A coordinate system composed of intersecting surfaces. If the intersections are all at right angles,
then the curvilinear coordinates are said to form an orthogonal
coordinate system. If not, they form a skew
coordinate system.
A general metric has a line element
|
(1)
|
where Einstein summation is being used. Orthogonal coordinates are defined as those with a diagonal metric
so that
|
(2)
|
where
is the Kronecker delta and is a so-called scale factor.
Orthogonal curvilinear coordinates therefore have a simple line
element
which is just the Pythagorean theorem, so
the differential vector is
|
(5)
|
or
|
(6)
|
where the scale factors are
|
(7)
|
and
Equation (◇) may therefore be re-expressed as
|
(10)
|
See also
Curve,
Divergence,
Gradient,
Metric,
Line
Element,
Orthogonal Coordinate System,
Scale Factor,
Skew
Coordinate System
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References
Byerly, W. E. "Orthogonal Curvilinear Coördinates." §130 in An
Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal
Harmonics, with Applications to Problems in Mathematical Physics. New York:
Dover, pp. 238-239, 1959.Moon, P. and Spencer, D. E. Foundations
of Electrodynamics. Princeton, NJ: Van Nostrand, 1960.Moon,
P. and Spencer, D. E. Field
Theory Handbook, Including Coordinate Systems, Differential Equations, and Their
Solutions, 2nd ed. New York: Springer-Verlag, pp. 1-3, 1988.Referenced
on Wolfram|Alpha
Curvilinear Coordinates
Cite this as:
Weisstein, Eric W. "Curvilinear Coordinates."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CurvilinearCoordinates.html
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