Inmathematics,orthogonal coordinates are defined as a set ofd coordinates${\displaystyle \mathbf{q}=(q^1,q^2,\dots ,q^d)}$ in which thecoordinate hypersurfaces all meet atright angles (note that superscripts areindices, notexponents). A coordinate surface for a particular coordinateq^k is thecurve,surface, orhypersurface on whichq^k is a constant. For example, the three-dimensionalCartesian coordinates(x,y,z) is an orthogonal coordinate system, since its coordinate surfacesx = constant,y = constant, andz = constant are planes that meet at right angles to one another, i.e., are perpendicular. Orthogonal coordinates are a special but extremely common case ofcurvilinear coordinates.

While vector operations and physical laws are normally easiest to derive inCartesian coordinates (with orthogonal and straight-line axes), curvilinear orthogonal coordinates are often used instead for the solution of various problems, especiallyboundary value problems, such as those arising in field theories ofquantum mechanics,fluid flow,electrodynamics,plasma physics and thediffusion ofchemical species orheat.

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 inspherical 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 incylindrical coordinates the problem becomes one-dimensional with anordinary differential equation instead of apartial differential equation.

The reason to prefer orthogonal coordinates instead of generalcurvilinear coordinates is simplicity: many complications arise when coordinates are not orthogonal. For example, in orthogonal coordinates many problems may be solved byseparation of variables. Separation of variables is a mathematical technique that converts a complexd-dimensional problem intod one-dimensional problems that can be solved in terms of known functions. Many equations can be reduced toLaplace's equation or theHelmholtz equation.Laplace's equation is separable in 13 orthogonal coordinate systems (the 14 listedin the table below with the exception oftoroidal), and theHelmholtz equation is separable in 11 orthogonal coordinate systems.^[1][2]

Orthogonal coordinates never have off-diagonal terms in theirmetric tensor. In other words, the infinitesimal squared distanceds² can always be written as a scaled sum of the squared infinitesimal coordinate displacements:

${\displaystyle d{s}^2=\sum _{k=1}^d{\left(h_{k}d{q}^k\right)}^2}$

whered is the dimension and the scaling functions (or scale factors):

${\displaystyle h_k(\mathbf{q})\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\sqrt{g_{kk}(\mathbf{q})}=|{\mathbf{e}}_k|}$

equal the square roots of the diagonal components of the metric tensor, or the lengths of the local basis vectors${\displaystyle {\mathbf{e}}_k}$ described below. These scaling functionsh_i are used to calculate differential operators in the new coordinates, e.g., thegradient, theLaplacian, thedivergence and thecurl.

A simple method for generating orthogonal coordinates systems in two dimensions is by aconformal mapping of a standard two-dimensional grid ofCartesian coordinates(x,y). Acomplex numberz =x +iy can be formed from the real coordinatesx andy, wherei represents theimaginary unit. Anyholomorphic functionw =f(z) with non-zero complex derivative will produce aconformal mapping; if the resulting complex number is writtenw =u +iv, then the curves of constantu andv intersect at right angles, just as the original lines of constantx andy 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 theellipsoidal coordinates. More general orthogonal coordinates may be obtained by starting with some necessary coordinate surfaces and considering theirorthogonal trajectories.

InCartesian coordinates, thebasis vectors are fixed (constant). In the more general setting ofcurvilinear 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 alwaysorthogonal with respect to each other. In other words,

${\displaystyle {\mathbf{e}}_i\cdot {\mathbf{e}}_j=0\text{if}i\ne j}$

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

${\displaystyle {\mathbf{e}}_i=\frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }{q}^i}}$

wherer is some point andqⁱ 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 aparametric 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${\displaystyle h_i}$ of the basis vectors${\displaystyle {\mathbf{e}}_i}$ (see table below). The scale factors are sometimes calledLamé coefficients, not to be confused withLamé parameters (solid mechanics).

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

${\displaystyle {\hat{\mathbf{e}}}_i=\frac{{\mathbf{e}}_i}{h_i}=\frac{{\mathbf{e}}_i}{\left|{\mathbf{e}}_i\right|}}$

Avector field may be specified by its components with respect to the basis vectors or the normalized basis vectors, and one must be sure which case is meant. Components in the normalized basis are most common in applications for clarity of the quantities (for example, one may want to deal with tangential velocity instead of tangential velocity times a scale factor); in derivations the normalized basis is less common since it is more complicated.

