Curvilinear coordinates

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1 Introduction
2 Terminology
- 2.1 Example
3 Line and surface integrals
- 3.1 Line integrals
- 3.2 Surface integrals
4 Grad, curl, div, Laplacian
5 References
6 See also
7 External links

[edit] Introduction

Curvilinear coordinates are a coordinate system for a Euclidean space based on some transformation of the standard Cartesian coordinate system. We need the same number of coordinates. If we consider the 2D case, then instead of Cartesian coordinates x and y we use e.g. p and q; the level curves of p and q in the xy-plane. Required is that the transformation is locally invertible (a one-to-one map) at each point. This means that we can convert a point given in one coordinate system to its curvilinear coordinates and back.

Depending on the application, a curvilinear coordinate system may be simpler to use than the Cartesian coordinate system. For instance, a physical problem with spherical symmetry defined in R³ (e.g., motion in the field of a point mass/charge), is usually easier to solve in spherical polar coordinates than in Cartesian coordinates. Also boundary conditions may enforce symmetry. One would describe the motion of a particle in a rectangular box in Cartesian coordinates, whereas one would prefer spherical coordinates for a particle in a sphere.

Many of the concepts in vector calculus, which are given in Cartesian or spherical polar coordinates, can be formulated in arbitrary curvilinear coordinates. This gives a certain economy of thought, as it is possible to derive general expressions—valid for any curvilinear coordinate system—for concepts as gradient, divergence, curl, and the Laplacian. Well-known examples of curvilinear systems are polar coordinates for R², and cylinder and spherical polar coordinates for R³.

The name curvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved . While a Cartesian coordinate surface is a plane, e.g., $z=0\;\,$ defines the x-y plane, the coordinate surface $r=1\,$ in spherical polar coordinates is the surface of a unit sphere in R³—which obviously is curved.

[edit] Terminology

In R³, for example, we have some transformation: $\mathbf{} x_i=x_i(x_1', x_2', x_3'); i=1,2,3$ giving curvilinear coordinates x₁′, x₂′,x₃′, for x₁, x₂, x₃. If this transformation is locally invertible everywhere, the Jacobian determinant

${\partial(x_1, x_2, x_3) \over \partial(x_1', x_2', x_3')} =\begin{vmatrix} \frac{\partial x_1}{\partial x'_1} & \frac{\partial x_2}{\partial x'_1} & \frac{\partial x_3}{\partial x'_1} \\ \frac{\partial x_1}{\partial x'_2} & \frac{\partial x_2}{\partial x'_2} & \frac{\partial x_3}{\partial x'_2} \\ \frac{\partial x_1}{\partial x'_3} & \frac{\partial x_2}{\partial x'_3} & \frac{\partial x_3}{\partial x'_3} \end{vmatrix}.$

is nonzero, and for this to happen, the vectors

${ \partial \mathbf{x} \over \partial x_i' }$

must form a basis for R³.

From these basis vectors, we define scale factors or Lamé coefficients (named after Gabriel Lamé),

$h_{x_i'}=h_i=\left|{\partial \mathbf{x} \over \partial{x_i'}} \right| =\sqrt{\sum_{k=1}^3 { \left(\frac{\partial{x_k}}{\partial{x_i'}}\right)^2}}$

and thus arrive at the unit basis vectors for the curvilinear coordinates to be

$\mathbf{e}_{x_i'}=\frac{1}{h_i} {\partial \mathbf{x} \over \partial{x_i'}}$

Note that the coordinate system we choose need not be orthogonal, but for the purposes of this article, they are treated as being so. The system is defined to be orthogonal when

$\mathbf{e}_{x_i'}\cdot\mathbf{e}_{x_j'} = \delta_{ij}$

where δ_ij is the Kronecker delta.

Cartesian coordinates $x 1, x 2, x 3$ which have the scalar product, are called Euclidean coordinates. It is often convenient to associate the points of Euclidean space with vectors, for example, with each point P we associate the vector (or arrow) with its tail at the origin of coordinates and its tip at P. This vector is called the radius vector with components $(x 1, x 2, x 3)$ . At any point P of Euclidean space we can construct the small line element

$d \bold{x} = (dx_1,dx_2,dx_3) \,\!$

which is vector too. Two vectors $h = (x 1, x 2, x 3)$ and $f = (y l, y 2, y 3)$ from the same origin can be added and result is the vector with coordinates $(x l + y l, x 2 + y 2, x 3 + y 3)$ . A vector can also be multiplied by any real number. The Euclidean scalar product of two (real) vectors is the number

$\lang f,h \rang =\sum_{i} x_{i} y_{i} \,\!$ .

