“Curvilinear Coordinates”的概念、定义、习题解题方法及翻译-数学辞典

单词

Curvilinear Coordinates

释义

Curvilinear Coordinates

A general Metric has a Line Element

(1)

where Einstein Summation is being used. Curvilinear coordinates are defined as those with a diagonal Metricso that

(2)

where

is the Kronecker Delta. Curvilinear coordinatestherefore have a simple Line Element

(3)

which is just the Pythagorean Theorem, so the differential Vectoris

(4)

(5)

where the Scale Factors are

(6)

and

(7)

Equation (5) may therefore be re-expressed as

(8)

The Gradient is

(9)

the Divergence is

(10)

and the Curl is




			(11)

Orthogonal curvilinear coordinates satisfy the additional constraint that

(12)

Therefore, the Line Element is

(13)

and the Volume Element is





			(14)

where the latter is the Jacobian.

Orthogonal curvilinear coordinate systems include Bipolar Cylindrical Coordinates, Bispherical Coordinates,Cartesian Coordinates, Confocal Ellipsoidal Coordinates, Confocal Paraboloidal Coordinates,Conical Coordinates, Cyclidic Coordinates, Cylindrical Coordinates, Ellipsoidal Coordinates,Elliptic Cylindrical Coordinates, Oblate Spheroidal Coordinates, Parabolic Coordinates,Parabolic Cylindrical Coordinates, Paraboloidal Coordinates, Polar Coordinates, ProlateSpheroidal Coordinates, Spherical Coordinates, and Toroidal Coordinates. These are degenerate cases of theConfocal Ellipsoidal Coordinates.

See also Change of Variables Theorem, Curl, Divergence, Gradient, Jacobian,Laplacian

References

Arfken, G. ``Curvilinear Coordinates'' and ``Differential Vector Operators.'' §2.1 and 2.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 86-90 and 90-94, 1985.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, pp. 1084-1088, 1980.

Morse, P. M. and Feshbach, H. ``Curvilinear Coordinates'' and ``Table of Properties of Curvilinear Coordinates.'' §1.3 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 21-31 and 115-117, 1953.

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