The basis vectors shown above arecovariant basis vectors (because they "co-vary" with vectors). In the case of orthogonal coordinates, the contravariant basis vectors are easy to find since they will be in the same direction as the covariant vectors butreciprocal length (for this reason, the two sets of basis vectors are said to be reciprocal with respect to each other):

${\displaystyle {\mathbf{e}}^i=\frac{{\hat{\mathbf{e}}}_i}{h_i}=\frac{{\mathbf{e}}_i}{h_i^2}}$

this follows from the fact that, by definition,${\displaystyle {\mathbf{e}}_i\cdot {\mathbf{e}}^j={\delta }_i^j}$, using theKronecker delta. Note that:

${\displaystyle {\hat{\mathbf{e}}}_i=\frac{{\mathbf{e}}_i}{h_i}=h_i{\mathbf{e}}^i\equiv {\hat{\mathbf{e}}}^i}$

We now face three different basis sets commonly used to describe vectors in orthogonal coordinates: the covariant basise_i, the contravariant basiseⁱ, and the normalized basisêⁱ. While a vector is anobjective quantity, meaning its identity is independent of any coordinate system, the components of a vector depend on what basis the vector is represented in.

To avoid confusion, the components of the vectorx with respect to thee_i basis are represented asxⁱ, while the components with respect to theeⁱ basis are represented asx_i:

${\displaystyle \mathbf{x}=\sum _i{x}^i{\mathbf{e}}_i=\sum _i{x}_i{\mathbf{e}}^i}$

The position of the indices represent how the components are calculated (upper indices should not be confused withexponentiation). Note that thesummation symbols Σ (capitalSigma) and the summation range, indicating summation over all basis vectors (i = 1, 2, ...,d), are oftenomitted. The components are related simply by:

${\displaystyle h_i^2{x}^i=x_i}$

There is no distinguishing widespread notation in use for vector components with respect to the normalized basis; in this article we'll use subscripts for vector components and note that the components are calculated in the normalized basis.

Vector addition and negation are done component-wise just as in Cartesian coordinates with no complication. Extra considerations may be necessary for other vector operations.

Note however, that all of these operations assume that two vectors in avector field are bound to the same point (in other words, the tails of vectors coincide). Since basis vectors generally vary in orthogonal coordinates, if two vectors are added whose components are calculated at different points in space, the different basis vectors require consideration.

Thedot product inCartesian coordinates (Euclidean space with anorthonormal basis set) is simply the sum of the products of components. In orthogonal coordinates, the dot product of two vectorsx andy takes this familiar form when the components of the vectors are calculated in the normalized basis:

${\displaystyle \mathbf{x}\cdot \mathbf{y}=\sum _i{x}_i{\hat{\mathbf{e}}}_i\cdot \sum _j{y}_j{\hat{\mathbf{e}}}_j=\sum _i{x}_i{y}_i}$

This is an immediate consequence of the fact that the normalized basis at some point can form a Cartesian coordinate system: the basis set isorthonormal.

For components in the covariant or contravariant bases,

${\displaystyle \mathbf{x}\cdot \mathbf{y}=\sum _i{h}_i^2{x}^i{y}^i=\sum _i\frac{x_i{y}_i}{h_i^2}=\sum _i{x}^i{y}_i=\sum _i{x}_i{y}^i}$

This can be readily derived by writing out the vectors in component form, normalizing the basis vectors, and taking the dot product. For example, in 2D:

where the fact that the normalized covariant and contravariant bases are equal has been used.

Thecross product in 3D Cartesian coordinates is:

${\displaystyle \mathbf{x}\times \mathbf{y}=(x_2{y}_3-x_3{y}_2){\hat{\mathbf{e}}}_1+(x_3{y}_1-x_1{y}_3){\hat{\mathbf{e}}}_2+(x_1{y}_2-x_2{y}_1){\hat{\mathbf{e}}}_3}$

The above formula then remains valid in orthogonal coordinates if the components are calculated in the normalized basis.