The scalar product of the vector with itself give the square of the vector length. The square of the length of a line element in space with scalar product is called the metric of the space. The metric of Euclidean space is

$\lang d\mathbf{x},d\mathbf{x} \rang = dx_1^2+dx_2^2+dx_3^2$ .

The same Euclidean metric in curvilinear coordinates is

$\lang d\mathbf{x},d\mathbf{x} \rang = \sum_{k=1}^3 \frac{\partial{x_k}}{\partial{x_i'}} \frac{\partial{x_k}}{\partial{x_j'}} dx_i' dx_j'$ .

The symmetric tensor

$g_{i,j}(x_i',x_j')= \sum_{k=1}^3 \frac{\partial{x_k}}{\partial{x_i'}} \frac{\partial{x_k}}{\partial{x_j'}}$

are called the fundamental (or metric) tensor of the Euclidean space in curvilinear coordinates. Connection between fundamental tensor and Lamé coefficients is $g_{i,i}(x_i',x_j')= h_i^2$ .

[edit] Example

If we consider polar coordinates for R², note that

$(x, y)=(r \cos \theta, r \sin \theta) \,\!$

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

The basis vectors are b_r = (cos θ, sin θ), b_θ = (−r sin θ, r cos θ), with unit basis vectors e_r = (cos θ, sin θ), e_θ = (−sin θ, cos θ) with scale factors h_r = 1 and h_θ= r. The fundamental tensor is g_1,1 =1, g_2,2 =r², g_1,2 = g_2,1 =0.

[edit] Line and surface integrals

Since we use curvilinear coordinates to aid in the calculation in vector calculus, there are adjustments we need to make in the calculation of line, surface and volume integrals.

[edit] Line integrals

Normally in the calculation of line integrals we are interested in calculating

$\int_C f \,ds = \int_a^b f(\mathbf{x}(t))\left|{\partial \mathbf{x} \over \partial t}\right|\; dt$

where x(t) parametrizes C in Cartesian coordinates. In curvilinear coordinates, the term

$\left|{\partial \mathbf{x} \over \partial t}\right| = \left| \sum {\partial \mathbf{x} \over \partial x_i'}{\partial x_i' \over \partial t}\right|$

by the chain rule. But from the definition of the curvilinear coordinates,

${\partial \mathbf{x} \over \partial x_i'} = h_i \mathbf{e}_{x_i'}$

and thus

$\left|{\partial \mathbf{x} \over \partial t}\right| = \sqrt{\sum h_i \mathbf{e}_{x_i'} {\partial x_i' \over \partial t}}$

and we can proceed normally.

[edit] Surface integrals

Likewise, if we are interested in a surface integral, the relevant calculation, with the parameterization of the surface in Cartesian coordinates is:

$\int_S f \,ds = \iint_T f(\mathbf{x}(s, t)) \left|{\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}\right| ds dt$

Again, in curvilinear coordinates, the term

$\left|{\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}\right| = \left|{\partial \mathbf{x} \over \partial x_i'}{\partial x_i' \over \partial s} \times {\partial \mathbf{x} \over \partial x_i'}{\partial x_i' \over \partial t}\right|$

and we make use of the definition of curvilinear coordinates again to yield

${\partial \mathbf{x} \over \partial x_i'}{\partial x_i' \over \partial s} = \sum {\partial x_i' \over \partial s} h_{x_i'} \mathbf{e}_{x_i'}$

and

${\partial \mathbf{x} \over \partial x_i'}{\partial x_i' \over \partial t} = \sum {\partial x_i' \over \partial t} h_{x_i'} \mathbf{e}_{x_i'}$

where the cross product, in terms of curvilinear coordinates, will be:

$\begin{vmatrix} \mathbf{e}_{x_1'} & \mathbf{e}_{x_2'} & \mathbf{e}_{x_3'} \\ && \\ h_1 {\partial x_1' \over \partial s} & h_2 {\partial x_2' \over \partial s} & h_3 {\partial x_3' \over \partial s} \\ && \\ h_1 {\partial x_1' \over \partial t} & h_2 {\partial x_2' \over \partial t} & h_3 {\partial x_3' \over \partial t} \end{vmatrix}$

[edit] Grad, curl, div, Laplacian

In orthogonal curvilinear coordinates, one can express the gradient, curl, divergence, and Laplacian of a function or vector field as follows:

$\nabla f = \sum_i {1 \over h_i} {\partial f \over \partial {x_i}} \hat e_{x_i}$

$\nabla\times {\vec v} = \frac{1}{\Omega} \sum_i \hat e_{x_i} \sum_{jk} \epsilon_{ijk} h_i \frac{\partial h_k v_k}{\partial x_j} \qquad (\hbox{only for } n=3)$