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Cylindrical Coordinates

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CylindricalCoordinates

Cylindrical coordinates are a generalization of two-dimensional polar coordinates to three dimensions by superposing a height (z) axis. Unfortunately, there are a number of different notations used for the other two coordinates. Either r or rho is used to refer to the radial coordinate and either phi or theta to the azimuthal coordinates. Arfken (1985), for instance, uses (rho,phi,z), while Beyer (1987) uses (r,theta,z). In this work, the notation (r,theta,z) is used.

The following table summarizes notational conventions used by a number of authors.

(radial, azimuthal, vertical)reference
(r,theta,z)this work, Beyer (1987, p. 212)
(Rr, Ttheta, Zz)SetCoordinates[Cylindrical] in the Wolfram Language package VectorAnalysis`
(rho,phi,z)Arfken (1985, p. 95)
(r,psi,z)Moon and Spencer (1988, p. 12)
(r^',phi,z)Korn and Korn (1968, p. 60)
(xi_1,xi_2,xi_3)Morse and Feshbach (1953)

In terms of the Cartesian coordinates (x,y,z),

r=sqrt(x^2+y^2)
(1)
theta=tan^(-1)(y/x)
(2)
z=z,
(3)

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

In terms of x, y, and z

x=rcostheta
(4)
y=rsintheta
(5)
z=z.
(6)

Note that Morse and Feshbach (1953) define the cylindrical coordinates by

x=xi_1xi_2
(7)
y=xi_1sqrt(1-xi_2^2)
(8)
z=xi_3,
(9)

where xi_1=r and xi_2=costheta.

The metric elements of the cylindrical coordinates are

g_(rr)=1
(10)
g_(thetatheta)=r^2
(11)
g_(zz)=1,
(12)

so the scale factors are

g_r=1
(13)
g_theta=r
(14)
g_z=1.
(15)

The line element is

ds=drr^^+rdthetatheta^^+dzz^^,
(16)

and the volume element is

dV=rdrdthetadz.
(17)

The Jacobian is

|(partial(x,y,z))/(partial(r,theta,z))|=r.
(18)

A Cartesian vector is given in cylindrical coordinates by

r=[rcostheta; rsintheta; z].
(19)

To find the unit vectors,

r^^=((dr)/(dr))/(|(dr)/(dr)|)=[costheta; sintheta; 0]
(20)
theta^^=((dr)/(dtheta))/(|(dr)/(dtheta)|)=[-sintheta; costheta; 0]
(21)
z^^=((dr)/(dz))/(|(dr)/(dz)|)=[0; 0; 1].
(22)

Derivatives of unit vectors with respect to the coordinates are

(partialr^^)/(partialr)=0
(23)
(partialr^^)/(partialtheta)=theta^^
(24)
(partialr^^)/(partialz)=0
(25)
(partialtheta^^)/(partialr)=0
(26)
(partialtheta^^)/(partialtheta)=-r^^
(27)
(partialtheta^^)/(partialz)=0
(28)
(partialz^^)/(partialr)=0
(29)
(partialz^^)/(partialtheta)=0
(30)
(partialz^^)/(partialz)=0.
(31)

The gradient operator in cylindrical coordinates is given by

del =r^^partial/(partialr)+theta^^1/rpartial/(partialtheta)+z^^partial/(partialz),
(32)

so the gradient components become

del _rr^^=0
(33)
del _thetar^^=1/rtheta^^
(34)
del _zr^^=0
(35)
del _rtheta^^=0
(36)
del _thetatheta^^=-1/rr^^
(37)
del _ztheta^^=0
(38)
del _rz^^=0
(39)
del _thetaz^^=0
(40)
del _zz^^=0.
(41)

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 0]
(42)
Gamma^theta=[0 1/r 0; 0 0 0; 0 0 0]
(43)
Gamma^z=[0 0 0; 0 0 0; 0 0 0].
(44)

The Christoffel symbols of the second kind in the definition of Arfken (1985) are given by

Gamma^r=[0 0 0; 0 -r 0; 0 0 0]
(45)
Gamma^theta=[0 1/r 0; 1/r 0 0; 0 0 0]
(46)
Gamma^z=[0 0 0; 0 0 0; 0 0 0]
(47)

(Walton 1967; Arfken 1985, p. 164, Ex. 3.8.10; Moon and Spencer 1988, p. 12a).

