Curvilinear coordinates

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thumb|upright=1.25|<span style="color:blue">Curvilinear</span> (top), [[Affine coordinate system|<span style="color:red">affine</span> (right), and <span style="color:black">Cartesian</span> (left) coordinates in two-dimensional space]]

In geometry, 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 (a one-to-one map) at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The name curvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved.

Well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space (R³) are cylindrical and spherical coordinates. A Cartesian coordinate surface in this space is a coordinate plane; for example z = 0 defines the x-y plane. In the same space, the coordinate surface r = 1 in spherical coordinates is the surface of a unit sphere, which is curved. The formalism of curvilinear coordinates provides a unified and general description of the standard coordinate systems.

Curvilinear coordinates are often used to define the location or distribution of physical quantities which may be, for example, scalars, vectors, or tensors. Mathematical expressions involving these quantities in vector calculus and tensor analysis (such as the gradient, divergence, curl, and Laplacian) can be transformed from one coordinate system to another, according to transformation rules for scalars, vectors, and tensors. Such expressions then become valid for any curvilinear coordinate system.

A curvilinear coordinate system may be simpler to use than the Cartesian coordinate system for some applications. The motion of particles under the influence of central forces is usually easier to solve in spherical coordinates than in Cartesian coordinates; this is true of many physical problems with spherical symmetry defined in R³. Equations with boundary conditions that follow coordinate surfaces for a particular curvilinear coordinate system may be easier to solve in that system. While one might describe the motion of a particle in a rectangular box using Cartesian coordinates, it is easier to describe the motion in a sphere with spherical coordinates. Spherical coordinates are the most common curvilinear coordinate systems and are used in Earth sciences, cartography, quantum mechanics, relativity, and engineering.

Orthogonal curvilinear coordinates in 3 dimensions

Coordinates, basis, and vectors

thumb|upright=1.35|Fig. 1 - Coordinate surfaces, coordinate lines, and coordinate axes of general curvilinear coordinates.

[[File:Spherical coordinate elements.svg|thumb|upright=1.35|Fig. 2 - Coordinate surfaces, coordinate lines, and coordinate axes of spherical coordinates. Surfaces: r - spheres, θ - cones, &Phi; - half-planes; Lines: r - straight beams, θ - vertical semicircles, &Phi; - horizontal circles;

Axes: r - straight beams, θ - tangents to vertical semicircles, &Phi; - tangents to horizontal circles]]

For now, consider 3-D space. A point P in 3-D space (or its position vector r) can be defined using Cartesian coordinates (x, y, z) [equivalently written (x¹, x², x³)], by <math>\mathbf{r} = x \mathbf{e}_x + y\mathbf{e}_y + z\mathbf{e}_z</math>, where e_x, e_y, e_z are the standard basis vectors.

It can also be defined by its curvilinear coordinates (q¹, q², q³) if this triplet of numbers defines a single point in an unambiguous way. The relation between the coordinates is then given by the invertible transformation functions:

:<math> x = f^1(q^1, q^2, q^3),\, y = f^2(q^1, q^2, q^3),\, z = f^3(q^1, q^2, q^3)</math>

:<math> q^1 = g^1(x,y,z),\, q^2 = g^2(x,y,z),\, q^3 = g^3(x,y,z)</math>

The surfaces q¹ = constant, q² = constant, q³ = constant are called the coordinate surfaces; and the space curves formed by their intersection in pairs are called the coordinate curves. The coordinate axes are determined by the tangents to the coordinate curves at the intersection of three surfaces. They are not in general fixed directions in space, which happens to be the case for simple Cartesian coordinates, and thus there is generally no natural global basis for curvilinear coordinates.

In the Cartesian system, the standard basis vectors can be derived from the derivative of the location of point P with respect to the local coordinate

:<math>\mathbf{e}_x = \dfrac{\partial\mathbf{r{\partial x}; \;

\mathbf{e}_y = \dfrac{\partial\mathbf{r{\partial y}; \;

\mathbf{e}_z = \dfrac{\partial\mathbf{r{\partial z}.</math>

Applying the same derivatives to the curvilinear system locally at point P defines the natural basis vectors:

:<math>\mathbf{h}_1 = \dfrac{\partial\mathbf{r{\partial q^1}; \;

\mathbf{h}_2 = \dfrac{\partial\mathbf{r{\partial q^2}; \;

\mathbf{h}_3 = \dfrac{\partial\mathbf{r{\partial q^3}.</math>

Such a basis, whose vectors change their direction and/or magnitude from point to point is called a local basis. All bases associated with curvilinear coordinates are necessarily local. Basis vectors that are the same at all points are global bases, and can be associated only with linear or affine coordinate systems.

