释义 |
Cylindrical CoordinatesCylindrical 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 usedto 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 isused.
where , , and . In terms of , , and 
Morse and Feshbach (1953) define the cylindrical coordinates by
where and . The Metric elements of the cylindrical coordinates are
so the Scale Factors are
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,
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) |
so the Gradient components become
 |  |  | (33) |  |  |  | (34) |  |  |  | (35) |  |  |  | (36) |  |  |  | (37) |  |  |  | (38) |  |  |  | (39) |  |  |  | (40) |  |  |  | (41) |
Now, since the Connection Coefficients are defined by
 | (42) |
The Covariant Derivatives, given by
 | (46) |
are
 |  |  | (47) |  |  |  | | |  |  | (48) |  |  |  | (49) |  |  |  | (50) |  |  |  | | |  |  | (51) |  |  |  | (52) |  |  |  | (53) |  |  |  | (54) |  |  |  | (55) |
Cross Products of the coordinate axes are
The Commutation Coefficients are given by
 | (59) |
But
 | (60) |
so , where . Also
 | (61) |
so , . Finally,
 | (62) |
Summarizing,
Time Derivatives of the Vector are
Speed is given by
 | (68) |
Time derivatives of the unit Vectors are
Cross Products of the axes are
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,
so, as expected,
 | (84) |
We have already computed , so combining all three pieces gives
The Divergence is
or, in Vector notation
 | (87) |
The Cross Product is
and the Laplacian is
The vector Laplacian is
 | (90) |
The Helmholtz Differential Equation is separable in cylindrical coordinates and has StäckelDeterminant (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. |