Curvilinear Coordinates -- from Wolfram MathWorld
- ️Weisstein, Eric W.
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.
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Cite this as:
Weisstein, Eric W. "Curvilinear Coordinates." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CurvilinearCoordinates.html