Curvilinear Coordinates

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

			(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)