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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 g_(munu) has a line element

 ds^2=g_(munu)du^mudu^nu,
(1)

where Einstein summation is being used. Orthogonal coordinates are defined as those with a diagonal metric so that

 g_(munu)=delta_nu^muh_mu^2,
(2)

where delta_nu^mu is the Kronecker delta and h_mu is a so-called scale factor. Orthogonal curvilinear coordinates therefore have a simple line element

ds^2=delta_nu^muh_mu^2du^mudu^nu
(3)
=h_mu^2(du^mu)^2,
(4)

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

 dr=h_mudu_muu_mu^^,
(5)

or

 dr=(partialr)/(partialu_1)du_1+(partialr)/(partialu_2)du_2+(partialr)/(partialu_3)du_3,
(6)

where the scale factors are

 h_i=|(partialr)/(partialu_i)|
(7)

and

u_i^^=((partialr)/(partialu_i))/(|(partialr)/(partialu_i)|)
(8)
=1/(h_i)(partialr)/(partialu_i).
(9)

Equation (◇) may therefore be re-expressed as

 dr=h_1du_1u_1^^+h_2du_2u_2^^+h_3du_3u_3^^.
(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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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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