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where

J = \left|\frac{\partial\mathbf{r}}{\partial q_1}\cdot\left(\frac{\partial\mathbf{r}}{\partial q_2}\times\frac{\partial\mathbf{r}}{\partial q_3} \right)\right| = \left|\frac{\partial(x,y,z)}{\partial(q_1,q_2,q_3)} \right| = h_1 h_2 h_3

is the Jacobian determinant, which has the geometric interpretation of the deformation in volume from the infinitesimal cube dxdydz to the infinitesimal curved volume in the orthogonal coordinates.

Integration

Using the line element shown above, the line integral along a path \scriptstyle \mathcal P of a vector F is:

\int_{\mathcal P} \mathbf F \cdot d\mathbf r = \int_{\mathcal P} \sum_i F_i \mathbf e^i \cdot \sum_j \mathbf e_j \, dq^j = \sum_i \int_{\mathcal P} F_i \, dq^i

An infinitesimal element of area for a surface described by holding one coordinate qk constant is:

dA = \prod_{i \neq k} ds_i = \prod_{i \neq k} h_i \, dq^i\,

Similarly, the volume element is:

dV = \prod_i ds_i = \prod_i h_i \, dq^i

where the large symbol Π (capital Pi) indicates a product the same way that a large Σ indicates summation. Note that the product of all the scale factors is the Jacobian determinant.

As an example, the surface integral of a vector function F over a q1 = constant surface \scriptstyle\mathcal S in 3D is:

\int_{\mathcal S} \mathbf F \cdot d\mathbf A = \int_{\mathcal S} \mathbf F \cdot \hat{\mathbf n} \ d A = \int_{\mathcal S} \mathbf F \cdot \hat{\mathbf e}_1 \ d A = \int_{\mathcal S} F^1 \frac{h_2 h_3}{h_1} \, dq^2 \, dq^3

Note that F1/h1 is the component of F normal to the surface.

Differential operators in three dimensions

Since these operations are common in application, all vector components in this section are presented with respect to the normalised basis: F_i = \mathbf{F} \cdot \hat{\mathbf{e}}_i.

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Orthogonal coordinates

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Orthogonal coordinates

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 (note: superscripts are indices, not exponents). 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 but extremely common case of curvilinear coordinates.

Contents

  • Motivation 1
  • Basis vectors 2
    • Covariant basis 2.1
    • Contravariant basis 2.2
  • Vector algebra 3
    • Dot product 3.1
    • Cross product 3.2
  • Vector calculus 4
    • Differentiation 4.1
    • Basis vector formulae 4.2
    • Integration 4.3
  • Differential operators in three dimensions 5
  • Table of orthogonal coordinates 6
  • See also 7
  • Notes 8
  • References 9

Motivation

A conformal map acting on a rectangular grid. Note that the orthogonality of the curved grid is retained.

While vector operations and physical laws are normally easiest to derive in Cartesian coordinates, non-Cartesian orthogonal coordinates are often used instead for the solution of various problems, especially boundary value problems, such as those arising in field theories of quantum mechanics, fluid flow, electrodynamics and the diffusion of chemical species or heat.

The chief advantage of non-Cartesian coordinates is that they can be chosen to match the symmetry of the problem. For example, the pressure wave due to an explosion far from the ground (or other barriers) depends on 3D space in Cartesian coordinates, however the pressure predominantly moves away from the center, so that in spherical coordinates the problem becomes very nearly one-dimensional (since the pressure wave dominantly depends only on time and the distance from the center). Another example is (slow) fluid in a straight circular pipe: in Cartesian coordinates, one has to solve a (difficult) two dimensional boundary value problem involving a partial differential equation, but in cylindrical coordinates the problem becomes one-dimensional with an ordinary differential equation instead of a partial differential equation.

The reason to prefer orthogonal coordinates instead of general curvilinear coordinates is simplicity: many complications arise when coordinates are not orthogonal. For example, in orthogonal coordinates many problems may 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. Laplace's equation is separable in 13 orthogonal coordinate systems, and the Helmholtz equation is separable in 11 orthogonal coordinate systems.[1][2]

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 (or scale factors)

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 hi are used to calculate differential operators in the new coordinates, e.g., the gradient, the Laplacian, the divergence and the curl.

