Orthogonal Coordinate System

An orthogonal coordinate system is a system of curvilinear coordinates in which each family of surfaces intersects the others at right angles. Orthogonal coordinates therefore satisfy the additional constraint that


where delta_(ij) is the Kronecker delta. Therefore, the line element becomes


and the volume element becomes

=|(partialx)/(partialu_1) (partialx)/(partialu_2) (partialx)/(partialu_3); (partialy)/(partialu_1) (partialy)/(partialu_2) (partialy)/(partialu_3); (partialz)/(partialu_1) (partialz)/(partialu_2) (partialz)/(partialu_3)|du_1du_2du_3

where the latter is the Jacobian.

The gradient of a function phi is given in orthogonal curvilinear coordinates by

grad(phi)=del phi

the divergence is

 div(F)=del ·F=1/(h_1h_2h_3)[partial/(partialu_1)(h_2h_3F_1)+partial/(partialu_2)(h_3h_1F_2)+partial/(partialu_3)(h_1h_2F_3)],

and the curl is

del xF=1/(h_1h_2h_3)|h_1u_1^^ h_2u_2^^ h_3u_3^^; partial/(partialu_1) partial/(partialu_2) partial/(partialu_3); h_1F_1 h_2F_2 h_3F_3|

For surfaces of first degree, the only three-dimensional coordinate system of surfaces having orthogonal intersections is Cartesian coordinates (Moon and Spencer 1988, p. 1). Including degenerate cases, there are 11 sets of quadratic surfaces having orthogonal coordinates. Furthermore, Laplace's equation and the Helmholtz differential equation are separable in all of these coordinate systems (Moon and Spencer 1988, p. 1).

Planar orthogonal curvilinear coordinate systems of degree two or less include two-dimensional Cartesian coordinates and polar coordinates.

Three-dimensional orthogonal curvilinear coordinate systems of degree two or less include bipolar cylindrical coordinates, bispherical coordinates, three-dimensional Cartesian coordinates, confocal ellipsoidal coordinates, confocal paraboloidal coordinates, conical coordinates, cyclidic coordinates, cylindrical coordinates, elliptic cylindrical coordinates, oblate spheroidal coordinates, parabolic coordinates, parabolic cylindrical coordinates, paraboloidal coordinates, prolate spheroidal coordinates, spherical coordinates, and toroidal coordinates. These are degenerate cases of the confocal ellipsoidal coordinates.

Orthogonal coordinate systems can also be built from fourth-order (in particular, cyclidic coordinates) and higher surfaces (Bôcher 1894), but are generally less important in solving physical problems than are quadratic surfaces (Moon and Spencer 1988, p. 1).

See also

Change of Variables Theorem, Curl, Curvilinear Coordinates, Cyclidic Coordinates, Divergence, Gradient, Jacobian, Laplacian, Skew Coordinate System

Explore with Wolfram|Alpha


Arfken, G. "Curvilinear Coordinates" and "Differential Vector Operators." §2.1 and 2.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 86-90 and 90-94, 1985.Bôcher, M. Über die Reihenentwicklungen der Potentialtheorie. Leipzig, Germany: Teubner, 1894.Darboux, G. Sur une classe remarquable de courbes et de surfaces algébriques et sur la théorie des imaginaires. Paris: Hermann, 1896.Darboux, G. Leçons sur les systemes orthogonaux et les coordonnées curvilignes. Paris: Gauthier-Villars, 1910.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1084-1088, 2000.Lamé, G. Leçons sur les coordonnées curvilignes et leurs diverses applications. Paris: Mallet-Bachelier, 1859.Moon, P. and Spencer, D. E. "Eleven Coordinate Systems." §1 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 1-48, 1988.Morse, P. M. and Feshbach, H. "Curvilinear Coordinates" and "Table of Properties of Curvilinear Coordinates." §1.3 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 21-31 and 115-117, 1953.Müller, E. "Die verschiedenen Koordinatensysteme." S. 596 in Encyk. Math. Wissensch., Bd. III.1.1. Leipzig, Germany: Teubner, 1907-1910.


Curvilinear Coordinates

Referenced on Wolfram|Alpha

Orthogonal Coordinate System

Cite this as:

Weisstein, Eric W. "Orthogonal Coordinate System." From MathWorld--A Wolfram Web Resource.

Subject classifications