For this article e is reserved for the standard basis (Cartesian) and h or b is for the curvilinear basis.

These may not have unit length, and may also not be orthogonal. In the case that they are orthogonal at all points where the derivatives are well-defined, we define the Lamé coefficients (after Gabriel Lamé) by

:<math>h_1 = |\mathbf{h}_1|; \; h_2 = |\mathbf{h}_2|; \; h_3 = |\mathbf{h}_3|</math>

and the curvilinear orthonormal basis vectors by

:<math>\mathbf{b}_1 = \dfrac{\mathbf{h}_1}{h_1}; \;

\mathbf{b}_2 = \dfrac{\mathbf{h}_2}{h_2}; \;

\mathbf{b}_3 = \dfrac{\mathbf{h}_3}{h_3}.</math>

These basis vectors may well depend upon the position of P; it is therefore necessary that they are not assumed to be constant over a region. (They technically form a basis for the tangent space of <math>\mathbb{R}^3</math> at P, and so are local to P.)

In general, curvilinear coordinates allow the natural basis vectors h_i not all mutually perpendicular to each other, and not required to be of unit length: they can be of arbitrary magnitude and direction. The use of an orthogonal basis makes vector manipulations simpler than for non-orthogonal. However, some areas of physics and engineering, particularly fluid mechanics and continuum mechanics, require non-orthogonal bases to describe deformations and fluid transport to account for complicated directional dependences of physical quantities. A discussion of the general case appears later on this page.

Vector calculus

Differential elements

In orthogonal curvilinear coordinates, since the total differential change in r is

:<math>d\mathbf{r}=\dfrac{\partial\mathbf{r{\partial q^1}dq^1 + \dfrac{\partial\mathbf{r{\partial q^2}dq^2 + \dfrac{\partial\mathbf{r{\partial q^3}dq^3 = h_1 dq^1 \mathbf{b}_1 + h_2 dq^2 \mathbf{b}_2 + h_3 dq^3 \mathbf{b}_3 </math>

so scale factors are <math>h_i = \left|\frac{\partial\mathbf{r{\partial q^i}\right|</math>

In non-orthogonal coordinates the length of <math>d\mathbf{r}= dq^1 \mathbf{h}_1 + dq^2 \mathbf{h}_2 + dq^3 \mathbf{h}_3 </math> is the positive square root of <math>d\mathbf{r} \cdot d\mathbf{r} = dq^i dq^j \mathbf{h}_i \cdot \mathbf{h}_j </math> (with Einstein summation convention). The six independent scalar products g_ij=h_i.h_j of the natural basis vectors generalize the three scale factors defined above for orthogonal coordinates. The nine g_ij are the components of the metric tensor, which has only three non zero components in orthogonal coordinates: g₁₁=h₁h₁, g₂₂=h₂h₂, g₃₃=h₃h₃.

Covariant and contravariant bases

[[File:Vector 1-form.svg|upright=1.5|thumb| A vector v (<span style="color:#CC0000;">red</span>) represented by

• a vector basis (<span style="color:orange;">yellow</span>, left: e₁, e₂, e₃), tangent vectors to coordinate curves (black) and

• a covector basis or cobasis (<span style="color:blue;">blue</span>, right: e¹, e², e³), normal vectors to coordinate surfaces (<span style="color:#3B444B;">grey</span>)

in general (not necessarily orthogonal) curvilinear coordinates (q¹, q², q³). The basis and cobasis do not coincide unless the coordinate system is orthogonal.]]

Spatial gradients, distances, time derivatives and scale factors are interrelated within a coordinate system by two groups of basis vectors:

1. basis vectors that are locally tangent to their associated coordinate pathline: <math display="block">\mathbf{b}_i=\dfrac{\partial\mathbf{r{\partial q^i}</math> are contravariant vectors (denoted by lowered indices), and
2. basis vectors that are locally normal to the isosurface created by the other coordinates: <math display="block">\mathbf{b}^i=\nabla q^i </math> are covariant vectors (denoted by raised indices), ∇ is the del operator.

Note that, because of Einstein's summation convention, the position of the indices of the vectors is the opposite of that of the coordinates.

Consequently, a general curvilinear coordinate system has two sets of basis vectors for every point: {b₁, b₂, b₃} is the contravariant basis, and {b¹, b², b³} is the covariant (a.k.a. reciprocal) basis. The covariant and contravariant basis vectors types have identical direction for orthogonal curvilinear coordinate systems, but as usual have inverted units with respect to each other.

