The Jacobian for Polar and Spherical Coordinates


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Spherical Coordinates

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SphericalCoordinates

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 theta to be the azimuthal
angle in the xy– plane from
the x-axis with 0<=theta<2pi
(denoted lambda when referred to as the longitude ),
phi to be the polar
angle (also known as the zenith angle and colatitude , with phi=90 degrees-delta
where delta is the latitude )
from the positive z-axis with 0<=phi<=pi,
and r 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 r, theta, and phi, respectively.
Note that this definition provides a logical extension of the usual polar
coordinates notation, with theta remaining the
angle in the xy– plane
and phi 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 theta and phi are reversed
(both in meaning and in order listed) is also frequently used, especially in physics.
This is especially confusing since the identical notation (r,theta,phi) typically
means (radial, azimuthal, polar) to a mathematician but (radial, polar, azimuthal)
to a physicist. The symbol rho is sometimes
also used in place of r, theta instead of
theta, and phi and psi instead of phi. The following table summarizes a number of conventions
used by various authors. Extreme care is therefore needed when consulting the literature.

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

The spherical coordinates (r,theta,phi) are
related to the Cartesian coordinates (x,y,z) by

r=sqrt(x^2+y^2+z^2)

(1)
theta=tan^(-1)(y/x)

(2)
phi=cos^(-1)(z/r),

(3)

where r in [0,infty), theta in [0,2pi),
and phi in [0,pi], and the inverse
tangent must be suitably defined to take the correct quadrant of (x,y) into account.

In terms of Cartesian coordinates ,

x=rcosthetasinphi

(4)
y=rsinthetasinphi

(5)
z=rcosphi.

(6)

The scale factors are

h_r=1

(7)
h_theta=rsinphi
(8)
h_phi=r,

(9)

so the metric coefficients
are

g_(rr)=1
(10)
g_(thetatheta)=r^2sin^2phi
(11)
g_(phiphi)=r^2.
(12)

The line element is

 ds=drr^^+rdphiphi^^+rsinphidthetatheta^^,

(13)

the area element

 da=r^2sinphidthetadphi,

(14)

and the volume element

 dV=r^2sinphidphidthetadr.

(15)

The Jacobian is

 |(partial(x,y,z))/(partial(r,theta,phi))|=r^2sinphi.

(16)

The radius vector is

 r=[rcosthetasinphi; rsinthetasinphi; rcosphi],

(17)

so the unit vectors are

r^^=((dr)/(dr))/(|(dr)/(dr)|)

(18)
=[costhetasinphi; sinthetasinphi; cosphi]

(19)
theta^^=((dr)/(dtheta))/(|(dr)/(dtheta)|)

(20)
=[-sintheta; costheta; 0]

(21)
phi^^=((dr)/(dphi))/(|(dr)/(dphi)|)

(22)
=[costhetacosphi; sinthetacosphi; -sinphi].

(23)

Derivatives of the unit vectors are

(partialr^^)/(partialr)=0
(24)
(partialtheta^^)/(partialr)=0
(25)
(partialphi^^)/(partialr)=0
(26)
(partialr^^)/(partialtheta)=sinphitheta^^
(27)
(partialtheta^^)/(partialtheta)=-cosphiphi^^-sinphir^^

(28)
(partialphi^^)/(partialtheta)=cosphitheta^^
(29)
(partialr^^)/(partialphi)=phi^^
(30)
(partialtheta^^)/(partialphi)=0
(31)
(partialphi^^)/(partialphi)=-r^^.
(32)

The gradient is

 del =r^^partial/(partialr)+1/rphi^^partial/(partialphi)+1/(rsinphi)theta^^partial/(partialtheta),

(33)

and its components are

del _rr^^=0
(34)
del _thetar^^=1/rtheta^^
(35)
del _phir^^=1/rphi^^
(36)
del _rtheta^^=0
(37)
del _thetatheta^^=-(cotphi)/rphi^^-1/rr^^

(38)
del _phitheta^^=0
(39)
del _rphi^^=0
(40)
del _thetaphi^^=1/rcotphitheta^^

(41)
del _phiphi^^=-1/rr^^
(42)

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

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

Gamma^r=[0 0 0; 0 -1/r 0; 0 0 -1/r]

(43)
Gamma^theta=[0 1/r 0; 0 0 0; 0 (cotphi)/r 0]

(44)
Gamma^phi=[0 0 1/r; 0 -(cotphi)/r 0; 0 0 0]

(45)

(Misner et al. 1973, p. 213, who however use the notation convention (r,phi,theta)). The Christoffel
symbols of the second kind in the definition of Arfken (1985) are given by

Gamma^r=[0 0 0; 0 -rsin^2phi 0; 0 0 -r]

(46)
Gamma^theta=[0 1/r 0; 1/r 0 cotphi; 0 cotphi 0]

(47)
Gamma^phi=[0 0 1/r; 0 -sinphicosphi 0; 1/r 0 0]

(48)

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

The divergence is

 del ·F=partial/(partialr)A^r+2/rA^r+1/(rsinphi)partial/(partialtheta)A^theta+1/rpartial/(partialphi)A^phi+(cotphi)/rA^phi,

(49)

or, in vector notation,

del ·F=(2/r+partial/(partialr))F_r+(1/rpartial/(partialphi)+(cotphi)/r)F_phi+1/(rsinphi)(partialF_theta)/(partialtheta)

(50)
=1/(r^2)partial/(partialr)(r^2F_r)+1/(rsinphi)partial/(partialphi)(sinphiF_phi)+1/(rsinphi)(partialF_theta)/(partialtheta).

(51)

The covariant derivatives are given by

 A_(j;k)=1/(g_(kk))(partialA_j)/(partialx_k)-Gamma_(jk)^iA_i,

(52)

so

A_(r;r)=(partialA_r)/(partialr)

(53)
A_(r;theta)=1/(rsinphi)(partialA_r)/(partialphi)-(A_theta)/r

(54)
A_(r;phi)=1/r((partialA_r)/(partialphi)-A_phi)

(55)
A_(theta;r)=(partialA_theta)/(partialr)

(56)
A_(theta;theta)=1/(rsinphi)(partialA_theta)/(partialtheta)+(cotphi)/rA_phi+(A_r)/r

(57)
A_(theta;phi)=1/r(partialA_theta)/(partialr)-Gamma_(phir)^iA_i(partialA_theta)/(partialphi)

(58)
A_(phi;r)=(partialA_phi)/(partialr)-Gamma_(phir)^iA_i=(partialA_phi)/r

(59)
A_(phi;theta)=1/(rsinphi)(partialA_phi)/(partialtheta)-(cotphi)/rA_theta

(60)
A_(phi;phi)=1/r(partialA_phi)/(partialphi)+(A_r)/r.