To construct the cross product in orthogonal coordinates with covariant or contravariant bases we again must simply normalize the basis vectors, for example:

${\displaystyle \mathbf{x}\times \mathbf{y}=\sum _i{x}^i{\mathbf{e}}_i\times \sum _j{y}^j{\mathbf{e}}_j=\sum _i{x}^i{h}_i{\hat{\mathbf{e}}}_i\times \sum _j{y}^j{h}_j{\hat{\mathbf{e}}}_j}$

which, written expanded out,

${\displaystyle \mathbf{x}\times \mathbf{y}=\left(x^2{y}^3-x^3{y}^2\right)\frac{h_2{h}_3}{h_1}{\mathbf{e}}_1+\left(x^3{y}^1-x^1{y}^3\right)\frac{h_1{h}_3}{h_2}{\mathbf{e}}_2+\left(x^1{y}^2-x^2{y}^1\right)\frac{h_1{h}_2}{h_3}{\mathbf{e}}_3}$

Terse notation for the cross product, which simplifies generalization to non-orthogonal coordinates and higher dimensions, is possible with theLevi-Civita tensor, which will have components other than zeros and ones if the scale factors are not all equal to one.

Looking at an infinitesimal displacement from some point, it's apparent that

${\displaystyle d\mathbf{r}=\sum _i\frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }{q}^i}d{q}^i=\sum _i{\mathbf{e}}_{i}d{q}^i}$

Bydefinition, the gradient of a function must satisfy (this definition remains true ifƒ is anytensor)

${\displaystyle df=\mathrm{\nabla }f\cdot d\mathbf{r}\Rightarrow df=\mathrm{\nabla }f\cdot \sum _i{\mathbf{e}}_{i}d{q}^i}$

It follows then thatdel operator must be:

${\displaystyle \mathrm{\nabla }=\sum _i{\mathbf{e}}^i\frac{\mathrm{\partial }}{\mathrm{\partial }{q}^i}}$

and this happens to remain true in general curvilinear coordinates. Quantities like thegradient andLaplacian follow through proper application of this operator.

From dr and normalized basis vectorsê_i, the following can be constructed.^[3][4]

where

${\displaystyle J=\left|\frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }{q}^1}\cdot \left(\frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }{q}^2}\times \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }{q}^3}\right)\right|=\left|\frac{\mathrm{\partial }(x,y,z)}{\mathrm{\partial }(q^1,q^2,q^3)}\right|=h_1{h}_2{h}_3}$

is theJacobian 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.

Using the line element shown above, theline integral along a path${\displaystyle \mathcal{P}}$ of a vectorF is:

${\displaystyle {\int }_{\mathcal{P}}\mathbf{F}\cdot d\mathbf{r}={\int }_{\mathcal{P}}\sum _i{F}_i{\mathbf{e}}^i\cdot \sum _j{\mathbf{e}}_{j}d{q}^j=\sum _i{\int }_{\mathcal{P}}{F}_{i}d{q}^i}$

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

${\displaystyle d{A}_k=\prod _{i\ne k}d{s}_i=\prod _{i\ne k}{h}_{i}d{q}^i}$

Similarly, the volume element is:

${\displaystyle dV=\prod _{i}d{s}_i=\prod _i{h}_{i}d{q}^i}$

where the large symbol Π (capitalPi) indicates aproduct the same way that a large Σ indicates summation. Note that the product of all the scale factors is theJacobian determinant.

As an example, thesurface integral of a vector functionF over aq¹ =constant surface${\displaystyle \mathcal{S}}$ in 3D is:

${\displaystyle {\int }_{\mathcal{S}}\mathbf{F}\cdot d\mathbf{A}={\int }_{\mathcal{S}}\mathbf{F}\cdot \hat{\mathbf{n}}dA={\int }_{\mathcal{S}}\mathbf{F}\cdot {\hat{\mathbf{e}}}_{1}dA={\int }_{\mathcal{S}}{F}^1\frac{h_2{h}_3}{h_1}d{q}^{2}d{q}^3}$

Note thatF¹/h₁ is the component ofF normal to the surface.