The covariant derivatives are then given by

A_(j;k)=1/(g^(kk))(partialA_j)/(partialx_k)-Gamma_(jk)^iA_i,
(48)

are

A_(r;r)=(partialA_r)/(partialr)
(49)
A_(r;theta)=1/r(partialA_r)/(partialtheta)-(A_theta)/r
(50)
A_(r;z)=(partialA_r)/(partialz)
(51)
A_(theta;r)=(partialA_theta)/(partialr)
(52)
A_(theta;theta)=1/r(partialA_theta)/(partialtheta)+(A_r)/r
(53)
A_(theta;z)=(partialA_theta)/(partialz)
(54)
A_(z;r)=(partialA_z)/(partialr)
(55)
A_(z;theta)=1/r(partialA_z)/(partialtheta)
(56)
A_(z;z)=(partialA_z)/(partialz).
(57)

Cross products of the coordinate axes are

r^^xz^^=-theta^^
(58)
theta^^xz^^=r^^
(59)
r^^xtheta^^=z^^.
(60)

The commutation coefficients are given by

c_(alphabeta)^mue^->_mu=[e^->_alpha,e^->_beta]=del _alphae^->_beta-del _betae^->_alpha,
(61)

But

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

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

[r^^,theta^^]=-[theta^^,r^^]=del _rtheta^^-del _thetar^^=0-1/rtheta^^=-1/rtheta^^,
(63)

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

[r^^,phi^^]=[theta^^,phi^^]=0.
(64)

Summarizing,

c^r=[0 0 0; 0 0 0; 0 0 0]
(65)
c^theta=[0 -1/r 0; 1/r 0 0; 0 0 0]
(66)
c^phi=[0 0 0; 0 0 0; 0 0 0].
(67)

Time derivatives of the vector are

r^.=[costhetar^.-rsinthetatheta^.; sinthetar^.+rcosthetatheta^.; z^.]=r^.r^^+rtheta^.theta^^+z^.z^^
(68)
r^..=[-sinthetar^.theta^.+costhetar^..-sinthetar^.theta^.-rcosthetatheta^.^2-rsinthetatheta^..; costhetar^.theta^.+sinthetar^..+costhetar^.theta^.-rsinthetatheta^.^2+rcosthetatheta^..; z^..]
(69)
(70)
=[-2sinthetar^.theta^.+costhetar^..-rcosthetatheta^.^2-rsinthetatheta^..; 2costhetar^.theta^.+sinthetar^..-rsinthetatheta^.^2+rcosthetatheta^..; z^..]
(71)
=(r^..-rtheta^.^2)r^^+(2r^.theta^.+rtheta^..)theta^^+z^..z^^.
(72)

Speed is given by

v=|r^.|
(73)
=sqrt(r^.^2+r^2theta^.^2+z^.^2).
(74)

Time derivatives of the unit vectors are

r^^^.=[-sinthetatheta^.; costhetatheta^.; 0]
(75)
=theta^.theta^^
(76)
theta^^^.=[-costhetatheta^.; -sinthetatheta^.; 0]
(77)
=-theta^.r^^
(78)
z^^^.=[0; 0; 0]
(79)
=0.
(80)

The convective derivative is

(Dr^.)/(Dt)=(partial/(partialt)+r^.·del )r^.
(81)
=(partialr^.)/(partialt)+r^.·del r^..
(82)

To rewrite this, use the identity

del (A·B)=Ax(del xB)+Bx(del xA)+(A·del )B+(B·del )A
(83)

and set A=B, to obtain

del (A·A)=2Ax(del xA)+2(A·del )A,
(84)

so

(A·del )A=del (1/2A^2)-Ax(del xA).
(85)

Then

(Dr^.)/(Dt)=r^..+del (1/2r^.^2)-r^.x(del xr^.)
(86)
=r^..+(del xr^.)xr^.+del (1/2r^.^2).
(87)

The curl in the above expression gives

del xr^.=1/rpartial/(partialr)(r^2theta^.)z^^
(88)
=2theta^.z^^,
(89)

so

-r^.x(del xr^.)=-2theta^.(r^.r^^xz^^+rtheta^.theta^^xz^^)
(90)
=-2theta^.(-r^.theta^^+rtheta^.r^^)
(91)
=2r^.theta^.theta^^-2rtheta^.^2r^^.
(92)

We expect the gradient term to vanish since speed does not depend on position. Check this using the identity del (f^2)=2fdel f,

del (1/2r^.^2)=1/2del (r^.^2+r^2theta^.^2+z^.^2)
(93)
=r^.del r^.+rtheta^.del (rtheta^.)+z^.del z^..
(94)