(61)

The commutation coefficients are given
by

 c_(alphabeta)^mue^->_mu=[e^->_alpha,e^->_beta]=del _alphae^->_beta-del _betae^->_alpha

(62)
 [r^^,r^^]=[theta^^,theta^^]=[phi^^,phi^^]=0,

(63)

so c_(rr)^alpha=c_(thetatheta)^alpha=c_(phiphi)^alpha=0,
where alpha=r,theta,phi.

 [r^^,theta^^]=-[theta^^,r^^]=del _rtheta^^-del _thetar^^=0-1/rtheta^^=-1/rtheta^^,

(64)

so c_(rtheta)^theta=-c_(thetar)^theta=-1/r,
c_(rtheta)^r=c_(rtheta)^phi=0.

 [r^^,phi^^]=-[phi^^,r^^]=0-1/rphi^^=-1/rphi^^,

(65)

so c_(rphi)^phi=-c_(phir)^phi=1/r.

 [theta^^,phi^^]=-[phi^^,theta^^]=1/rcotphitheta^^-0=1/rcotphitheta^^,

(66)

so

 c_(thetaphi)^theta=-c_(phitheta)^theta=1/rcotphi.

(67)

Summarizing,

c^r=[0 0 0; 0 0 0; 0 0 0]

(68)
c^theta=[0 -1/r 0; 1/r 0 1/rcotphi; 0 -1/rcotphi 0]

(69)
c^phi=[0 0 -1/r; 0 0 0; 1/r 0 0].

(70)

Time derivatives of the radius vector are

r^.=[costhetasinphir^.-rsinthetasinphitheta^.+rcosthetacosphiphi^.; sinthetasinphir^.+rcosthetasinphitheta^.+rsinthetacosphiphi^.; cosphir^.-rsinphiphi^.]

(71)
=[costhetasinphi; sinthetasinphi; cosphi]r^.+rsinphi[-sintheta; costheta; 0]theta^.+r[costhetacosphi; sinthetacosphi; -sinphi]phi^.

(72)
=r^.r^^+rsinphitheta^.theta^^+rphi^.phi^^.

(73)

The speed is therefore given by

 v=|r^.|=sqrt(r^.^2+r^2sin^2phitheta^.^2+r^2phi^.^2).

(74)

The acceleration is

x^..=(-sinthetasinphitheta^.r^.+costhetacosphir^.phi^.+costhetasinphir^..)-(sinthetasinphir^.theta^.+rcosthetasinphitheta^.^2+rsinthetacosphitheta^.phi^.+rsinthetasinphitheta^..)+(costhetacosphir^.phi^.-rsinthetacostheta^.phi^.-rcosthetasinphiphi^.^2+rcosthetacosphiphi^..)

(75)
=-2sinthetasinphitheta^.r^.+2costhetacosphir^.phi^.-2rsinthetacosphitheta^.phi^.+costhetasinphir^..-rsinthetasinphitheta^..+rcosthetacosphiphi^..-rcosthetasinphi(theta^.^2+phi^.^2)

(76)
y^..=(sinthetasinphir^..+rcosthetasinphitheta^.+rcosphisinthetaphi^.)+(costhetasinphir^.theta^.-rsinthetasinphitheta^.^2+rcosthetacosphitheta^.phi^.+rcosthetasinphitheta^..)+(sinthetacosphir^.phi^.+rcosthetacosphitheta^.phi^.-rsinthetasinphiphi^.^2+rsinthetacosphiphi^..)

(77)
=2costhetasinphitheta^.r^.+2sinthetacosphir^.phi^.+2rcosthetacosphitheta^.phi^.+sinthetasinphir^..+rcosthetasinphitheta^..+rsinthetacosphiphi^..-rsinthetasinphi(theta^.^2+phi^.^2)

(78)
z^..=(cosphir^..-sinphir^.phi^.)-(r^.sinphiphi^.+rcosphiphi^.^2+rsinphiphi^..)

(79)
=-rcosphiphi^.^2+cosphir^..-2sinphiphi^.r^.-rsinphiphi^...

(80)

Plugging these in gives

r^..=(r^..-rphi^.^2)[costhetasinphi; sinthetasinphi; cosphi]+(2rcosphitheta^.phi^.+2sinphitheta^.r^.+rsinphitheta^..)[-sintheta; costheta; 0]+(2r^.phi^.+rphi^..)[costhetacosphi; sinthetacosphi; -sinphi]-rsinphitheta^.^2[costheta; sintheta; 0],

(81)

but

sinphir^^+cosphiphi^^=[costhetasin^2phi+costhetacos^2phi; sinthetasin^2phi+sinthetacos^2phi; 0]

(82)
=[costheta; sintheta; 0],

(83)

so

r^..=(r^..-rphi^.^2)r^^+(2rcosphitheta^.phi^.+2sinphitheta^.r^.+rsinphitheta^..)theta^^+(2r^.phi^.+rphi^..)phi^^-rsinphitheta^.^2(sinphir^^+cosphiphi^^)

(84)
=(r^..-rphi^.^2-rsin^2phitheta^.^2)r^^+(2sinphitheta^.r^.+2rcosphitheta^.phi^.+rsinphitheta^..)theta^^+(2r^.phi^.+rphi^..-rsinphicosphitheta^.^2)phi^^.

(85)

Time derivatives of the unit
vectors are

r^^^.=sinphitheta^.theta^^+phi^.phi^^

(86)
theta^^^.=-theta^.(sinphir^^+cosphiphi^^)

(87)
phi^^^.=-phi^.r^^+cosphitheta^.theta^^.

(88)

The curl is

 del ×F=1/(rsinphi)[partial/(partialphi)(sinphiF_theta)-(partialF_phi)/(partialtheta)]r^^+1/r[1/(sinphi)(partialF_r)/(partialtheta)-partial/(partialr)(rF_theta)]phi^^+1/r[partial/(partialr)(rF_phi)-(partialF_r)/(partialphi)]theta^^.

