misner et al. (1973, p. 205) (radial, polar, azimuthal) Arfken (1985, p. 102) (radial, polar, azimuthal) Moon and Spencer (1988, p. 24) (radial, polar, azimuthal) Korn and Korn (1968, p. 60), Bronshtein et al. (2004, pp. 209-210) (radial, polar, azimuthal) Zwillinger (1996, pp. 297-299)

The spherical coordinates are
related to the Cartesian coordinates by

 (1) (2) (3)

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

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)

 (17)

so the unit vectors are

 (18) (19) (20) (21) (22) (23)

Derivatives of the unit vectors are

 (24) (25) (26) (27) (28) (29) (30) (31) (32)

 (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 ).

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 kind in 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 divergence is

 (49)

or, in vector notation,

 (50) (51)

The covariant derivatives are given by

 (52)

so

 (53) (54) (55) (56) (57) (58) (59) (60) (61)

The commutation coefficients are given
by

 (62)
 (63)

so ,
where .

 (64)

so ,
.

 (65)

so .

 (66)

so

 (67)

Summarizing,

 (68) (69) (70)

Time derivatives of the radius vector are

 (71) (72) (73)

The speed is therefore given by

 (74)

The acceleration is

 (75) (76) (77) (78) (79) (80)

Plugging these in gives

 (81)

but

 (82) (83)

so

 (84) (85)

Time derivatives of the unit
vectors are

 (86) (87) (88)

The curl is

 (89)

The Laplacian is

 (90) (91) (92)

The vector Laplacian in spherical coordinates
is given by

 (93)

To express partial derivatives with respect to Cartesian axes in terms of partial derivatives
of the spherical coordinates,

 (94) (95) (96)

Upon inversion, the result is

 (97)

The Cartesian partial derivatives in spherical
coordinates are therefore

 (98) (99) (100)

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

The Helmholtz differential equation
is separable in spherical coordinates.

Differential Equation–Spherical Coordinates , Latitude ,
Longitude , Oblate
Spheroidal Coordinates , Polar Angle , Polar
Coordinates , Polar Plot , Polar
Vector , Prolate Spheroidal Coordinates ,
Zenith Angle

REFERENCES:

Anton, H. Calculus
with Analytic Geometry, 2nd ed.
New York: Wiley, 1984.

Apostol, T. M. Calculus, 2nd ed., Vol. 2: Multi-Variable Calculus and Linear Algebra, with Applications
to Differential Equations and Probability.
Waltham, MA: Blaisdell, 1969.

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.

Bronshtein, I. N.; Semendyayev, K. A.; Musiol, G.; and Muehlig, H. Handbook
of Mathematics, 4th ed.
New York: Springer-Verlag, 2004.

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.

Referenced on Wolfram|Alpha: Spherical Coordinates

CITE THIS AS:

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

# Wolfram Web Resources

 Mathematica » The #1 tool for creating Demonstrations and anything technical. Wolfram|Alpha » Explore anything with the first computational knowledge engine. Wolfram Demonstrations Project » Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Computerbasedmath.org » Join the initiative for modernizing math education. Online Integral Calculator » Solve integrals with Wolfram|Alpha. Step-by-step Solutions » Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own. Wolfram Problem Generator » Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Wolfram Education Portal » Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. Wolfram Language » Knowledge-based programming for everyone.
THINGS TO TRY:

parametric equations
spherical coordinates
of vector (1, 2, 3)
spherical coordinates
for 2,1,-2

 Damped Spherical Spring Pendulum Erik Mahieu Surface Morphing Yu-Sung Chang Zebra-Painted Egg Michael Schreiber Cone Placed on a Sphere Sándor Kabai
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
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
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
(both in meaning and in order listed) is also frequently used, especially in physics.
This is especially confusing since the identical notation typically
to a physicist. The symbol is sometimes
also used in place of , instead of
, and and instead of . The following table summarizes a number of conventions
used by various authors. Extreme care is therefore needed when consulting the literature.

 order notation reference (radial, azimuthal, polar) this work (radial, azimuthal, polar) Apostol (1969, p. 95), Anton (1984, p. 859), Beyer (1987, p. 212) (radial, polar, azimuthal) ISO 31-11, Misner et al. (1973, p. 205) (radial, polar, azimuthal) Arfken (1985, p. 102) (radial, polar, azimuthal) Moon and Spencer (1988, p. 24) (radial, polar, azimuthal) Korn and Korn (1968, p. 60), Bronshtein et al. (2004, pp. 209-210) (radial, polar, azimuthal) Zwillinger (1996, pp. 297-299)

The spherical coordinates are
related to the Cartesian coordinates by

 (1) (2) (3)

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

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)

 (17)

so the unit vectors are

 (18) (19) (20) (21) (22) (23)

Derivatives of the unit vectors are

 (24) (25) (26) (27) (28) (29) (30) (31) (32)

 (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 ).

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 kind in 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 divergence is

 (49)

or, in vector notation,

 (50) (51)

The covariant derivatives are given by

 (52)

so

 (53) (54) (55) (56) (57) (58) (59) (60) (61)

The commutation coefficients are given
by

 (62)
 (63)

so ,
where .

 (64)

so ,
.

 (65)

so .

 (66)

so

 (67)

Summarizing,

 (68) (69) (70)

Time derivatives of the radius vector are

 (71) (72) (73)

The speed is therefore given by

 (74)

The acceleration is

 (75) (76) (77) (78) (79) (80)

Plugging these in gives

 (81)

but

 (82) (83)

so

 (84) (85)

Time derivatives of the unit
vectors are

 (86) (87) (88)

The curl is

 (89)

The Laplacian is

 (90) (91) (92)

The vector Laplacian in spherical coordinates
is given by

 (93)

To express partial derivatives with respect to Cartesian axes in terms of partial derivatives
of the spherical coordinates,

 (94) (95) (96)

Upon inversion, the result is

 (97)

The Cartesian partial derivatives in spherical
coordinates are therefore

 (98) (99) (100)

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

The Helmholtz differential equation
is separable in spherical coordinates.

Differential Equation–Spherical Coordinates , Latitude ,
Longitude , Oblate
Spheroidal Coordinates , Polar Angle , Polar
Coordinates , Polar Plot , Polar
Vector , Prolate Spheroidal Coordinates ,
Zenith Angle

REFERENCES:

Anton, H. Calculus
with Analytic Geometry, 2nd ed.
New York: Wiley, 1984.

Apostol, T. M. Calculus, 2nd ed., Vol. 2: Multi-Variable Calculus and Linear Algebra, with Applications
to Differential Equations and Probability.
Waltham, MA: Blaisdell, 1969.

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.

Bronshtein, I. N.; Semendyayev, K. A.; Musiol, G.; and Muehlig, H. Handbook
of Mathematics, 4th ed.
New York: Springer-Verlag, 2004.

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.

Referenced on Wolfram|Alpha: Spherical Coordinates

CITE THIS AS:

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

# Wolfram Web Resources

 Mathematica » The #1 tool for creating Demonstrations and anything technical. Wolfram|Alpha » Explore anything with the first computational knowledge engine. Wolfram Demonstrations Project » Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Computerbasedmath.org » Join the initiative for modernizing math education. Online Integral Calculator » Solve integrals with Wolfram|Alpha. Step-by-step Solutions » Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own. Wolfram Problem Generator » Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Wolfram Education Portal » Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. Wolfram Language » Knowledge-based programming for everyone.