Examining this term by term,

r^.del r^.=r^.partial/(partialt)del r
(95)
=r^.partial/(partialt)r^^
(96)
=r^.r^^^.
(97)
=r^.theta^.theta^^
(98)
rtheta^.del (rtheta^.)=rtheta^.[rpartial/(partialt)del theta+theta^.del r]
(99)
=rtheta^.[rpartial/(partialt)(1/rtheta^^)+theta^.r^^]
(100)
=rtheta^.[r(-1/(r^2)r^.theta^^+1/rtheta^^^.)+theta^.r^^]
(101)
=-theta^.r^.theta^^+rtheta^.(-theta^.r^^)+rtheta^.^2r^^
(102)
=-theta^.r^.theta^^
(103)
z^.del z^.=z^.partial/(partialt)del z
(104)
=z^.partial/(partialt)z^^
(105)
=z^.z^^^.
(106)
=0,
(107)

so, as expected,

del (1/2r^.^2)=0.
(108)

We have already computed r^.., so combining all three pieces gives

(Dr^.)/(Dt)=(r^..-rtheta^.^2-2rtheta^.^2)r^^+(2r^.theta^.+2r^.theta^.+rtheta^..)theta^^+z^..z^^
(109)
(110)
=(r^..-3rtheta^.^2)r^^+(4r^.theta^.+rtheta^..)theta^^+z^..z^^.
(111)

The divergence is

del ·A=A_(;r)^r=A_(,r)^r+(Gamma_(rr)^rA^t+Gamma_(thetar)^rA^theta+Gamma_(zr)^rA^z)+A_(,theta)^theta+(Gamma_(rtheta)^thetaA^r+Gamma_(thetatheta)^thetaA^theta+Gamma_(ztheta)^thetaA^z)+A_(,z)^z+(Gamma_(rz)^zA^r+Gamma_(thetaz)^zA^theta+Gamma_(zz)^zA^z)
(112)
=A_(,r)^r+A_(,theta)^theta+A_(,z)^z+(0+0+0)+(1/r+0+0)+(0+0+0)
(113)
=1/(g_r)partial/(partialr)A^r+1/(g_theta)partial/(partialtheta)A^theta+1/(g_z)partial/(partialz)A^z+1/rA^r
(114)
=(partial/(partialr)+1/r)A^r+1/rpartial/(partialtheta)A^theta+partial/(partialz)A^z,
(115)

or, in vector notation

del ·F=1/rpartial/(partialr)(rF_r)+1/r(partialF_theta)/(partialtheta)+(partialF_z)/(partialz).
(116)

The curl is

del xF=(1/r(partialF_z)/(partialtheta)-(partialF_theta)/(partialz))r^^+((partialF_r)/(partialz)-(partialF_z)/(partialr))theta^^+1/r[partial/(partialr)(rF_theta)-(partialF_r)/(partialtheta)]z^^.
(117)

The scalar Laplacian is

del ^2f=1/rpartial/(partialr)(r(partialf)/(partialr))+1/(r^2)(partial^2f)/(partialtheta^2)+(partial^2f)/(partialz^2)
(118)
=(partial^2f)/(partialr^2)+1/r(partialf)/(partialr)+1/(r^2)(partial^2f)/(partialtheta^2)+(partial^2f)/(partialz^2).
(119)

The vector Laplacian is

del ^2v=[(partial^2v_r)/(partialr^2)+1/(r^2)(partial^2v_r)/(partialtheta^2)+(partial^2v_r)/(partialz^2)+1/r(partialv_r)/(partialr)-2/(r^2)(partialv_theta)/(partialtheta)-(v_r)/(r^2); (partial^2v_theta)/(partialr^2)+1/(r^2)(partial^2v_theta)/(partialtheta^2)+(partial^2v_theta)/(partialz^2)+1/r(partialv_theta)/(partialr)+2/(r^2)(partialv_r)/(partialtheta)-(v_theta)/(r^2); (partial^2v_z)/(partialr^2)+1/(r^2)(partial^2v_z)/(partialtheta^2)+(partial^2v_z)/(partialz^2)+1/r(partialv_z)/(partialr)].
(120)

The Helmholtz differential equation is separable in cylindrical coordinates and has Stäckel determinant S=1 (for r, theta, z) or S=1/(1-xi_2^2) (for Morse and Feshbach's xi_1, xi_2, and xi_3).

See also

Cartesian Coordinates, Elliptic Cylindrical Coordinates, Helmholtz Differential Equation--Circular Cylindrical Coordinates, Polar Coordinates, Spherical Coordinates

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References

Arfken, G. "Circular Cylindrical Coordinates." §2.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 95-101, 1985.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987.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: W. H. Freeman, 1973.Moon, P. and Spencer, D. E. "Circular-Cylinder Coordinates (r,psi,z)." Table 1.02 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 12-17, 1988.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 657, 1953.Walton, J. J. "Tensor Calculations on Computer: Appendix." Comm. ACM 10, 183-186, 1967.

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Cylindrical Coordinates

Cite this as:

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

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