A simple method for generating orthogonal coordinates systems 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 angles, 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. More general orthogonal coordinates may be obtained by starting with some necessary coordinate surfaces and considering their orthogonal trajectories.

Basis vectors

Covariant basis

In Cartesian coordinates, the basis vectors are fixed (constant). In the more general setting of curvilinear coordinates, a point in space is specified by the coordinates, and at every such point there is bound a set of basis vectors, which generally are not constant: this is the essence of curvilinear coordinates in general and is a very important concept. What distinguishes orthogonal coordinates is that, though the basis vectors vary, they are always orthogonal with respect to each other. In other words,

\mathbf e_i \cdot \mathbf e_j = 0 \quad \text{if} \quad i \neq j

These basis vectors are by definition the tangent vectors of the curves obtained by varying one coordinate, keeping the others fixed:

Visualization of 2D orthogonal coordinates. Curves obtained by holding all but one coordinate constant are shown, along with basis vectors. Note that the basis vectors aren't of equal length: they need not be, they only need to be orthogonal.
\mathbf e_i = \frac{\partial \mathbf r}{\partial q^i}

where r is some point and qi is the coordinate for which the basis vector is extracted. In other words, a curve is obtained by fixing all but one coordinate; the unfixed coordinate is varied as in a parametric curve, and the derivative of the curve with respect to the parameter (the varying coordinate) is the basis vector for that coordinate.

Note that the vectors are not necessarily of equal length. The useful functions known as scale factors of the coordinates are simply the lengths h_i of the basis vectors \hat{\mathbf e}_i (see table below). The scale factors are sometimes called Lamé coefficients, but this terminology is best avoided since some more well known coefficients in linear elasticity carry the same name.

The normalized basis vectors are notated with a hat and obtained by dividing by the length:

\hat{\mathbf e}_i = \frac_i = \frac{\partial \mathbf{r}}{\partial q_i}dq_i

Infinitesimal length

d\ell = \sqrt{d\mathbf{r}\cdot d\mathbf{r}}= \sqrt{h_1^2 \, dq_1^2 + h_2^2 \, dq_2^2 + h_3^2 \, dq_3^2}

Surface element Normal to coordinate surface qk = constant:

\begin{align} d\mathbf{S} & = (h_idq_i\hat{\mathbf{e}}_i)\times(h_jdq_j\hat{\mathbf{e}}_j) \\ & = dq_idq_j\left(\frac{\partial \mathbf{r}}{\partial q_i}\times\frac{\partial \mathbf{r}}{\partial q_j}\right)\\ & = h_ih_jdq_idq_j \hat{\mathbf{e}}_k \end{align}

Infinitesimal surface

dS_k = h_ih_j \, dq^i \, dq^j

Volume element N/A Infinitesimal volume

\begin{align} dV & = |(h_1 \, dq_1\hat{\mathbf{e}}_1)\cdot(h_2 \, dq_2\hat{\mathbf{e}}_2)\times(h_3 \, dq_3\hat{\mathbf{e}}_3)| \\ & = |\hat{\mathbf{e}}_1\cdot\hat{\mathbf{e}}_2\times\hat{\mathbf{e}}_3| h_1h_2h_3 \, dq_1 \, dq_2 \, dq_3\\ & = J \, dq_1 \, dq_2 \, dq_3 \\ & = h_1 h_2 h_3 \, dq_1 \, dq_2 \, dq_3 \end{align}

Operator Expression
Gradient of a scalar field \nabla \phi = \frac{\hat{ \mathbf e}_1}{h_1} \frac{\partial \phi}{\partial q^1} + \frac{\hat{ \mathbf e}_2}{h_2} \frac{\partial \phi}{\partial q^2} + \frac{\hat{ \mathbf e}_3}{h_3} \frac{\partial \phi}{\partial q^3}
Divergence of a vector field \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]
Curl of a vector field \begin{align} \nabla \times \mathbf F & = \frac{\hat{ \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{\hat{ \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] \\[10pt] & + \frac{\hat{ \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] =\frac{1}{h_1 h_2 h_3} \begin{vmatrix} h_1\hat{\mathbf{e}}_1 & h_2\hat{\mathbf{e}}_2 & h_3\hat{\mathbf{e}}_3 \\ \dfrac{\partial}{\partial q^1} & \dfrac{\partial}{\partial q^2} & \dfrac{\partial}{\partial q^3} \\ h_1 F_1 & h_2 F_2 & h_3 F_3 \end{vmatrix} \end{align}
Laplacian of a scalar field \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 above expressions can be written in a more compact form using the Levi-Civita symbol, defining H = h_1 h_2 h_3, and assuming summation over repeated indices:

