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Cylindrical Coordinates

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

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

(radial, azimuthal, vertical)	reference
	this work, Beyer (1987, p. 212)
(`Rr`, `Ttheta`, `Zz`)	`SetCoordinates`[`Cylindrical`] in the Wolfram Language package VectorAnalysis`
	Arfken (1985, p. 95)
	Moon and Spencer (1988, p. 12)
	Korn and Korn (1968, p. 60)
	Morse and Feshbach (1953)

In terms of the Cartesian coordinates ,

			(1)
			(2)
			(3)

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

In terms of , , and

			(4)
			(5)
			(6)

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

			(7)
			(8)
			(9)

where and .

The metric elements of the cylindrical coordinates are

			(10)
			(11)
			(12)

so the scale factors are

			(13)
			(14)
			(15)

The line element is

(16)

and the volume element is

(17)

The Jacobian is

(18)

A Cartesian vector is given in cylindrical coordinates by

(19)

To find the unit vectors,

			(20)
			(21)
			(22)

Derivatives of unit vectors with respect to the coordinates are

			(23)
			(24)
			(25)
			(26)
			(27)
			(28)
			(29)
			(30)
			(31)

The gradient operator in cylindrical coordinates is given by

(32)

so the gradient components become

			(33)
			(34)
			(35)
			(36)
			(37)
			(38)
			(39)
			(40)
			(41)

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

			(42)
			(43)
			(44)

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

			(45)
			(46)
			(47)

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

The covariant derivatives are then given by

(48)

are

			(49)
			(50)
			(51)
			(52)
			(53)
			(54)
			(55)
			(56)
			(57)

Cross products of the coordinate axes are

			(58)
			(59)
			(60)

The commutation coefficients are given by

(61)

But

(62)

so , where . Also

(63)

so , . Finally,

(64)

Summarizing,

			(65)
			(66)
			(67)

Time derivatives of the vector are

			(68)
			(69)
			(70)
			(71)
			(72)

Speed is given by

			(73)
			(74)

Time derivatives of the unit vectors are

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

The convective derivative is

			(81)
			(82)

To rewrite this, use the identity

(83)

and set , to obtain

(84)

(85)

Then

			(86)
			(87)

The curl in the above expression gives

			(88)
			(89)

			(90)
			(91)
			(92)

We expect the gradient term to vanish since speed does not depend on position. Check this using the identity ,

			(93)
			(94)

Examining this term by term,

			(95)
			(96)
			(97)
			(98)
			(99)
			(100)
			(101)
			(102)
			(103)
			(104)
			(105)
			(106)
			(107)

so, as expected,

(108)

We have already computed , so combining all three pieces gives

			(109)
			(110)
			(111)

The divergence is

			(112)
			(113)
			(114)
			(115)

or, in vector notation

(116)

The curl is

(117)

The scalar Laplacian is

			(118)
			(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 (for , , ) or (for Morse and Feshbach's , , and ).

Cylindrical Coordinates

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