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. Curvilinear coordinates are defined as those with a diagonal metric so that

(2)

where is the Kronecker delta and is a so-called scale factor. Curvilinear coordinates therefore have a simple line element

			(3)
			(4)

which is just the Pythagorean theorem, so the differential vector is

(5)

or

(6)

where the scale factors are

(7)

and

			(8)
			(9)

Equation (◇) may therefore be re-expressed as

(10)

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.

CITE THIS AS:

Weisstein, Eric W. "Curvilinear Coordinates." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CurvilinearCoordinates.html