Operator Expression
Gradient of a scalar field (\nabla \phi)_k = \frac{\hat{ \mathbf e}_k}{h_k} \frac{\partial \phi}{\partial q^k}
Divergence of a vector field \nabla \cdot \mathbf F = \frac{1}{H}\frac{\partial}{\partial q^k} \left(\frac{H}{h_k} F_k\right)
Curl of a vector field \left(\nabla \times \mathbf F\right)_k = \frac{h_k \hat{ \mathbf e}_k}{H} \epsilon_{ijk}\frac{\partial}{\partial q^i}\left(h_j F_j\right)
Laplacian of a scalar field \nabla^2 \phi = \frac{1}{H} \frac{\partial}{\partial q^k}\left(\frac{H}{h_k^2}\frac{\partial \phi}{\partial q^k}\right)

Table of orthogonal coordinates

Besides the usual cartesian coordinates, several others are tabulated below.[5] Interval notation is used for compactness in the coordinates column.

Curvillinear coordinates (q1, q2, q3) Transformation from cartesian (x, y, z) Scale factors
Spherical polar coordinates

(r, \theta, \phi)\in[0,\infty)\times[0,\pi]\times[0,2\pi)

\begin{align} x&=r\sin\theta\cos\phi \\ y&=r\sin\theta\sin\phi \\ z&=r\cos\theta \end{align} \begin{align} h_1&=1 \\ h_2&=r \\ h_3&=r\sin\theta \end{align}
Cylindrical polar coordinates

(r, \phi, z)\in[0,\infty)\times[0,2\pi)\times(-\infty,\infty)

\begin{align} x&=r\cos\phi \\ y&=r\sin\phi \\ z&=z \end{align} \begin{align} h_1&=h_3=1 \\ h_2&=r \end{align}
Parabolic cylindrical coordinates

(u, v, z)\in(-\infty,\infty)\times[0,\infty)\times(-\infty,\infty)

\begin{align} x&=\frac{1}{2}(u^2-v^2)\\ y&=uv\\ z&=z \end{align} \begin{align} h_1&=h_2=\sqrt{u^2+v^2} \\ h_3&=1 \end{align}
Paraboloidal coordinates

(u, v, \phi)\in[0,\infty)\times[0,\infty)\times[0,2\pi)

\begin{align} x&=uv\cos\phi\\ y&=uv\sin\phi\\ z&=\frac{1}{2}(u^2-v^2) \end{align} \begin{align} h_1&=h_2=\sqrt{u^2+v^2} \\ h_3&=uv \end{align}
Elliptic cylindrical coordinates

(u, v, z)\in[0,\infty)\times[0,2\pi)\times(-\infty,\infty)

\begin{align} x&=a\cosh u \cos v\\ y&=a\sinh u \sin v\\ z&=z \end{align} \begin{align} h_1&=h_2=a\sqrt{\sinh^2u+\sin^2v} \\ h_3&=1 \end{align}
Prolate spheroidal coordinates

(\xi, \eta, \phi)\in[0,\infty)\times[0,\pi]\times[0,2\pi)

\begin{align} x&=a\sinh\xi\sin\eta\cos\phi\\ y&=a\sinh\xi\sin\eta\sin\phi\\ z&=a\cosh\xi\cos\eta \end{align} \begin{align} h_1&=h_2=a\sqrt{\sinh^2\xi+\sin^2\eta} \\ h_3&=a\sinh\xi\sin\eta \end{align}
Oblate spheroidal coordinates

