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Spherical Coordinates


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)SphericalPlot3D in the Wolfram Language
(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 Explore this topic in the MathWorld classroom

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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.

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Spherical Coordinates

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

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