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In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, orthogonal coordinates are defined as a set of d coordinates q = (q¹, q², ..., q^d) in which the coordinate surfaces all meet at right angles (note: superscripts are indices

Einstein notation

In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae...

, not exponents). A coordinate surface for a particular coordinate q^k is the curve, surface, or hypersurface on which q_k is a constant. For example, the three-dimensional Cartesian coordinates

Cartesian coordinate system

A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

(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

Curvilinear coordinates

Curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible at each point. This means that one can convert a point given...

. 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 problem

Boundary value problem

In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions...

s, such as those arising in field theories of quantum mechanics

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, fluid flow, electrodynamics and the diffusion

Diffusion

Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...

of chemical species

Chemical species

Chemical species are atoms, molecules, molecular fragments, ions, etc., being subjected to a chemical process or to a measurement. Generally, a chemical species can be defined as an ensemble of chemically identical molecular entities that can explore the same set of molecular energy levels on a...

or heat

Heat

In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

. 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 epicenter). 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

Ordinary differential equation

In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....

instead of a partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

. The reason to prefer orthogonal coordinates instead of general curvilinear coordinates

Curvilinear coordinates

Curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible at each point. This means that one can convert a point given...

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

In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation....

. 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

Laplace's equation

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...

or the Helmholtz equation

Helmholtz equation

The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation\nabla^2 A + k^2 A = 0where ∇2 is the Laplacian, k is the wavenumber, and A is the amplitude.-Motivation and uses:...

. Laplace's equation

Laplace's equation

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...

is separable in 13 orthogonal coordinate systems, and the Helmholtz equation

Helmholtz equation

The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation\nabla^2 A + k^2 A = 0where ∇2 is the Laplacian, k is the wavenumber, and A is the amplitude.-Motivation and uses:...

is separable in 11 orthogonal coordinate systems. Orthogonal coordinates never have off-diagonal terms in their metric tensor

Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

. 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 h_i are used to calculate differential operators in the new coordinates, e.g., the gradient

Gradient

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

, the Laplacian

Vector Laplacian

In mathematics and physics, the vector Laplace operator, denoted by \scriptstyle \nabla^2, named after Pierre-Simon Laplace, is a differential operator defined over a vector field. The vector Laplacian is similar to the scalar Laplacian...

, the divergence

Divergence

In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

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

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

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

Ellipsoidal coordinates

Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system that generalizes the two-dimensional elliptic coordinate system...

. More general orthogonal coordinates may be obtained by starting with some necessary coordinate surfaces and considering their orthogonal trajectories.

Basis vectors

In Cartesian coordinates, the basis vectors are fixed (constant). In the more general setting of curvilinear coordinates

Curvilinear coordinates

Curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible at each point. This means that one can convert a point given...

, 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 \mbox{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:

\mathbf e_i = \frac{\partial \mathbf r}{\partial q^i}

where

\mathbf r

is some point and

q^i

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

\hat{\mathbf e}_i = \frac{\mathbf e_i}{\left|\mathbf e_i\right|}

A vector field

Vector field

In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

may be specified by its components with respect to the basis vectors or the normalized basis vectors, one must be sure which case is dealt. 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 useful functions known as scale factors (sometimes called Lamé coefficients, this should be avoided since some more well known coefficients in linear elasticity

Linear elasticity

Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...

carry the same name) of the coordinates are simply the lengths of the basis vectors:

h_i = \left|\mathbf e_i\right|

In cylindrical coordinates,the scale factors are

h_1=1,h_2=\rho,h_3=1

In spherical coordinates,the scale factors are

h_1=1,h_2=r,h_3=r\sin\theta

Contravariant basis

The basis vectors shown above are covariant 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 but reciprocal length

Reciprocal length

Reciprocal length or inverse length is a measurement used in several branches of science and mathematics. As the reciprocal of length, common units used for this measurement include the reciprocal metre or inverse metre , the reciprocal centimetre or inverse centimetre , and, in optics, the...

(for this reason, the two sets of basis vectors are said to be reciprocal with respect to each other):

\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,

\mathbf e_i \cdot \mathbf e_j = \delta^j_i

, using the Kronecker delta. Note that:

\hat{\mathbf e}_i = \frac{\mathbf e_i}{h_i} = h_i \mathbf e^i = \hat{\mathbf e}^i

We now face three different basis sets commonly used to describe vectors in orthogonal coordinates: the covariant basis

\mathbf e_i

, the contravariant basis

\mathbf e^i

, and the normalized basis

\hat{\mathbf e}_i

. While a vector is an objective 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 vector

\mathbf x

with respect to the

\mathbf e_i

basis are represented as

x^i

, while the components with respect to the

\mathbf e^i

basis are represented as

x_i

:

\mathbf x = \sum x^i \mathbf e_i = \sum x_i \mathbf e^i

The position of the indices represent how the components are calculated (upper indices should not be confused with exponentiation

Exponentiation

Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...