(89)

The Laplacian is

del ^2=1/(r^2)partial/(partialr)(r^2partial/(partialr))+1/(r^2sin^2phi)(partial^2)/(partialtheta^2)+1/(r^2sinphi)partial/(partialphi)(sinphipartial/(partialphi))

(90)
=1/(r^2)(r^2(partial^2)/(partialr^2)+2rpartial/(partialr))+1/(r^2sin^2phi)(partial^2)/(partialtheta^2)+1/(r^2sinphi)(cosphipartial/(partialphi)+sinphi(partial^2)/(partialphi^2))

(91)
=(partial^2)/(partialr^2)+2/rpartial/(partialr)+1/(r^2sin^2phi)(partial^2)/(partialtheta^2)+(cosphi)/(r^2sinphi)partial/(partialphi)+1/(r^2)(partial^2)/(partialphi^2).

(92)

The vector Laplacian in spherical coordinates
is given by

 del ^2v=[1/r(partial^2(rv_r))/(partialr^2)+1/(r^2)(partial^2v_r)/(partialtheta^2)+1/(r^2sin^2theta)(partial^2v_r)/(partialphi^2)+(cottheta)/(r^2)(partialv_r)/(partialtheta)-2/(r^2)(partialv_theta)/(partialtheta)-2/(r^2sintheta)(partialv_phi)/(partialphi)-(2v_r)/(r^2)-(2cottheta)/(r^2)v_theta
1/r(partial^2(rv_(theta)))/(partialr^2)+1/(r^2)(partial^2v_(theta))/(partialtheta^2)+1/(r^2sin^2theta)(partial^2v_(theta))/(partialphi^2)+(cottheta)/(r^2)(partialv_(theta))/(partialtheta)-2/(r^2)(cottheta)/(sintheta)(partialv_(phi))/(partialphi)+2/(r^2)(partialv_r)/(partialtheta)-(v_(theta))/(r^2sin^2theta)
1/r(partial^2(rv_(phi)))/(partialr^2)+1/(r^2)(partial^2v_(phi))/(partialtheta^2)+1/(r^2sin^2theta)(partial^2v_(phi))/(partialphi^2)+(cottheta)/(r^2)(partialv_(phi))/(partialtheta)+2/(r^2sintheta)(partialv_r)/(partialphi)+(2cottheta)/(r^2sintheta)(partialv_(theta))/(partialphi)-(v_(phi))/(r^2sin^2theta) ].

(93)

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

[x; y; z]=[rcosthetasinphi; rsinthetasinphi; rcosphi]

(94)
[dx; dy; dz]=[costhetasinphidr-rsinthetasinphidtheta+rcosthetacosphidphi; sinthetasinphidr+rsinphicosthetadtheta+rsinthetacosphidphi; cosphidr-rsinphidphi]

(95)
=[costhetasinphi -rsinthetasinphi rcosthetacosphi; sinthetasinphi rcosthetasinphi rsinthetacosphi; cosphi 0 -rsinphi][dr; dtheta; dphi].

(96)

Upon inversion, the result is

 [dr; dtheta; dphi]=[costhetasinphi sinthetasinphi cosphi; -(sintheta)/(rsinphi) (costheta)/(rsinphi) 0; (costhetacosphi)/r (sinthetacosphi)/r -(sinphi)/r][dx; dy; dz].

(97)

The Cartesian partial derivatives in spherical
coordinates are therefore

partial/(partialx)=costhetasinphipartial/(partialr)-(sintheta)/(rsinphi)partial/(partialtheta)+(costhetacosphi)/rpartial/(partialphi)

(98)
partial/(partialy)=sinthetasinphipartial/(partialr)+(costheta)/(rsinphi)partial/(partialtheta)+(sinthetacosphi)/rpartial/(partialphi)

(99)
partial/(partialz)=cosphipartial/(partialr)-(sinphi)/rpartial/(partialphi)

(100)

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

The Helmholtz differential equation
is separable in spherical coordinates.

SEE ALSO: Azimuth , Colatitude , Great Circle , Helmholtz
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 (r,theta,psi)."
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

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Weisstein, Eric W. "Spherical Coordinates."
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Spherical Coordinates

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EXPLORE THIS TOPIC IN the MathWorld Classroom

SphericalCoordinates

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 theta to be the azimuthal
angle in the xy– plane from
the x-axis with 0<=theta<2pi
(denoted lambda when referred to as the longitude ),
phi to be the polar
angle (also known as the zenith angle and colatitude , with phi=90 degrees-delta
where delta is the latitude )
from the positive z-axis with 0<=phi<=pi,
and r 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 r, theta, and phi, respectively.
Note that this definition provides a logical extension of the usual polar
coordinates notation, with theta remaining the
angle in the xy– plane
and phi 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 theta and phi are reversed
(both in meaning and in order listed) is also frequently used, especially in physics.
This is especially confusing since the identical notation (r,theta,phi) typically
means (radial, azimuthal, polar) to a mathematician but (radial, polar, azimuthal)
to a physicist. The symbol rho is sometimes
also used in place of r, theta instead of
theta, and phi and psi instead of phi. The following table summarizes a number of conventions
used by various authors. Extreme care is therefore needed when consulting the literature.

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

The spherical coordinates (r,theta,phi) are
related to the Cartesian coordinates (x,y,z) by

r=sqrt(x^2+y^2+z^2)

(1)
theta=tan^(-1)(y/x)

(2)
phi=cos^(-1)(z/r),

(3)

where r in [0,infty), theta in [0,2pi),
and phi in [0,pi], and the inverse
tangent must be suitably defined to take the correct quadrant of (x,y) into account.

In terms of Cartesian coordinates ,

x=rcosthetasinphi

(4)
y=rsinthetasinphi

(5)
z=rcosphi.

(6)

The scale factors are

h_r=1

(7)
h_theta=rsinphi
(8)
h_phi=r,

(9)

so the metric coefficients
are

g_(rr)=1
(10)
g_(thetatheta)=r^2sin^2phi
(11)
g_(phiphi)=r^2.
(12)

The line element is

 ds=drr^^+rdphiphi^^+rsinphidthetatheta^^,

(13)

the area element

 da=r^2sinphidthetadphi,

(14)

and the volume element

 dV=r^2sinphidphidthetadr.

