Orthogonal coordinates

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In mathematics, orthogonal coordinates are defined as a set of coordinates <math>\mathbf{q}</math> that have no off-diagonal terms in their metric tensor, i.e., the infinitesimal squared distance <math>ds^{2}</math> can be written as a scaled sum of the squared infinitesimal coordinate displacements

<math>

ds^{2} = \sum_{k=1}^{D} \left( h_{k} dq_{k} \right)^{2} </math>

where <math>D</math> is the dimension and the scaling functions <math>h_{k}(\mathbf{q})\ \stackrel{\mathrm{def}}{=}\ \sqrt{g_{kk}(\mathbf{q})}</math> equal the square roots of the diagonal components of the metric tensor. More intuitively, orthogonal coordinate systems are those in which the coordinate surfaces meet at right angles.

1 Vectors and integrals
2 Differential operators in three dimensions
3 Examples of orthogonal coordinate systems in two dimensions
4 Examples of orthogonal coordinate systems in three dimensions
5 References
6 External links

[edit] Vectors and integrals

The distance formula above shows that an infinitesimal change in an orthogonal coordinate <math>dq_{m}</math> is associated with a length <math>ds_{m} = h_{k} dq_{k}</math>. Hence, a differential displacement vector <math>d\mathbf{r}</math> equals

<math>

d\mathbf{r} = \sum_{k=1}^{D} h_{k} dq_{k} \mathbf{e}_{k} </math>

where the <math>\mathbf{e}_{k}</math> are the unit vectors normal to their respective surfaces of constant <math>q_{k}</math>. These unit vectors are tangent to the coordinate lines and form the coordinate axes of a local Cartesian coordinate system.

The formulae for the vector dot product and vector cross product remain the same in orthogonal coordinate systems, e.g.,

<math>

\mathbf{A} \cdot \mathbf{B} = \sum_{k=1}^{D} A_{k} B_{k} </math> Thus, a line integral along a contour <math>\mathcal{C}</math> in orthogonal coordinates equals

<math>

\int_{\mathcal{C}} \mathbf{F} \cdot d\mathbf{r} = \sum_{k=1}^{D} \int_{\mathcal{C}} F_{k} h_{k} dq_{k} </math>

where <math>F_{k}</math> is the component of the vector <math>\mathbf{F}</math> in the direction of the <math>\mathrm{k^{th}}</math> unit vector <math>\mathbf{e}_{k}</math>

<math>

F_{k} \ \stackrel{\mathrm{def}}{=}\ \mathbf{e}_{k} \cdot \mathbf{F} </math>

Similarly, an infinitesimal element of area <math>dA = ds_{i} ds_{j} = h_{i} h_{j} dq_{i} dq_{j}</math> (where <math>i \neq j</math>) and the infinitesimal volume <math>dV = ds_{i} ds_{j} ds_{k} = h dq_{i} dq_{j} dq_{k}</math>, where <math>h \ \stackrel{\mathrm{def}}{=}\ h_{i} h_{j} h_{k}</math> and <math>i \neq j \neq k</math>. For illustration, a surface integral over a surface <math>\mathcal{S}</math> in three-dimensional orthogonal coordinates equals

<math>

\int_{\mathcal{S}} \mathbf{F} \cdot d\mathbf{A} = \int_{\mathcal{S}} F_{1} h_{2} h_{3} dq_{2} dq_{3} + \int_{\mathcal{S}} F_{2} h_{3} h_{1} dq_{3} dq_{1} + \int_{\mathcal{S}} F_{3} h_{1} h_{2} dq_{1} dq_{2} </math>

[edit] Differential operators in three dimensions

The gradient equals

<math>

\nabla \Phi = \frac{\mathbf{e}_{1}}{h_{1}} \frac{\partial \Phi}{\partial q_{1}} + \frac{\mathbf{e}_{2}}{h_{2}} \frac{\partial \Phi}{\partial q_{2}} + \frac{\mathbf{e}_{3}}{h_{3}} \frac{\partial \Phi}{\partial q_{3}} </math>

The Laplacian equals

<math>

\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] </math>

The divergence equals

<math>

\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] </math>

where <math>F_{k}</math> is again the <math>\mathrm{k^{th}}</math> component of the vector <math>\mathbf{F}</math>.

Similarly, the curl equals

<math>

\nabla \times \mathbf{F} = \frac{\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{\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{\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] </math>

[edit] Examples of orthogonal coordinate systems in two dimensions

[edit] Examples of orthogonal coordinate systems in three dimensions

The following coordinate systems are characterized by surfaces of degree two (i.e., quadratic) or lower. They are often used to solve Laplace's equation or the Helmholtz equation, since they allow these equations to be solved by separation of variables.

With the exception of ellipsoidal coordinates, most of these coordinate systems are generated from a two-dimensional orthogonal coordinate system, either by rotating it about a symmetry axis, or by simply projecting it perpendicularly into a third dimension. For example, projecting the two-dimensional polar coordinate system results in the cylindrical coordinate system, whereas rotating it produces the spherical coordinate system.