). Note that the summation symbols (

\Sigma

) and the summation range, indicating summation over all basis vectors (

i = 1, 2, ..., d

), are often omitted

Einstein notation

In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae...

. The components are related simply by:

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 algebra

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 a vector field

Vector field

In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

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.

Dot product

The dot product

Dot product

In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

in Cartesian coordinates (Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

with an orthonormal basis set) is simply the sum of the products of components. In orthogonal coordinates, the dot product of two vectors

\mathbf x

and

\mathbf y

takes this familiar form when the components of the vectors are calculated in the normalized basis:

\mathbf x \cdot \mathbf y = \sum x_i \hat{\mathbf e}_i \cdot \sum y_i \hat{\mathbf e}_i = \sum 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 is orthonormal. For components in the covariant or contraviant bases,

\mathbf x \cdot \mathbf y = \sum h_i^2 x^i y^i = \sum \frac{x_i y_i}{h_i^2} = \sum x^i y_i = \sum 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:

\mathbf x \cdot \mathbf y =
\left(x^1 \mathbf e_1 + x^2 \mathbf e_2\right) \cdot \left(y_1 \mathbf e^1 + y_2 \mathbf e^2\right) =
\left(x^1 h_1 \hat{ \mathbf e}_1 + x^2 h_2 \hat{ \mathbf e}_2\right) \cdot \left(y_1 \frac{\hat{ \mathbf e}^1}{h_1} + y_2 \frac{\hat{ \mathbf e}^2}{h_2}\right) = x^1 y_1 + x ^2 y_2

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

Cross product

The cross product

Cross product

In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...

in 3D Cartesian coordinates is:

\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:

\mathbf x \times \mathbf y = \sum x^i \mathbf e_i \times \sum y^i \mathbf e_i =
\sum x^i h_i \hat{\mathbf e}_i \times \sum y^i h_i \hat{\mathbf e}_i

which, written expanded out,

\mathbf x \times \mathbf y =
(x^2 y^3 - x^3 y^2) \frac{h_2 h_3}{h_1} \mathbf e_1 + (x^3 y^1 - x^1 y^3) \frac{h_1 h_3}{h_2} \mathbf e_2 + (x^1 y^2 - x^2 y^1) \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 the Levi-Civita tensor, which will have components other than zeros and ones if the scale factors are not all equal to one.

Differentiation

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

d\mathbf r = \sum \frac{\partial \mathbf r}{\partial q^i} dq^i = \sum \mathbf e_i dq^i

By definition, the gradient of a function must satisfy (this definition remains true if

f

is any tensor

Tensor

Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

)

df = \nabla f \cdot d\mathbf r \quad \Rightarrow \quad df = \nabla f \cdot \sum \mathbf e_i dq^i

It follows then that del operator must be:

\nabla = \sum \mathbf e^i \frac{\partial}{\partial q^i}

and this happens to remain true in general curvilinear coordinates. Quantities like the gradient

Gradient

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

and Laplacian follow through proper application of this operator.

Integration

Using the line element shown above, the line integral

Line integral

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...

along a path

\mathcal P

of a vector

\mathbf F

is:

\int_{\mathcal P} \mathbf F \cdot d\mathbf r =
\int_{\mathcal P} \sum F_i \mathbf e^i \cdot \sum \mathbf e_i dq^i = \sum \int_{\mathcal P} F_i dq^i

An infinitesimal element of area for a surface described by holding one coordinate

q_k

constant is:

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

Similarly, the volume element is:

dV = \prod ds_i = \prod h_i dq^i

where the large uppercase Pi symbol indicates a product the same way that a large Sigma indicates summation. Note that the product of all the scale factors is the Jacobian determinant. As an example, the surface integral

Surface integral

In mathematics, a surface integral is a definite integral taken over a surface ; it can be thought of as the double integral analog of the line integral...

of a vector function

\mathbf F

over a

q^1 = constant

surface

\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

\frac{F^1}{h_1}

is the component of

\mathbf 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 normalized basis. The gradient

Gradient

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

of a scalar equals

\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}.

The Laplacian of a scalar 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

Divergence

In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

of a vector 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].

The curl

Curl

In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field. At every point in the field, the curl is represented by a vector...

equals

\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] +
\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 \\
\frac{\partial}{\partial q^1} & \frac{\partial}{\partial q^2} & \frac{\partial}{\partial q^3} \\
h_1 F_1 & h_2 F_2 & h_3 F_3
\end{vmatrix}. \\
\end{align}

Orthogonal coordinates

Basis vectors

Contravariant basis

Vector algebra

Dot product

Cross product

Differentiation

Integration

Differential operators in three dimensions

See also