Note the following important equality:

<math display="block"> \mathbf{b}^i\cdot\mathbf{b}_j = \delta^i_j </math>

wherein <math> \delta^i_j </math> denotes the generalized Kronecker delta.

{\partial q^j} dq^j = \mathbf{b}^i \cdot \mathbf{b}_j dq^j</math>

and the Einstein summation convention is implied.

A vector v can be specified in terms of either basis, i.e.,

:<math> \mathbf{v} = v^1\mathbf{b}_1 + v^2\mathbf{b}_2 + v^3\mathbf{b}_3 = v_1\mathbf{b}^1 + v_2\mathbf{b}^2 + v_3\mathbf{b}^3 </math>

Using the Einstein summation convention, the basis vectors relate to the components by Two diffeomorphic coordinate patches on a differential manifold need not overlap differentiably. With this simple definition of a curvilinear coordinate system, all the results that follow below are simply applications of standard theorems in differential topology.

The transformation functions are such that there's a one-to-one relationship between points in the "old" and "new" coordinates, that is, those functions are bijections, and fulfil the following requirements within their domains:

Vector and tensor algebra in three-dimensional curvilinear coordinates

Elementary vector and tensor algebra in curvilinear coordinates is used in some of the older scientific literature in mechanics and physics and can be indispensable to understanding work from the early and mid-1900s, for example the text by Green and Zerna. Some useful relations in the algebra of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden, Naghdi, Simmonds, Green and Zerna, and Ciarlet.

Tensors in curvilinear coordinates

A second-order tensor can be expressed as

:<math>

\boldsymbol{S} = S^{ij}\mathbf{b}_i\otimes\mathbf{b}_j = S^i{}_j\mathbf{b}_i\otimes\mathbf{b}^j = S_i{}^j\mathbf{b}^i\otimes\mathbf{b}_j = S_{ij}\mathbf{b}^i\otimes\mathbf{b}^j

</math>

where <math>\scriptstyle\otimes</math> denotes the tensor product. The components S^ij are called the contravariant components, Sⁱ _j the mixed right-covariant components, S_i ^j the mixed left-covariant components, and S_ij the covariant components of the second-order tensor. The components of the second-order tensor are related by

:<math> S^{ij} = g^{ik}S_k{}^j = g^{jk}S^i{}_k = g^{ik}g^{j\ell}S_{k\ell} </math>

The metric tensor in orthogonal curvilinear coordinates

At each point, one can construct a small line element , so the square of the length of the line element is the scalar product dx • dx and is called the metric of the space, given by:

:<math>d\mathbf{x}\cdot d\mathbf{x} = \cfrac{\partial x_i}{\partial q^j}\cfrac{\partial x_i}{\partial q^k}dq^jdq^k

</math>.

The following portion of the above equation

:<math> \cfrac{\partial x_k}{\partial q^i}\cfrac{\partial x_k}{\partial q^j} = g_{ij}(q^i,q^j) = \mathbf{b}_i\cdot\mathbf{b}_j </math>

is a symmetric tensor called the fundamental (or metric) tensor of the Euclidean space in curvilinear coordinates.

Indices can be raised and lowered by the metric:

:<math> v^i = g^{ik}v_k </math>

Relation to Lamé coefficients

Defining the scale factors h_i by

:<math> h_ih_j = g_{ij} = \mathbf{b}_i\cdot\mathbf{b}_j \quad \Rightarrow \quad h_i =\sqrt{g_{ii= \left|\mathbf{b}_i\right|=\left|\cfrac{\partial\mathbf{x{\partial q^i}\right| </math>

gives a relation between the metric tensor and the Lamé coefficients, and

:<math> g_{ij} = \cfrac{\partial\mathbf{x{\partial q^i}\cdot\cfrac{\partial\mathbf{x{\partial q^j}

= \left( h_{ki}\mathbf{e}_k\right)\cdot\left( h_{mj}\mathbf{e}_m\right)

= h_{ki}h_{kj} </math>

where h_ij are the Lamé coefficients. For an orthogonal basis we also have:

:<math> g = g_{11}g_{22}g_{33} = h_1^2h_2^2h_3^2 \quad \Rightarrow \quad \sqrt{g} = h_1h_2h_3 = J </math>

Example: Polar coordinates

If we consider polar coordinates for R²,

:<math> (x, y)=(r \cos \theta, r \sin \theta) </math>

(r, θ) are the curvilinear coordinates, and the Jacobian determinant of the transformation (r,θ) → (r cos θ, r sin θ) is r.

