Cylindrical Coordinates

Cylindrical coordinates are a generalization of 2-D Polar Coordinates to 3-D 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.

 (1) (2) (3)

where , , and . In terms of , , and
 (4) (5) (6)

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 of a Vector Field in cylindrical coordinates is given by
 (32)

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

Now, since the Connection Coefficients are defined by
 (42)

 (43) (44) (45)

The Covariant Derivatives, given by
 (46)

are
 (47) (48) (49) (50) (51) (52) (53) (54) (55)

Cross Products of the coordinate axes are
 (56) (57) (58)

The Commutation Coefficients are given by
 (59)

But
 (60)

so , where . Also
 (61)

so , . Finally,
 (62)

Summarizing,
 (63) (64) (65)

Time Derivatives of the Vector are

 (66) (67)

Speed is given by

 (68)

Time derivatives of the unit Vectors are
 (69)

Cross Products of the axes are
 (70) (71) (72)

The Convective Derivative is
 (73)

To rewrite this, use the identity
 (74)

and set , to obtain
 (75)

so
 (76)

Then
 (77)

The Curl in the above expression gives
 (78)

so

 (79)

We expect the gradient term to vanish since Speed does not depend on position. Check this using the identity ,
 (80)

Examining this term by term,
 (81) (82) (83)

so, as expected,
 (84)

We have already computed , so combining all three pieces gives
 (85)

The Divergence is
 (86)

or, in Vector notation
 (87)

The Cross Product is
 (88)

and the Laplacian is
 (89)

The vector Laplacian is

 (90)

The Helmholtz Differential Equation is separable in cylindrical coordinates and has Stäckel Determinant (for , , ) or (for Morse and Feshbach's , , ).

See also Elliptic Cylindrical Coordinates, Helmholtz Differential Equation--Circular Cylindrical Coordinates, Polar Coordinates, Spherical Coordinates

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, p. 212, 1987.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 657, 1953.