curvilinear coordinates

Curvilinear coordinates are a coordinate system based on some transformation of the standard coordinate system, such that the transformation is locally invertible at each point. This means that we can convert a point given in one coordinate system to its curvilinear coordinates and back at will. This is useful in that working with the curvilinear coordinate system in one situation may be more simple than in the Cartesian coordinate system.This also has consequences that we can express many of the concepts in vector calculus which are given in Cartesian or spherical coordinates or any other arbitrary coordinate system, also in curvilinear coordinates.

Terminology

In R³, for example, if we have some transformation

H(x_1', x_2', x_3')=(x_1, x_2, x_3)=\mathbf{x}

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')

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

h_{x_i'}=h_i=\left|{\partial \mathbf{x} \over \partial{x_i'}}\right|

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

\mathbf{e}_{x_i'}=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 orthogonal iff

{\partial \mathbf{x} \over \partial x_i'} \cdot {\partial \mathbf{x} \over \partial x_j'} = \delta_{ij}

where δ_ij is the Kronecker delta.

Example

If we consider polar coordinates for R², note that

(r \mathrm{cos}\theta, r \mathrm{sin} \theta) = (x, y)\,\!

(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.

Line, surface, and volume 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.

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|{\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.

Surface integrals

Likewise, if we are interested in a surface integral, the relevant calculation, with the parametrisation 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}

Div, curl, grad

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

\nabla f = \sum {1 \over h_i} {\partial f \over \partial {x_i'}}

\nabla\times F = {1 \over {\Pi h_i}} \begin{pmatrix} h_1 \partial_1 \\ \vdots \\ h_n \partial_n \end{pmatrix}\times F

\nabla\cdot F = \sum 1/h_i{\partial F_{x_i'} \over \partial {x_i'}}

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