The orthogonal basis vectors are b_r = (cos θ, sin θ), b_θ = (−r sin θ, r cos θ). The scale factors are h_r = 1 and h_θ= r. The fundamental tensor is g₁₁ =1, g₂₂ =r², g₁₂ = g₂₁ =0.

The alternating tensor

In an orthonormal right-handed basis, the third-order alternating tensor is defined as

:<math> \boldsymbol{\mathcal{E = \varepsilon_{ijk}\mathbf{e}^i\otimes\mathbf{e}^j\otimes\mathbf{e}^k </math>

In a general curvilinear basis the same tensor may be expressed as

:<math>

\boldsymbol{\mathcal{E = \mathcal{E}_{ijk}\mathbf{b}^i\otimes\mathbf{b}^j\otimes\mathbf{b}^k

= \mathcal{E}^{ijk}\mathbf{b}_i\otimes\mathbf{b}_j\otimes\mathbf{b}_k

</math>

It can also be shown that

:<math>

\mathcal{E}^{ijk} = \cfrac{1}{J}\varepsilon_{ijk} = \cfrac{1}{+\sqrt{g\varepsilon_{ijk}

</math>

Christoffel symbols

;Christoffel symbols of the first kind <math>\Gamma_{kij}</math>:

:<math>

\mathbf{b}_{i,j} = \frac{\partial \mathbf{b}_i}{\partial q^j} = \mathbf{b}^k \Gamma_{kij} \quad \Rightarrow \quad

\mathbf{b}_k \cdot \mathbf{b}_{i,j} = \Gamma_{kij}

</math>

where the comma denotes a partial derivative (see Ricci calculus). To express Γ_kij in terms of g_ij,

:<math>

\begin{align}

g_{ij,k} & = (\mathbf{b}_i\cdot\mathbf{b}_j)_{,k} = \mathbf{b}_{i,k}\cdot\mathbf{b}_j + \mathbf{b}_i\cdot\mathbf{b}_{j,k}

= \Gamma_{jik} + \Gamma_{ijk}\\

g_{ik,j} & = (\mathbf{b}_i\cdot\mathbf{b}_k)_{,j} = \mathbf{b}_{i,j}\cdot\mathbf{b}_k + \mathbf{b}_i\cdot\mathbf{b}_{k,j}

= \Gamma_{kij} + \Gamma_{ikj}\\

g_{jk,i} & = (\mathbf{b}_j\cdot\mathbf{b}_k)_{,i} = \mathbf{b}_{j,i}\cdot\mathbf{b}_k + \mathbf{b}_j\cdot\mathbf{b}_{k,i}

= \Gamma_{kji} + \Gamma_{jki}

\end{align}

</math>

Since

:<math>\mathbf{b}_{i,j} = \mathbf{b}_{j,i}\quad\Rightarrow\quad\Gamma_{kij} = \Gamma_{kji}</math>

using these to rearrange the above relations gives

:<math>\Gamma_{kij} = \frac{1}{2}(g_{ik,j} + g_{jk,i} - g_{ij,k}) = \frac{1}{2}[(\mathbf{b}_i\cdot\mathbf{b}_k)_{,j} + (\mathbf{b}_j\cdot\mathbf{b}_k)_{,i} - (\mathbf{b}_i\cdot\mathbf{b}_j)_{,k}]

</math>

;Christoffel symbols of the second kind <math>\Gamma^k{}_{ji}</math>:

:<math>\Gamma^k{}_{ij} = g^{kl}\Gamma_{lij} = \Gamma^k{}_{ji},\quad \cfrac{\partial \mathbf{b}_i}{\partial q^j} = \mathbf{b}_k \Gamma^k{}_{ij} </math>

This implies that

:<math> \Gamma^k{}_{ij} = \cfrac{\partial \mathbf{b}_i}{\partial q^j}\cdot\mathbf{b}^k = -\mathbf{b}_i\cdot\cfrac{\partial \mathbf{b}^k}{\partial q^j}\quad </math> since <math> \quad\cfrac{\partial}{\partial q^j}(\mathbf{b}_i\cdot\mathbf{b}^k)=0</math>.

Other relations that follow are

:<math>

\cfrac{\partial \mathbf{b}^i}{\partial q^j} = -\Gamma^i{}_{jk}\mathbf{b}^k,\quad

\boldsymbol{\nabla}\mathbf{b}_i = \Gamma^k{}_{ij}\mathbf{b}_k\otimes\mathbf{b}^j,\quad

\boldsymbol{\nabla}\mathbf{b}^i = -\Gamma^i{}_{jk}\mathbf{b}^k\otimes\mathbf{b}^j

</math>

Vector operations