Orthogonal coordinates
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In mathematics, orthogonal coordinates are defined as a set of coordinates that have no off-diagonal terms in their metric tensor, i.e., the infinitesimal squared distance ds2 can be written as a scaled sum of the squared infinitesimal coordinate displacements
where D is the dimension and the scaling functions equal the square roots of the diagonal components of the metric tensor. More intuitively, orthogonal coordinate systems are those in which the coordinate surfaces meet at right angles.
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[edit] Vectors and integrals
The distance formula above shows that an infinitesimal change in an orthogonal coordinate dqm is associated with a length dsm = hkdqk. Hence, a differential displacement vector equals
where the are the unit vectors normal to their respective surfaces of constant qk. These unit vectors are tangent to the coordinate lines and form the coordinate axes of a local Cartesian coordinate system.
The formulae for the vector dot product and vector cross product remain the same in orthogonal coordinate systems, e.g.,
Thus, a line integral along a contour in orthogonal coordinates equals
where Fk is the component of the vector in the direction of the kth unit vector
Similarly, an infinitesimal element of area dA = dsidsj = hihjdqidqj (where ) and the infinitesimal volume dV = dsidsjdsk = hdqidqjdqk, where and . For illustration, a surface integral over a surface in three-dimensional orthogonal coordinates equals
[edit] Differential operators in three dimensions
The gradient equals
The Laplacian equals
The divergence equals
where Fk is again the kth component of the vector .
Similarly, the curl equals
[edit] Examples of orthogonal coordinate systems in two dimensions
- Cartesian coordinates (two-dimensional)
- Polar coordinates
- Elliptic coordinates
- Parabolic coordinates (two-dimensional)
- Bipolar coordinates
[edit] Examples of orthogonal coordinate systems in three dimensions
The following coordinate systems are characterized by surfaces of degree two (i.e., quadratic) or lower. They are often used to solve Laplace's equation or the Helmholtz equation, since they allow these equations to be solved by separation of variables.
With the exception of ellipsoidal coordinates, most of these coordinate systems are generated from a two-dimensional orthogonal coordinate system, either by rotating it about a symmetry axis, or by simply projecting it perpendicularly into a third dimension. For example, projecting the two-dimensional polar coodinate system results in the cylindrical coordinate system, whereas rotating it produces the spherical coordinate system.
- Cartesian coordinates (three-dimensional)
- Spherical coordinates
- Cylindrical coordinates
- Elliptic cylindrical coordinates
- Ellipsoidal coordinates
- Prolate spheroidal coordinates
- Oblate spheroidal coordinates
- Conical coordinates
- Parabolic cylindrical coordinates
- Parabolic coordinates (three-dimensional)
- Paraboloidal coordinates
- Bipolar cylindrical coordinates
- Toroidal coordinates
- Bispherical coordinates
[edit] References
- Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers, McGraw-Hill, pp. 164-182.
- Morse PM and Feshbach H. (1953) Methods of Theoretical Physics, McGraw-Hill, pp. 494-523, 655-666.
- Margenau H. and Murphy GM. (1956) The Mathematics of Physics and Chemistry, 2nd. ed., Van Nostrand, pp.172-192.