Orthogonal coordinates

in which the coordinate hypersurfaces all meet at right angles (note that superscripts are indices, not exponents).While vector operations and physical laws are normally easiest to derive in Cartesian coordinates, non-Cartesian orthogonal coordinates are often used instead for the solution of various problems, especially boundary value problems, such as those arising in field theories of quantum mechanics, fluid flow, electrodynamics, plasma physics and the diffusion of chemical species or heat.The chief advantage of non-Cartesian coordinates is that they can be chosen to match the symmetry of the problem.For example, the pressure wave due to an explosion far from the ground (or other barriers) depends on 3D space in Cartesian coordinates, however the pressure predominantly moves away from the center, so that in spherical coordinates the problem becomes very nearly one-dimensional (since the pressure wave dominantly depends only on time and the distance from the center).Another example is (slow) fluid in a straight circular pipe: in Cartesian coordinates, one has to solve a (difficult) two dimensional boundary value problem involving a partial differential equation, but in cylindrical coordinates the problem becomes one-dimensional with an ordinary differential equation instead of a partial differential equation.Laplace's equation is separable in 13 orthogonal coordinate systems (the 14 listed in the table below with the exception of toroidal), and the Helmholtz equation is separable in 11 orthogonal coordinate systems.[1][2] Orthogonal coordinates never have off-diagonal terms in their metric tensor.In other words, the infinitesimal squared distance ds2 can always be written as a scaled sum of the squared infinitesimal coordinate displacements where d is the dimension and the scaling functions (or scale factors) equal the square roots of the diagonal components of the metric tensor, or the lengths of the local basis vectorsThese scaling functions hi are used to calculate differential operators in the new coordinates, e.g., the gradient, the Laplacian, the divergence and the curl.A complex number z = x + iy can be formed from the real coordinates x and y, where i represents the imaginary unit.Any holomorphic function w = f(z) with non-zero complex derivative will produce a conformal mapping; if the resulting complex number is written w = u + iv, then the curves of constant u and v intersect at right angles, just as the original lines of constant x and y did.In Cartesian coordinates, the basis vectors are fixed (constant).The useful functions known as scale factors of the coordinates are simply the lengthsComponents in the normalized basis are most common in applications for clarity of the quantities (for example, one may want to deal with tangential velocity instead of tangential velocity times a scale factor); in derivations the normalized basis is less common since it is more complicated.To avoid confusion, the components of the vector x with respect to the ei basis are represented as xi, while the components with respect to the ei basis are represented as xi: The position of the indices represent how the components are calculated (upper indices should not be confused with exponentiation).Vector addition and negation are done component-wise just as in Cartesian coordinates with no complication.In orthogonal coordinates, the dot product of two vectors x and y takes this familiar form when the components of the vectors are calculated in the normalized basis: This is an immediate consequence of the fact that the normalized basis at some point can form a Cartesian coordinate system: the basis set is orthonormal.For components in the covariant or contravariant bases, This can be readily derived by writing out the vectors in component form, normalizing the basis vectors, and taking the dot product.For example, in 2D: where the fact that the normalized covariant and contravariant bases are equal has been used.To construct the cross product in orthogonal coordinates with covariant or contravariant bases we again must simply normalize the basis vectors, for example: which, written expanded out, Terse notation for the cross product, which simplifies generalization to non-orthogonal coordinates and higher dimensions, is possible with the Levi-Civita tensor, which will have components other than zeros and ones if the scale factors are not all equal to one.Looking at an infinitesimal displacement from some point, it's apparent that By definition, the gradient of a function must satisfy (this definition remains true if ƒ is any tensor) It follows then that del operator must be: and this happens to remain true in general curvilinear coordinates.Quantities like the gradient and Laplacian follow through proper application of this operator.From dr and normalized basis vectors êi, the following can be constructed.of a vector F is: An infinitesimal element of area for a surface described by holding one coordinate qk constant is: Similarly, the volume element is: where the large symbol Π (capital Pi) indicates a product the same way that a large Σ indicates summation.Note that the product of all the scale factors is the Jacobian determinant.Since these operations are common in application, all vector components in this section are presented with respect to the normalised basis:The above expressions can be written in a more compact form using the Levi-Civita symbol, assuming summation over repeated indices: Also notice the gradient of a scalar field can be expressed in terms of the Jacobian matrix J containing canonical partial derivatives: upon a change of basis: where the rotation and scaling matrices are:[5] Interval notation is used for compactness in the curvilinear coordinates column, and the entries are grouped by their interval signatures, e.g. COxCCxCO for spherical coordinates, with the x in each signature indicating the Cartesian product, with a theoretical limit of 27 products.After the grouping of the entries by interval signature, the sort order here is alphabetic by the curvilinear coordinate system name.