(15)

The Jacobian is

 |(partial(x,y,z))/(partial(r,theta,phi))|=r^2sinphi.

(16)

The radius vector is

 r=[rcosthetasinphi; rsinthetasinphi; rcosphi],

(17)

so the unit vectors are

r^^=((dr)/(dr))/(|(dr)/(dr)|)

(18)
=[costhetasinphi; sinthetasinphi; cosphi]

(19)
theta^^=((dr)/(dtheta))/(|(dr)/(dtheta)|)

(20)
=[-sintheta; costheta; 0]

(21)
phi^^=((dr)/(dphi))/(|(dr)/(dphi)|)

(22)
=[costhetacosphi; sinthetacosphi; -sinphi].

(23)

Derivatives of the unit vectors are

(partialr^^)/(partialr)=0
(24)
(partialtheta^^)/(partialr)=0
(25)
(partialphi^^)/(partialr)=0
(26)
(partialr^^)/(partialtheta)=sinphitheta^^
(27)
(partialtheta^^)/(partialtheta)=-cosphiphi^^-sinphir^^

(28)
(partialphi^^)/(partialtheta)=cosphitheta^^
(29)
(partialr^^)/(partialphi)=phi^^
(30)
(partialtheta^^)/(partialphi)=0
(31)
(partialphi^^)/(partialphi)=-r^^.
(32)

The gradient is

 del =r^^partial/(partialr)+1/rphi^^partial/(partialphi)+1/(rsinphi)theta^^partial/(partialtheta),

(33)

and its components are

del _rr^^=0
(34)
del _thetar^^=1/rtheta^^
(35)
del _phir^^=1/rphi^^
(36)
del _rtheta^^=0
(37)
del _thetatheta^^=-(cotphi)/rphi^^-1/rr^^

(38)
del _phitheta^^=0
(39)
del _rphi^^=0
(40)
del _thetaphi^^=1/rcotphitheta^^

(41)
del _phiphi^^=-1/rr^^
(42)

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

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

Gamma^r=[0 0 0; 0 -1/r 0; 0 0 -1/r]

(43)
Gamma^theta=[0 1/r 0; 0 0 0; 0 (cotphi)/r 0]

(44)
Gamma^phi=[0 0 1/r; 0 -(cotphi)/r 0; 0 0 0]

(45)

(Misner et al. 1973, p. 213, who however use the notation convention (r,phi,theta)). The Christoffel
symbols of the second kind in the definition of Arfken (1985) are given by

Gamma^r=[0 0 0; 0 -rsin^2phi 0; 0 0 -r]

(46)
Gamma^theta=[0 1/r 0; 1/r 0 cotphi; 0 cotphi 0]

(47)
Gamma^phi=[0 0 1/r; 0 -sinphicosphi 0; 1/r 0 0]

(48)

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

The divergence is

 del ·F=partial/(partialr)A^r+2/rA^r+1/(rsinphi)partial/(partialtheta)A^theta+1/rpartial/(partialphi)A^phi+(cotphi)/rA^phi,

(49)

or, in vector notation,

del ·F=(2/r+partial/(partialr))F_r+(1/rpartial/(partialphi)+(cotphi)/r)F_phi+1/(rsinphi)(partialF_theta)/(partialtheta)

(50)
=1/(r^2)partial/(partialr)(r^2F_r)+1/(rsinphi)partial/(partialphi)(sinphiF_phi)+1/(rsinphi)(partialF_theta)/(partialtheta).

(51)

The covariant derivatives are given by

 A_(j;k)=1/(g_(kk))(partialA_j)/(partialx_k)-Gamma_(jk)^iA_i,

(52)

so

A_(r;r)=(partialA_r)/(partialr)

(53)
A_(r;theta)=1/(rsinphi)(partialA_r)/(partialphi)-(A_theta)/r

(54)
A_(r;phi)=1/r((partialA_r)/(partialphi)-A_phi)

(55)
A_(theta;r)=(partialA_theta)/(partialr)

(56)
A_(theta;theta)=1/(rsinphi)(partialA_theta)/(partialtheta)+(cotphi)/rA_phi+(A_r)/r

(57)
A_(theta;phi)=1/r(partialA_theta)/(partialr)-Gamma_(phir)^iA_i(partialA_theta)/(partialphi)

(58)
A_(phi;r)=(partialA_phi)/(partialr)-Gamma_(phir)^iA_i=(partialA_phi)/r

(59)
A_(phi;theta)=1/(rsinphi)(partialA_phi)/(partialtheta)-(cotphi)/rA_theta

(60)
A_(phi;phi)=1/r(partialA_phi)/(partialphi)+(A_r)/r.

(61)

The commutation coefficients are given
by

 c_(alphabeta)^mue^->_mu=[e^->_alpha,e^->_beta]=del _alphae^->_beta-del _betae^->_alpha

(62)
 [r^^,r^^]=[theta^^,theta^^]=[phi^^,phi^^]=0,

(63)

so c_(rr)^alpha=c_(thetatheta)^alpha=c_(phiphi)^alpha=0,
where alpha=r,theta,phi.

 [r^^,theta^^]=-[theta^^,r^^]=del _rtheta^^-del _thetar^^=0-1/rtheta^^=-1/rtheta^^,

(64)

so c_(rtheta)^theta=-c_(thetar)^theta=-1/r,
c_(rtheta)^r=c_(rtheta)^phi=0.

 [r^^,phi^^]=-[phi^^,r^^]=0-1/rphi^^=-1/rphi^^,

(65)

so c_(rphi)^phi=-c_(phir)^phi=1/r.

 [theta^^,phi^^]=-[phi^^,theta^^]=1/rcotphitheta^^-0=1/rcotphitheta^^,

(66)

so

 c_(thetaphi)^theta=-c_(phitheta)^theta=1/rcotphi.

(67)

Summarizing,

c^r=[0 0 0; 0 0 0; 0 0 0]

(68)
c^theta=[0 -1/r 0; 1/r 0 1/rcotphi; 0 -1/rcotphi 0]

(69)
c^phi=[0 0 -1/r; 0 0 0; 1/r 0 0].