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

Operator	Expression
Gradient of ascalar field	${\displaystyle \mathrm{\nabla }\varphi =\frac{{\hat{\mathbf{e}}}_1}{h_1}\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{q}^1}+\frac{{\hat{\mathbf{e}}}_2}{h_2}\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{q}^2}+\frac{{\hat{\mathbf{e}}}_3}{h_3}\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{q}^3}}$
Divergence of avector field	${\displaystyle \mathrm{\nabla }\cdot \mathbf{F}=\frac{1}{h_1{h}_2{h}_3}\left[\frac{\mathrm{\partial }}{\mathrm{\partial }{q}^1}\left({\hat{F}}_1{h}_2{h}_3\right)+\frac{\mathrm{\partial }}{\mathrm{\partial }{q}^2}\left({\hat{F}}_2{h}_3{h}_1\right)+\frac{\mathrm{\partial }}{\mathrm{\partial }{q}^3}\left({\hat{F}}_3{h}_1{h}_2\right)\right]}$
Curl of a vector field
Laplacian of a scalar field	${\displaystyle {\mathrm{\nabla }}^2\varphi =\frac{1}{h_1{h}_2{h}_3}\left[\frac{\mathrm{\partial }}{\mathrm{\partial }{q}^1}\left(\frac{h_2{h}_3}{h_1}\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{q}^1}\right)+\frac{\mathrm{\partial }}{\mathrm{\partial }{q}^2}\left(\frac{h_3{h}_1}{h_2}\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{q}^2}\right)+\frac{\mathrm{\partial }}{\mathrm{\partial }{q}^3}\left(\frac{h_1{h}_2}{h_3}\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{q}^3}\right)\right]}$

The above expressions can be written in a more compact form using theLevi-Civita symbol${\displaystyle {ϵ}_{ijk}}$ and the Jacobian determinant${\displaystyle J=h_1{h}_2{h}_3}$, assuming summation over repeated indices:

Operator	Expression
Gradient of ascalar field	${\displaystyle \mathrm{\nabla }\varphi =\frac{{\hat{\mathbf{e}}}_k}{h_k}\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{q}^k}}$
Divergence of avector field	${\displaystyle \mathrm{\nabla }\cdot \mathbf{F}=\frac{1}{J}\frac{\mathrm{\partial }}{\mathrm{\partial }{q}^k}\left(\frac{J}{h_k}{\hat{F}}_k\right)}$
Curl of a vector field (3D only)	${\displaystyle \mathrm{\nabla }\times \mathbf{F}=\frac{h_k{\hat{\mathbf{e}}}_k}{J}{ϵ}_{ijk}\frac{\mathrm{\partial }}{\mathrm{\partial }{q}^i}\left(h_j{\hat{F}}_j\right)}$
Laplacian of a scalar field	${\displaystyle {\mathrm{\nabla }}^2\varphi =\frac{1}{J}\frac{\mathrm{\partial }}{\mathrm{\partial }{q}^k}\left(\frac{J}{h_k^2}\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{q}^k}\right)}$

Also notice the gradient of a scalar field can be expressed in terms of theJacobian matrixJ containing canonical partial derivatives:

${\displaystyle \mathbf{J}=\left[\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{q}^1},\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{q}^2},\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{q}^3}\right]}$

upon achange of basis:

${\displaystyle \mathrm{\nabla }\varphi =\mathbf{S}\mathbf{R}{\mathbf{J}}^T}$

where the rotation and scaling matrices are:

${\displaystyle \mathbf{R}=[{\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3]}$

${\displaystyle \mathbf{S}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}([h_1^{-1},h_2^{-1},h_3^{-1}]).}$

System	Complex Transform ${\displaystyle x+iy=f(u+iv)}$	Shape of${\displaystyle u}$ and${\displaystyle v}$ isolines	Comment
Cartesian	${\displaystyle u+iv}$	line, line
Log-polar	${\displaystyle \exp (u+iv)}$	circle, line	for${\displaystyle u=\ln r}$ becomesPolar
Parabolic	${\displaystyle \frac{1}{2}(u+iv{)}^2}$	parabola, parabola
Point dipole	${\displaystyle (u+iv{)}^{-1}}$	circle, circle
Elliptic	${\displaystyle \cosh (u+iv)}$	ellipse, hyperbola	field of a needle, appears Log-polar for large distances
Bipolar	${\displaystyle \coth (u+iv)}$	circle, circle	appears like point dipole for large distances
	${\displaystyle \sqrt{u+iv}}$	hyperbola, hyperbola	field of an inner edge
	${\displaystyle u=x^2+2y^2,y=v{x}^2}$	ellipse, parabola

cartesian

polar

logpolar

ellipse parabola

parabolic

point dipole

sqrt(u+iv)

elliptic

bipolar

inverse logpolar

Examples of two-dimensional orthogonal coordinates.^[5]

