Orthogonal coordinates

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In mathematics, orthogonal coordinates are defined as a set of d coordinates q = (q1, q2, ..., qd) in which the coordinate surfaces all meet at right angles. A coordinate surface for a particular coordinate qk is the curve, surface, or hypersurface on which qk is a constant. For example, the three-dimensional Cartesian coordinates (x, y, z) is an orthogonal coordinate system, since its coordinate surfaces x=constant, y=constant, and z=constant are planes that meet at right angles to one another, i.e., are perpendicular. Orthogonal coordinates are a special case of curvilinear coordinates.

Orthogonal coordinates are often used instead of Cartesian coordinates in the solution of partial differential equations, such as those arising in field theories of quantum mechanics, fluid flow, electrodynamics and the diffusion of molecules or heat. For example, it might be easier to solve for the wavefunction of a hydrogen ion H2– or the flow of water from a firehose in a well-chosen orthogonal coordinate system than in Cartesian coordinates. The chief advantages of orthogonal coordinates are that they can be chosen to match the symmetry of the problem, and that they allow the equation to be solved by separation of variables. Separation of variables is a mathematical technique that converts a complex d-dimensional problem into d one-dimensional problems that can be solved in terms of known functions. Many equations can be reduced to Laplace's equation or the Helmholtz equation, which are separable in many orthogonal coordinates.

In technical language, orthogonal coordinates never have off-diagonal terms in their metric tensor. In other words, the infinitesimal squared distance ds2 can always be written as a scaled sum of the squared infinitesimal coordinate displacements

ds^{2} = \sum_{k=1}^{d} \left( h_{k} dq_{k} \right)^{2}

where d is the dimension and the scaling functions

h_{k}(\mathbf{q})\ \stackrel{\mathrm{def}}{=}\ \sqrt{g_{kk}(\mathbf{q})} = |\mathbf e_k|

equal the square roots of the diagonal components of the metric tensor, or the lengths of the local basis vectors \mathbf e_k described below. These scaling functions h are used to calculate differential operators in the new coordinates, e.g., the gradient, the Laplacian, the divergence and the curl.

There are an infinite number of orthogonal coordinate systems. A simple method for generating them in two dimensions is by a conformal mapping of a standard two-dimensional grid of Cartesian coordinates (x, y). A complex number z = x + iy can be formed from the real coordinates x and y, where i represents the square root of -1. Any holomorphic function w = f(z) with non-zero complex derivative will produce a conformal mapping; if the resulting complex number is written w = u + iv, then the curves of constant u and v intersect at right rangles, just as the original lines of constant x and y did.

Orthogonal coordinates in three and higher dimensions can be generated from an orthogonal two-dimensional coordinate system, either by projecting it into a new dimension (cylindrical coordinates) or by rotating the two-dimensional system about one of its symmetry axes. However, there are other orthogonal coordinate systems in three dimensions that cannot be obtained by projecting or rotating a two-dimensional system, such as the ellipsoidal coordinates.

[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 d\mathbf{r} equals

d\mathbf{r} = \sum_{k=1}^{D} h_{k} dq_{k} \mathbf{e}_{k}

where the \mathbf{e}_{k} are the vectors normal to their respective surfaces of constant qk, not necessarily unit length. These vectors are tangent to the coordinate lines and form the basis vectors of a local Cartesian coordinate system (with possible stretched axes), they are sometimes called local basis vectors since they generally vary in space.

The formulae for the vector dot product and vector cross product remain the same in orthogonal coordinate systems, e.g.,

\mathbf{A} \cdot \mathbf{B} = \sum_{k=1}^{D} A_{k} B_{k}

Thus, a line integral along a contour \mathcal{C} in orthogonal coordinates equals

\int_{\mathcal{C}} \mathbf{F} \cdot d\mathbf{r} = \sum_{k=1}^{D} \int_{\mathcal{C}} F_{k} h_{k} dq_{k}

where Fk is the component of the vector \mathbf{F} in the direction of the kth basis vector \mathbf{e}_{k}

F_{k} \ \stackrel{\mathrm{def}}{=}\ \mathbf{e}_{k} \cdot \mathbf{F}

Similarly, an infinitesimal element of area dA = dsidsj = hihjdqidqj (where i \neq j) and the infinitesimal volume dV = dsidsjdsk = hdqidqjdqk, where h \ \stackrel{\mathrm{def}}{=}\ h_{i} h_{j} h_{k} and i \neq j \neq k. For illustration, a surface integral over a surface \mathcal{S} in three-dimensional orthogonal coordinates equals

\int_{\mathcal{S}} \mathbf{F} \cdot d\mathbf{A} = \int_{\mathcal{S}} F_{1} h_{2} h_{3} dq_{2} dq_{3} + \int_{\mathcal{S}} F_{2} h_{3} h_{1} dq_{3} dq_{1} + \int_{\mathcal{S}} F_{3} h_{1} h_{2} dq_{1} dq_{2}

Note that the product hihjhk is the Jacobian determinant.

[edit] Differential operators in three dimensions

The gradient equals

\nabla \Phi = \frac{\mathbf{e}_{1}}{h_{1}} \frac{\partial \Phi}{\partial q_{1}} + \frac{\mathbf{e}_{2}}{h_{2}} \frac{\partial \Phi}{\partial q_{2}} + \frac{\mathbf{e}_{3}}{h_{3}} \frac{\partial \Phi}{\partial q_{3}}

The Laplacian equals

\nabla^{2} \Phi = \frac{1}{h_{1} h_{2} h_{3}} \left[ \frac{\partial}{\partial q_{1}} \left( \frac{h_{2} h_{3}}{h_{1}} \frac{\partial \Phi}{\partial q_{1}} \right) + \frac{\partial}{\partial q_{2}} \left( \frac{h_{3} h_{1}}{h_{2}} \frac{\partial \Phi}{\partial q_{2}} \right) + \frac{\partial}{\partial q_{3}} \left( \frac{h_{1} h_{2}}{h_{3}} \frac{\partial \Phi}{\partial q_{3}} \right) \right]

The divergence equals

\nabla \cdot \mathbf{F} = \frac{1}{h_{1} h_{2} h_{3}} \left[ \frac{\partial}{\partial q_{1}} \left( F_{1} h_{2} h_{3} \right) + \frac{\partial}{\partial q_{2}} \left( F_{2} h_{3} h_{1} \right) + \frac{\partial}{\partial q_{3}} \left( F_{3} h_{1} h_{2} \right) \right]

where Fk is again the kth component of the vector \mathbf{F}.

Similarly, the curl equals

\nabla \times \mathbf{F} = \frac{\mathbf{e}_{1}}{h_{2} h_{3}} \left[ \frac{\partial}{\partial q_{2}} \left( h_{3} F_{3} \right) - \frac{\partial}{\partial q_{3}} \left( h_{2} F_{2} \right) \right] + \frac{\mathbf{e}_{2}}{h_{3} h_{1}} \left[ \frac{\partial}{\partial q_{3}} \left( h_{1} F_{1} \right) - \frac{\partial}{\partial q_{1}} \left( h_{3} F_{3} \right) \right] + \frac{\mathbf{e}_{3}}{h_{1} h_{2}} \left[ \frac{\partial}{\partial q_{1}} \left( h_{2} F_{2} \right) - \frac{\partial}{\partial q_{2}} \left( h_{1} F_{1} \right) \right]


[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.
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