(70)

Time derivatives of the radius vector are

r^.=[costhetasinphir^.-rsinthetasinphitheta^.+rcosthetacosphiphi^.; sinthetasinphir^.+rcosthetasinphitheta^.+rsinthetacosphiphi^.; cosphir^.-rsinphiphi^.]

(71)
=[costhetasinphi; sinthetasinphi; cosphi]r^.+rsinphi[-sintheta; costheta; 0]theta^.+r[costhetacosphi; sinthetacosphi; -sinphi]phi^.

(72)
=r^.r^^+rsinphitheta^.theta^^+rphi^.phi^^.

(73)

The speed is therefore given by

 v=|r^.|=sqrt(r^.^2+r^2sin^2phitheta^.^2+r^2phi^.^2).

(74)

The acceleration is

x^..=(-sinthetasinphitheta^.r^.+costhetacosphir^.phi^.+costhetasinphir^..)-(sinthetasinphir^.theta^.+rcosthetasinphitheta^.^2+rsinthetacosphitheta^.phi^.+rsinthetasinphitheta^..)+(costhetacosphir^.phi^.-rsinthetacostheta^.phi^.-rcosthetasinphiphi^.^2+rcosthetacosphiphi^..)

(75)
=-2sinthetasinphitheta^.r^.+2costhetacosphir^.phi^.-2rsinthetacosphitheta^.phi^.+costhetasinphir^..-rsinthetasinphitheta^..+rcosthetacosphiphi^..-rcosthetasinphi(theta^.^2+phi^.^2)

(76)
y^..=(sinthetasinphir^..+rcosthetasinphitheta^.+rcosphisinthetaphi^.)+(costhetasinphir^.theta^.-rsinthetasinphitheta^.^2+rcosthetacosphitheta^.phi^.+rcosthetasinphitheta^..)+(sinthetacosphir^.phi^.+rcosthetacosphitheta^.phi^.-rsinthetasinphiphi^.^2+rsinthetacosphiphi^..)

(77)
=2costhetasinphitheta^.r^.+2sinthetacosphir^.phi^.+2rcosthetacosphitheta^.phi^.+sinthetasinphir^..+rcosthetasinphitheta^..+rsinthetacosphiphi^..-rsinthetasinphi(theta^.^2+phi^.^2)

(78)
z^..=(cosphir^..-sinphir^.phi^.)-(r^.sinphiphi^.+rcosphiphi^.^2+rsinphiphi^..)

(79)
=-rcosphiphi^.^2+cosphir^..-2sinphiphi^.r^.-rsinphiphi^...

(80)

Plugging these in gives

r^..=(r^..-rphi^.^2)[costhetasinphi; sinthetasinphi; cosphi]+(2rcosphitheta^.phi^.+2sinphitheta^.r^.+rsinphitheta^..)[-sintheta; costheta; 0]+(2r^.phi^.+rphi^..)[costhetacosphi; sinthetacosphi; -sinphi]-rsinphitheta^.^2[costheta; sintheta; 0],

(81)

but

sinphir^^+cosphiphi^^=[costhetasin^2phi+costhetacos^2phi; sinthetasin^2phi+sinthetacos^2phi; 0]

(82)
=[costheta; sintheta; 0],

(83)

so

r^..=(r^..-rphi^.^2)r^^+(2rcosphitheta^.phi^.+2sinphitheta^.r^.+rsinphitheta^..)theta^^+(2r^.phi^.+rphi^..)phi^^-rsinphitheta^.^2(sinphir^^+cosphiphi^^)

(84)
=(r^..-rphi^.^2-rsin^2phitheta^.^2)r^^+(2sinphitheta^.r^.+2rcosphitheta^.phi^.+rsinphitheta^..)theta^^+(2r^.phi^.+rphi^..-rsinphicosphitheta^.^2)phi^^.

(85)

Time derivatives of the unit
vectors are

r^^^.=sinphitheta^.theta^^+phi^.phi^^

(86)
theta^^^.=-theta^.(sinphir^^+cosphiphi^^)

(87)
phi^^^.=-phi^.r^^+cosphitheta^.theta^^.

(88)

The curl is

 del ×F=1/(rsinphi)[partial/(partialphi)(sinphiF_theta)-(partialF_phi)/(partialtheta)]r^^+1/r[1/(sinphi)(partialF_r)/(partialtheta)-partial/(partialr)(rF_theta)]phi^^+1/r[partial/(partialr)(rF_phi)-(partialF_r)/(partialphi)]theta^^.

(89)

The Laplacian is

del ^2=1/(r^2)partial/(partialr)(r^2partial/(partialr))+1/(r^2sin^2phi)(partial^2)/(partialtheta^2)+1/(r^2sinphi)partial/(partialphi)(sinphipartial/(partialphi))

(90)
=1/(r^2)(r^2(partial^2)/(partialr^2)+2rpartial/(partialr))+1/(r^2sin^2phi)(partial^2)/(partialtheta^2)+1/(r^2sinphi)(cosphipartial/(partialphi)+sinphi(partial^2)/(partialphi^2))

(91)
=(partial^2)/(partialr^2)+2/rpartial/(partialr)+1/(r^2sin^2phi)(partial^2)/(partialtheta^2)+(cosphi)/(r^2sinphi)partial/(partialphi)+1/(r^2)(partial^2)/(partialphi^2).

(92)

The vector Laplacian in spherical coordinates
is given by

 del ^2v=[1/r(partial^2(rv_r))/(partialr^2)+1/(r^2)(partial^2v_r)/(partialtheta^2)+1/(r^2sin^2theta)(partial^2v_r)/(partialphi^2)+(cottheta)/(r^2)(partialv_r)/(partialtheta)-2/(r^2)(partialv_theta)/(partialtheta)-2/(r^2sintheta)(partialv_phi)/(partialphi)-(2v_r)/(r^2)-(2cottheta)/(r^2)v_theta
1/r(partial^2(rv_(theta)))/(partialr^2)+1/(r^2)(partial^2v_(theta))/(partialtheta^2)+1/(r^2sin^2theta)(partial^2v_(theta))/(partialphi^2)+(cottheta)/(r^2)(partialv_(theta))/(partialtheta)-2/(r^2)(cottheta)/(sintheta)(partialv_(phi))/(partialphi)+2/(r^2)(partialv_r)/(partialtheta)-(v_(theta))/(r^2sin^2theta)
1/r(partial^2(rv_(phi)))/(partialr^2)+1/(r^2)(partial^2v_(phi))/(partialtheta^2)+1/(r^2sin^2theta)(partial^2v_(phi))/(partialphi^2)+(cottheta)/(r^2)(partialv_(phi))/(partialtheta)+2/(r^2sintheta)(partialv_r)/(partialphi)+(2cottheta)/(r^2sintheta)(partialv_(theta))/(partialphi)-(v_(phi))/(r^2sin^2theta) ].

