Curvilinear coordinates
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Curvilinear coordinates are a coordinate system based on some transformation of the standard 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, as well as those of x and y in the pq-plane are in general curved. Required is 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. Depending on the application, a curvilinear coordinate system may be simpler than 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.
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Terminology
In R3, for example, if we have some transformation
- <math>H(x_1', x_2', x_3')=(x_1, x_2, x_3)=\mathbf{x}</math>
giving curvilinear coordinates x1′, x2′,x3′, for x1, x2, x3, if this transformation is locally invertible everywhere, the Jacobian determinant
- <math> \partial(x_1, x_2, x_3) \over \partial(x_1', x_2', x_3')</math>
is nonzero, and for this to happen, the vectors
- <math>{ \partial \mathbf{x} \over \partial x_i' }</math>
must form a basis for R3.
From these basis vectors, we define scale factors
- <math> h_{x_i'}=h_i=\left|{\partial \mathbf{x} \over \partial{x_i'}}\right|</math>
and thus arrive at the unit basis vectors for the curvilinear coordinates to be
- <math> \mathbf{e}_{x_i'}=1/h_i {\partial \mathbf{x} \over \partial{x_i'}}</math>
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
- <math> \mathbf{e}_{x_i'}\cdot\mathbf{e}_{x_j'} = \delta_{ij}</math>
where δij is the Kronecker delta.
Example
If we consider polar coordinates for R2, note that
- <math> (r \mathrm{cos}\theta, r \mathrm{sin} \theta) = (x, y)\,\!</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 basis vectors are br = (cos θ, sin θ), bθ = (-r sin θ, r cos θ), with unit basis vectors er = (cos θ, sin θ), eθ = (- sin θ, cos θ) with scale factors hr = 1 and hθ= 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
- <math> \int_C f \,ds = \int_a^b f(\mathbf{x}(t))\left|{\partial \mathbf{x} \over \partial t}\right|\; dt</math>
where x(t) parametrizes C in Cartesian coordinates. In curvilinear coordinates, the term
- <math> \left|{\partial \mathbf{x} \over \partial t}\right| = \left|{\partial \mathbf{x} \over \partial x_i'}{\partial x_i' \over \partial t}\right|</math>
by the chain rule. But from the definition of the curvilinear coordinates,
- <math> {\partial \mathbf{x} \over \partial x_i'} = h_i \mathbf{e}_{x_i'} </math>
and thus
- <math> \left|{\partial \mathbf{x} \over \partial t}\right| = \sqrt{\sum h_i \mathbf{e}_{x_i'} {\partial x_i' \over \partial t}}</math>
and we can proceed normally.
Surface integrals
Likewise, if we are interested in a surface integral, the relevant calculation, with the parameterization of the surface in Cartesian coordinates is:
- <math>\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</math>
Again, in curvilinear coordinates, the term
- <math>\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|</math>
and we make use of the definition of curvilinear coordinates again to yield
- <math>{\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'}</math>
and
- <math>{\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'}</math>
where the cross product, in terms of curvilinear coordinates, will be:
- <math>\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}</math>
Grad, curl, div
In orthogonal curvilinear coordinates, one can express the gradient, curl and divergence of a function or vector field as follows:
- <math> \nabla f = \sum {1 \over h_i} {\partial f \over \partial {x_i}} \hat e_{x_i} </math>
- <math> \nabla\times {\vec v} = {1 \over {\Pi h_i}} \begin{pmatrix} h_1 \partial_1 \\ \vdots \\ h_n \partial_n \end{pmatrix}\times {\vec v} </math>
- <math> \nabla\cdot {\vec v} = \sum {1 \over \Pi} {\partial \over {\partial_{x_i}}} \left ({\Pi \cdot v_i \over h_i} \right ),</math>
where <math>\Pi</math> is the product of all <math>h_i.</math>