(\xi, \eta, \phi)\in[0,\infty)\times\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\times[0,2\pi)

\begin{align} x&=a\cosh\xi\cos\eta\cos\phi\\ y&=a\cosh\xi\cos\eta\sin\phi\\ z&=a\sinh\xi\sin\eta \end{align} \begin{align} h_1&=h_2=a\sqrt{\sinh^2\xi+\sin^2\eta} \\ h_3&=a\cosh\xi\cos\eta \end{align}
Ellipsoidal coordinates

\begin{align} & (\lambda, \mu, \nu)\\ & \lambda < c^2 < b^2 < a^2,\\ & c^2 < \mu < b^2 < a^2,\\ & c^2 < b^2 < \nu < a^2, \end{align}

\frac{x^2}{a^2 - q_i} + \frac{y^2}{b^2 - q_i} + \frac{z^2}{c^2 - q_i} = 1

where (q_1,q_2,q_3)=(\lambda,\mu,\nu)

h_i=\frac{1}{2} \sqrt{\frac{(q_j-q_i)(q_k-q_i)}{(a^2-q_i)(b^2-q_i)(c^2-q_i)}}
Bipolar coordinates

(u,v,z)\in[0,2\pi)\times(-\infty,\infty)\times(-\infty,\infty)

\begin{align} x&=\frac{a\sinh v}{\cosh v - \cos u}\\ y&=\frac{a\sin u}{\cosh v - \cos u}\\ z&=z \end{align} \begin{align} h_1&=h_2=\frac{a}{\cosh v - \cos u}\\ h_3&=1 \end{align}
Toroidal coordinates

(u,v,\phi)\in(-\pi,\pi]\times[0,\infty)\times[0,2\pi)

\begin{align} x &= \frac{a\sinh v \cos\phi}{\cosh v - \cos u}\\ y &= \frac{a\sinh v \sin\phi}{\cosh v - \cos u} \\ z &= \frac{a\sin u}{\cosh v - \cos u} \end{align} \begin{align} h_1&=h_2=\frac{a}{\cosh v - \cos u}\\ h_3&=\frac{a\sinh v}{\cosh v - \cos u} \end{align}
Conical coordinates

\begin{align} & (\lambda,\mu,\nu)\\ & \nu^2 < b^2 < \mu^2 < a^2 \\ & \lambda \in [0,\infty) \end{align}

\begin{align} x &= \frac{\lambda\mu\nu}{ab}\\ y &= \frac{\lambda}{a}\sqrt{\frac{(\mu^2-a^2)(\nu^2-a^2)}{a^2-b^2}} \\ z &= \frac{\lambda}{b}\sqrt{\frac{(\mu^2-b^2)(\nu^2-b^2)}{a^2-b^2}} \end{align} \begin{align} h_1&=1\\ h_2^2&=\frac{\lambda^2(\mu^2-\nu^2)}{(\mu^2-a^2)(b^2-\mu^2)}\\ h_3^2&=\frac{\lambda^2(\mu^2-\nu^2)}{(\nu^2-a^2)(\nu^2-b^2)} \end{align}

See also

Notes

  1. ^  
  2. ^ Morse and Feshbach 1953, Volume 1, pp. 494-523, 655-666.
  3. ^ Mathematical Handbook of Formulas and Tables (3rd edition), S. Lipschutz, M.R. Spiegel, J. Liu, Schuam's Outline Series, 2009, ISBN 978-0-07-154855-7.
  4. ^ Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipschutz, D. Spellman, Schaum’s Outlines, McGraw Hill (USA), 2009, ISBN 978-0-07-161545-7
  5. ^ Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipschutz, D. Spellman, Schaum’s Outlines, McGraw Hill (USA), 2009, ISBN 978-0-07-161545-7

References

  • Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers, McGraw-Hill, pp. 164–182.
  • Morse and Feshbach (1953). "Methods of Theoretical Physics, Volume 1". McGraw-Hill. 
  • Margenau H. and Murphy GM. (1956) The Mathematics of Physics and Chemistry, 2nd. ed., Van Nostrand, pp. 172–192.
  • Leonid P. Lebedev and Michael J. Cloud (2003) Tensor Analysis, pp. 81 – 88.
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