(93)

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

[x; y; z]=[rcosthetasinphi; rsinthetasinphi; rcosphi]

(94)
[dx; dy; dz]=[costhetasinphidr-rsinthetasinphidtheta+rcosthetacosphidphi; sinthetasinphidr+rsinphicosthetadtheta+rsinthetacosphidphi; cosphidr-rsinphidphi]

(95)
=[costhetasinphi -rsinthetasinphi rcosthetacosphi; sinthetasinphi rcosthetasinphi rsinthetacosphi; cosphi 0 -rsinphi][dr; dtheta; dphi].

(96)

Upon inversion, the result is

 [dr; dtheta; dphi]=[costhetasinphi sinthetasinphi cosphi; -(sintheta)/(rsinphi) (costheta)/(rsinphi) 0; (costhetacosphi)/r (sinthetacosphi)/r -(sinphi)/r][dx; dy; dz].

(97)

The Cartesian partial derivatives in spherical
coordinates are therefore

partial/(partialx)=costhetasinphipartial/(partialr)-(sintheta)/(rsinphi)partial/(partialtheta)+(costhetacosphi)/rpartial/(partialphi)

(98)
partial/(partialy)=sinthetasinphipartial/(partialr)+(costheta)/(rsinphi)partial/(partialtheta)+(sinthetacosphi)/rpartial/(partialphi)

(99)
partial/(partialz)=cosphipartial/(partialr)-(sinphi)/rpartial/(partialphi)

(100)

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

The Helmholtz differential equation
is separable in spherical coordinates.

SEE ALSO: Azimuth , Colatitude , Great Circle , Helmholtz
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 (r,theta,psi)."
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

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Spherical Coordinates

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SphericalCoordinates

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 theta to be the azimuthal
angle in the xy– plane from
the x-axis with 0<=theta<2pi
(denoted lambda when referred to as the longitude ),
phi to be the polar
angle (also known as the zenith angle and colatitude , with phi=90 degrees-delta
where delta is the latitude )
from the positive z-axis with 0<=phi<=pi,
and r 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 r, theta, and phi, respectively.
Note that this definition provides a logical extension of the usual polar
coordinates notation, with theta remaining the
angle in the xy– plane
and phi 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 theta and phi are reversed
(both in meaning and in order listed) is also frequently used, especially in physics.
This is especially confusing since the identical notation (r,theta,phi) typically
means (radial, azimuthal, polar) to a mathematician but (radial, polar, azimuthal)
to a physicist. The symbol rho is sometimes
also used in place of r, theta instead of
theta, and phi and psi instead of phi. The following table summarizes a number of conventions
used by various authors. Extreme care is therefore needed when consulting the literature.

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

The spherical coordinates (r,theta,phi) are
related to the Cartesian coordinates (x,y,z) by

r=sqrt(x^2+y^2+z^2)

(1)
theta=tan^(-1)(y/x)

(2)
phi=cos^(-1)(z/r),

(3)

where r in [0,infty), theta in [0,2pi),
and phi in [0,pi], and the inverse
tangent must be suitably defined to take the correct quadrant of (x,y) into account.

In terms of Cartesian coordinates ,

x=rcosthetasinphi

(4)
y=rsinthetasinphi

(5)
z=rcosphi.

(6)

The scale factors are

h_r=1

(7)
h_theta=rsinphi
(8)
h_phi=r,

(9)

so the metric coefficients
are

g_(rr)=1
(10)
g_(thetatheta)=r^2sin^2phi
(11)
g_(phiphi)=r^2.
(12)

The line element is

 ds=drr^^+rdphiphi^^+rsinphidthetatheta^^,

(13)

the area element

 da=r^2sinphidthetadphi,

(14)

and the volume element

 dV=r^2sinphidphidthetadr.

(15)

The Jacobian is

 |(partial(x,y,z))/(partial(r,theta,phi))|=r^2sinphi.

(16)

The radius vector is

 r=[rcosthetasinphi; rsinthetasinphi; rcosphi],

(17)

so the unit vectors are

r^^=((dr)/(dr))/(|(dr)/(dr)|)

(18)
=[costhetasinphi; sinthetasinphi; cosphi]

(19)
theta^^=((dr)/(dtheta))/(|(dr)/(dtheta)|)

(20)
=[-sintheta; costheta; 0]

(21)
phi^^=((dr)/(dphi))/(|(dr)/(dphi)|)

(22)
=[costhetacosphi; sinthetacosphi; -sinphi].

(23)

Derivatives of the unit vectors are

(partialr^^)/(partialr)=0
(24)
(partialtheta^^)/(partialr)=0
(25)
(partialphi^^)/(partialr)=0
(26)
(partialr^^)/(partialtheta)=sinphitheta^^
(27)
(partialtheta^^)/(partialtheta)=-cosphiphi^^-sinphir^^

(28)
(partialphi^^)/(partialtheta)=cosphitheta^^
(29)
(partialr^^)/(partialphi)=phi^^
(30)
(partialtheta^^)/(partialphi)=0
(31)
(partialphi^^)/(partialphi)=-r^^.
(32)

The gradient is

 del =r^^partial/(partialr)+1/rphi^^partial/(partialphi)+1/(rsinphi)theta^^partial/(partialtheta),

(33)

and its components are

del _rr^^=0
(34)
del _thetar^^=1/rtheta^^
(35)
del _phir^^=1/rphi^^
(36)
del _rtheta^^=0
(37)
del _thetatheta^^=-(cotphi)/rphi^^-1/rr^^

(38)
del _phitheta^^=0
(39)
del _rphi^^=0
(40)
del _thetaphi^^=1/rcotphitheta^^

(41)
del _phiphi^^=-1/rr^^
(42)

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

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

Gamma^r=[0 0 0; 0 -1/r 0; 0 0 -1/r]

