Cylindrical Coordinates (original) (raw)

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CylindricalCoordinates

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 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 is used.

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

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

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 ,, and

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

where xi_1=r and xi_2=costheta .

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

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

A Cartesian vectoris given in cylindrical coordinates by

	(19)

To find the unit vectors,

Derivatives of unit vectors with respect to the coordinates are

The gradient operator in cylindrical coordinates is given by

	(32)

so the gradient components become

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

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

(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

Cross products of the coordinate axes are

The commutation coefficients are given by

	(61)

But

	(62)

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

	(63)

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

	(64)

Summarizing,

Time derivatives of the vectorare

Speed is given by

Time derivatives of the unit vectors are

The convective derivative is

To rewrite this, use the identity

	(83)

and set A=B , to obtain

	(84)

	(85)

Then

The curl in the above expression gives

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

Examining this term by term,

so, as expected,

	(108)

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

The divergence is

or, in vector notation

	(116)

The curl is

The scalar Laplacian is

The vector Laplacian is

	(120)

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

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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

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

Cylindrical Coordinates (original) (raw)

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