Besides the usual Cartesian coordinates, 13 others are tabulated below.^[6]

Curvillinear coordinates (q1,q2,q3)	Transformation from cartesian (x,y,z)	Scale factors
Spherical coordinates ${\displaystyle (r,\theta ,\varphi )\in [0,\mathrm{\infty })\times [0,\pi ]\times [0,2\pi )}$
Parabolic coordinates ${\displaystyle (u,v,\varphi )\in [0,\mathrm{\infty })\times [0,\mathrm{\infty })\times [0,2\pi )}$
Bipolar cylindrical coordinates ${\displaystyle (u,v,z)\in [0,2\pi )\times (-\mathrm{\infty },\mathrm{\infty })\times (-\mathrm{\infty },\mathrm{\infty })}$
Ellipsoidal coordinates	${\displaystyle \frac{x^2}{a^2-q_i}+\frac{y^2}{b^2-q_i}+\frac{z^2}{c^2-q_i}=1}$ where${\displaystyle (q_1,q_2,q_3)=(\lambda ,\mu ,\nu )}$	${\displaystyle h_i=\frac{1}{2}\sqrt{\frac{(q_j-q_i)(q_k-q_i)}{(a^2-q_i)(b^2-q_i)(c^2-q_i)}}}$
Paraboloidal coordinates	${\displaystyle \frac{x^2}{q_i-a^2}+\frac{y^2}{q_i-b^2}=2z+q_i}$ where${\displaystyle (q_1,q_2,q_3)=(\lambda ,\mu ,\nu )}$	${\displaystyle h_i=\frac{1}{2}\sqrt{\frac{(q_j-q_i)(q_k-q_i)}{(a^2-q_i)(b^2-q_i)}}}$
Cylindrical polar coordinates ${\displaystyle (r,\varphi ,z)\in [0,\mathrm{\infty })\times [0,2\pi )\times (-\mathrm{\infty },\mathrm{\infty })}$
Elliptic cylindrical coordinates ${\displaystyle (u,v,z)\in [0,\mathrm{\infty })\times [0,2\pi )\times (-\mathrm{\infty },\mathrm{\infty })}$
Oblate spheroidal coordinates ${\displaystyle (\xi ,\eta ,\varphi )\in [0,\mathrm{\infty })\times \left[-\frac{\pi }{2},\frac{\pi }{2}\right]\times [0,2\pi )}$
Prolate spheroidal coordinates ${\displaystyle (\xi ,\eta ,\varphi )\in [0,\mathrm{\infty })\times [0,\pi ]\times [0,2\pi )}$
Bispherical coordinates ${\displaystyle (u,v,\varphi )\in (-\pi ,\pi ]\times [0,\mathrm{\infty })\times [0,2\pi )}$
Toroidal coordinates ${\displaystyle (u,v,\varphi )\in (-\pi ,\pi ]\times [0,\mathrm{\infty })\times [0,2\pi )}$
Parabolic cylindrical coordinates ${\displaystyle (u,v,z)\in (-\mathrm{\infty },\mathrm{\infty })\times [0,\mathrm{\infty })\times (-\mathrm{\infty },\mathrm{\infty })}$
Conical coordinates

An example of more general but still analytical orthogonal coordinate system is Similar Oblate Spheroidal (SOS) system,^[7][8] in which the transformation from the Cartesian coordinates as well as the scale factors are expressed as infinite converging sums employing generalized binomial coefficients.

• Curvilinear coordinates
• Geodetic coordinates
• Orthogonal basis
• Skew coordinates
• Tensor
• Vector field

• Korn GA andKorn 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.

• Orthogonal coordinate systems

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