(43)
Gamma^theta=[0 1/r 0; 0 0 0; 0 (cotphi)/r 0]

(44)
Gamma^phi=[0 0 1/r; 0 -(cotphi)/r 0; 0 0 0]

(45)

(Misner et al. 1973, p. 213, who however use the notation convention (r,phi,theta)). The Christoffel
symbols of the second kind in the definition of Arfken (1985) are given by

Gamma^r=[0 0 0; 0 -rsin^2phi 0; 0 0 -r]

(46)
Gamma^theta=[0 1/r 0; 1/r 0 cotphi; 0 cotphi 0]

(47)
Gamma^phi=[0 0 1/r; 0 -sinphicosphi 0; 1/r 0 0]

(48)

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

The divergence is

 del ·F=partial/(partialr)A^r+2/rA^r+1/(rsinphi)partial/(partialtheta)A^theta+1/rpartial/(partialphi)A^phi+(cotphi)/rA^phi,

(49)

or, in vector notation,

del ·F=(2/r+partial/(partialr))F_r+(1/rpartial/(partialphi)+(cotphi)/r)F_phi+1/(rsinphi)(partialF_theta)/(partialtheta)

(50)
=1/(r^2)partial/(partialr)(r^2F_r)+1/(rsinphi)partial/(partialphi)(sinphiF_phi)+1/(rsinphi)(partialF_theta)/(partialtheta).

(51)

The covariant derivatives are given by

 A_(j;k)=1/(g_(kk))(partialA_j)/(partialx_k)-Gamma_(jk)^iA_i,

(52)

so

A_(r;r)=(partialA_r)/(partialr)

(53)
A_(r;theta)=1/(rsinphi)(partialA_r)/(partialphi)-(A_theta)/r

(54)
A_(r;phi)=1/r((partialA_r)/(partialphi)-A_phi)

(55)
A_(theta;r)=(partialA_theta)/(partialr)

(56)
A_(theta;theta)=1/(rsinphi)(partialA_theta)/(partialtheta)+(cotphi)/rA_phi+(A_r)/r

(57)
A_(theta;phi)=1/r(partialA_theta)/(partialr)-Gamma_(phir)^iA_i(partialA_theta)/(partialphi)

(58)
A_(phi;r)=(partialA_phi)/(partialr)-Gamma_(phir)^iA_i=(partialA_phi)/r

(59)
A_(phi;theta)=1/(rsinphi)(partialA_phi)/(partialtheta)-(cotphi)/rA_theta

(60)
A_(phi;phi)=1/r(partialA_phi)/(partialphi)+(A_r)/r.

(61)

The commutation coefficients are given
by

 c_(alphabeta)^mue^->_mu=[e^->_alpha,e^->_beta]=del _alphae^->_beta-del _betae^->_alpha

(62)
 [r^^,r^^]=[theta^^,theta^^]=[phi^^,phi^^]=0,

(63)

so c_(rr)^alpha=c_(thetatheta)^alpha=c_(phiphi)^alpha=0,
where alpha=r,theta,phi.

 [r^^,theta^^]=-[theta^^,r^^]=del _rtheta^^-del _thetar^^=0-1/rtheta^^=-1/rtheta^^,

(64)

so c_(rtheta)^theta=-c_(thetar)^theta=-1/r,
c_(rtheta)^r=c_(rtheta)^phi=0.

 [r^^,phi^^]=-[phi^^,r^^]=0-1/rphi^^=-1/rphi^^,

(65)

so c_(rphi)^phi=-c_(phir)^phi=1/r.

 [theta^^,phi^^]=-[phi^^,theta^^]=1/rcotphitheta^^-0=1/rcotphitheta^^,

(66)

so

 c_(thetaphi)^theta=-c_(phitheta)^theta=1/rcotphi.

(67)

Summarizing,

c^r=[0 0 0; 0 0 0; 0 0 0]

(68)
c^theta=[0 -1/r 0; 1/r 0 1/rcotphi; 0 -1/rcotphi 0]

(69)
c^phi=[0 0 -1/r; 0 0 0; 1/r 0 0].

(70)

Time derivatives of the radius vector are

r^.=[costhetasinphir^.-rsinthetasinphitheta^.+rcosthetacosphiphi^.; sinthetasinphir^.+rcosthetasinphitheta^.+rsinthetacosphiphi^.; cosphir^.-rsinphiphi^.]

(71)
=[costhetasinphi; sinthetasinphi; cosphi]r^.+rsinphi[-sintheta; costheta; 0]theta^.+r[costhetacosphi; sinthetacosphi; -sinphi]phi^.

(72)
=r^.r^^+rsinphitheta^.theta^^+rphi^.phi^^.

(73)

The speed is therefore given by

 v=|r^.|=sqrt(r^.^2+r^2sin^2phitheta^.^2+r^2phi^.^2).

(74)

The acceleration is

x^..=(-sinthetasinphitheta^.r^.+costhetacosphir^.phi^.+costhetasinphir^..)-(sinthetasinphir^.theta^.+rcosthetasinphitheta^.^2+rsinthetacosphitheta^.phi^.+rsinthetasinphitheta^..)+(costhetacosphir^.phi^.-rsinthetacostheta^.phi^.-rcosthetasinphiphi^.^2+rcosthetacosphiphi^..)

(75)
=-2sinthetasinphitheta^.r^.+2costhetacosphir^.phi^.-2rsinthetacosphitheta^.phi^.+costhetasinphir^..-rsinthetasinphitheta^..+rcosthetacosphiphi^..-rcosthetasinphi(theta^.^2+phi^.^2)

(76)
y^..=(sinthetasinphir^..+rcosthetasinphitheta^.+rcosphisinthetaphi^.)+(costhetasinphir^.theta^.-rsinthetasinphitheta^.^2+rcosthetacosphitheta^.phi^.+rcosthetasinphitheta^..)+(sinthetacosphir^.phi^.+rcosthetacosphitheta^.phi^.-rsinthetasinphiphi^.^2+rsinthetacosphiphi^..)

(77)
=2costhetasinphitheta^.r^.+2sinthetacosphir^.phi^.+2rcosthetacosphitheta^.phi^.+sinthetasinphir^..+rcosthetasinphitheta^..+rsinthetacosphiphi^..-rsinthetasinphi(theta^.^2+phi^.^2)

(78)
z^..=(cosphir^..-sinphir^.phi^.)-(r^.sinphiphi^.+rcosphiphi^.^2+rsinphiphi^..)

(79)
=-rcosphiphi^.^2+cosphir^..-2sinphiphi^.r^.-rsinphiphi^...

(80)

Plugging these in gives

r^..=(r^..-rphi^.^2)[costhetasinphi; sinthetasinphi; cosphi]+(2rcosphitheta^.phi^.+2sinphitheta^.r^.+rsinphitheta^..)[-sintheta; costheta; 0]+(2r^.phi^.+rphi^..)[costhetacosphi; sinthetacosphi; -sinphi]-rsinphitheta^.^2[costheta; sintheta; 0],

(81)

but

sinphir^^+cosphiphi^^=[costhetasin^2phi+costhetacos^2phi; sinthetasin^2phi+sinthetacos^2phi; 0]

(82)
=[costheta; sintheta; 0],

(83)

so

r^..=(r^..-rphi^.^2)r^^+(2rcosphitheta^.phi^.+2sinphitheta^.r^.+rsinphitheta^..)theta^^+(2r^.phi^.+rphi^..)phi^^-rsinphitheta^.^2(sinphir^^+cosphiphi^^)

(84)
=(r^..-rphi^.^2-rsin^2phitheta^.^2)r^^+(2sinphitheta^.r^.+2rcosphitheta^.phi^.+rsinphitheta^..)theta^^+(2r^.phi^.+rphi^..-rsinphicosphitheta^.^2)phi^^.

(85)

Time derivatives of the unit
vectors are

r^^^.=sinphitheta^.theta^^+phi^.phi^^

(86)
theta^^^.=-theta^.(sinphir^^+cosphiphi^^)

(87)
phi^^^.=-phi^.r^^+cosphitheta^.theta^^.

(88)

The curl is

 del ×F=1/(rsinphi)[partial/(partialphi)(sinphiF_theta)-(partialF_phi)/(partialtheta)]r^^+1/r[1/(sinphi)(partialF_r)/(partialtheta)-partial/(partialr)(rF_theta)]phi^^+1/r[partial/(partialr)(rF_phi)-(partialF_r)/(partialphi)]theta^^.

(89)

The Laplacian is

del ^2=1/(r^2)partial/(partialr)(r^2partial/(partialr))+1/(r^2sin^2phi)(partial^2)/(partialtheta^2)+1/(r^2sinphi)partial/(partialphi)(sinphipartial/(partialphi))

(90)
=1/(r^2)(r^2(partial^2)/(partialr^2)+2rpartial/(partialr))+1/(r^2sin^2phi)(partial^2)/(partialtheta^2)+1/(r^2sinphi)(cosphipartial/(partialphi)+sinphi(partial^2)/(partialphi^2))

(91)
=(partial^2)/(partialr^2)+2/rpartial/(partialr)+1/(r^2sin^2phi)(partial^2)/(partialtheta^2)+(cosphi)/(r^2sinphi)partial/(partialphi)+1/(r^2)(partial^2)/(partialphi^2).

(92)

The vector Laplacian in spherical coordinates
is given by

 del ^2v=[1/r(partial^2(rv_r))/(partialr^2)+1/(r^2)(partial^2v_r)/(partialtheta^2)+1/(r^2sin^2theta)(partial^2v_r)/(partialphi^2)+(cottheta)/(r^2)(partialv_r)/(partialtheta)-2/(r^2)(partialv_theta)/(partialtheta)-2/(r^2sintheta)(partialv_phi)/(partialphi)-(2v_r)/(r^2)-(2cottheta)/(r^2)v_theta
1/r(partial^2(rv_(theta)))/(partialr^2)+1/(r^2)(partial^2v_(theta))/(partialtheta^2)+1/(r^2sin^2theta)(partial^2v_(theta))/(partialphi^2)+(cottheta)/(r^2)(partialv_(theta))/(partialtheta)-2/(r^2)(cottheta)/(sintheta)(partialv_(phi))/(partialphi)+2/(r^2)(partialv_r)/(partialtheta)-(v_(theta))/(r^2sin^2theta)
1/r(partial^2(rv_(phi)))/(partialr^2)+1/(r^2)(partial^2v_(phi))/(partialtheta^2)+1/(r^2sin^2theta)(partial^2v_(phi))/(partialphi^2)+(cottheta)/(r^2)(partialv_(phi))/(partialtheta)+2/(r^2sintheta)(partialv_r)/(partialphi)+(2cottheta)/(r^2sintheta)(partialv_(theta))/(partialphi)-(v_(phi))/(r^2sin^2theta) ].

(93)

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

[x; y; z]=[rcosthetasinphi; rsinthetasinphi; rcosphi]

(94)
[dx; dy; dz]=[costhetasinphidr-rsinthetasinphidtheta+rcosthetacosphidphi; sinthetasinphidr+rsinphicosthetadtheta+rsinthetacosphidphi; cosphidr-rsinphidphi]

(95)
=[costhetasinphi -rsinthetasinphi rcosthetacosphi; sinthetasinphi rcosthetasinphi rsinthetacosphi; cosphi 0 -rsinphi][dr; dtheta; dphi].

(96)

Upon inversion, the result is

 [dr; dtheta; dphi]=[costhetasinphi sinthetasinphi cosphi; -(sintheta)/(rsinphi) (costheta)/(rsinphi) 0; (costhetacosphi)/r (sinthetacosphi)/r -(sinphi)/r][dx; dy; dz].

(97)

The Cartesian partial derivatives in spherical
coordinates are therefore

partial/(partialx)=costhetasinphipartial/(partialr)-(sintheta)/(rsinphi)partial/(partialtheta)+(costhetacosphi)/rpartial/(partialphi)

(98)
partial/(partialy)=sinthetasinphipartial/(partialr)+(costheta)/(rsinphi)partial/(partialtheta)+(sinthetacosphi)/rpartial/(partialphi)

(99)
partial/(partialz)=cosphipartial/(partialr)-(sinphi)/rpartial/(partialphi)

(100)

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

The Helmholtz differential equation
is separable in spherical coordinates.

SEE ALSO: Azimuth , Colatitude , Great Circle , Helmholtz
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 (r,theta,psi)."
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."
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