Curvilinear coordinates

In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The name curvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved.
Well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space are cylindrical and spherical coordinates. A Cartesian coordinate surface in this space is a coordinate plane; for example z = 0 defines the x-''y plane. In the same space, the coordinate surface r'' = 1 in spherical coordinates is the surface of a unit sphere, which is curved. The formalism of curvilinear coordinates provides a unified and general description of the standard coordinate systems.
Curvilinear coordinates are often used to define the location or distribution of physical quantities which may be, for example, scalars, vectors, or tensors. Mathematical expressions involving these quantities in vector calculus and tensor analysis can be transformed from one coordinate system to another, according to transformation rules for scalars, vectors, and tensors. Such expressions then become valid for any curvilinear coordinate system.
A curvilinear coordinate system may be simpler to use than the Cartesian coordinate system for some applications. The motion of particles under the influence of central forces is usually easier to solve in spherical coordinates than in Cartesian coordinates; this is true of many physical problems with spherical symmetry defined in R³. Equations with boundary conditions that follow coordinate surfaces for a particular curvilinear coordinate system may be easier to solve in that system. While one might describe the motion of a particle in a rectangular box using Cartesian coordinates, it is easier to describe the motion in a sphere with spherical coordinates. Spherical coordinates are the most common curvilinear coordinate systems and are used in Earth sciences, cartography, quantum mechanics, relativity, and engineering.

Orthogonal curvilinear coordinates in 3 dimensions

Coordinates, basis, and vectors

For now, consider 3-D space. A point P in 3-D space can be defined using Cartesian coordinates , by, where e_x, e_y, e_z are the standard basis vectors.
It can also be defined by its curvilinear coordinates if this triplet of numbers defines a single point in an unambiguous way. The relation between the coordinates is then given by the invertible transformation functions:
The surfaces q¹ = constant, q² = constant, q³ = constant are called the coordinate surfaces; and the space curves formed by their intersection in pairs are called the coordinate curves. The coordinate axes are determined by the tangents to the coordinate curves at the intersection of three surfaces. They are not in general fixed directions in space, which happens to be the case for simple Cartesian coordinates, and thus there is generally no natural global basis for curvilinear coordinates.
In the Cartesian system, the standard basis vectors can be derived from the derivative of the location of point P with respect to the local coordinate
Applying the same derivatives to the curvilinear system locally at point P defines the natural basis vectors:
Such a basis, whose vectors change their direction and/or magnitude from point to point is called a local basis. All bases associated with curvilinear coordinates are necessarily local. Basis vectors that are the same at all points are global bases, and can be associated only with linear or affine coordinate systems.
For this article e is reserved for the standard basis and h or b is for the curvilinear basis.
These may not have unit length, and may also not be orthogonal. In the case that they are orthogonal at all points where the derivatives are well-defined, we define the Lamé coefficients by
and the curvilinear orthonormal basis vectors by
These basis vectors may well depend upon the position of P; it is therefore necessary that they are not assumed to be constant over a region.
In general, curvilinear coordinates allow the natural basis vectors h_i not all mutually perpendicular to each other, and not required to be of unit length: they can be of arbitrary magnitude and direction. The use of an orthogonal basis makes vector manipulations simpler than for non-orthogonal. However, some areas of physics and engineering, particularly fluid mechanics and continuum mechanics, require non-orthogonal bases to describe deformations and fluid transport to account for complicated directional dependences of physical quantities. A discussion of the general case appears later on this page.

Vector calculus

Differential elements

In orthogonal curvilinear coordinates, since the total differential change in r is
so scale factors are
In non-orthogonal coordinates the length of is the positive square root of . The six independent scalar products g_ij=h_i.h_j of the natural basis vectors generalize the three scale factors defined above for orthogonal coordinates. The nine g_ij are the components of the metric tensor, which has only three non zero components in orthogonal coordinates: g₁₁=h₁h₁, g₂₂=h₂h₂, g₃₃=h₃h₃.

Covariant and contravariant bases

Spatial gradients, distances, time derivatives and scale factors are interrelated within a coordinate system by two groups of basis vectors:

basis vectors that are locally tangent to their associated coordinate pathline: are contravariant vectors, and
basis vectors that are locally normal to the isosurface created by the other coordinates: are covariant vectors, ∇ is the del operator.

Note that, because of Einstein's summation convention, the position of the indices of the vectors is the opposite of that of the coordinates.
Consequently, a general curvilinear coordinate system has two sets of basis vectors for every point: is the contravariant basis, and is the covariant basis. The covariant and contravariant basis vectors types have identical direction for orthogonal curvilinear coordinate systems, but as usual have inverted units with respect to each other.
Note the following important equality:
wherein denotes the generalized Kronecker delta.
A vector v can be specified in terms of either basis, i.e.,
Using the Einstein summation convention, the basis vectors relate to the components by
and
where g is the metric tensor.
A vector can be specified with covariant coordinates or contravariant coordinates. From the above vector sums, it can be seen that contravariant coordinates are associated with covariant basis vectors, and covariant coordinates are associated with contravariant basis vectors.
A key feature of the representation of vectors and tensors in terms of indexed components and basis vectors is invariance in the sense that vector components which transform in a covariant manner are paired with basis vectors that transform in a contravariant manner.

Integration

Constructing a covariant basis in one dimension

Consider the one-dimensional curve shown in Fig. 3. At point P, taken as an origin, x is one of the Cartesian coordinates, and q¹ is one of the curvilinear coordinates. The local basis vector is b₁ and it is built on the q¹ axis which is a tangent to that coordinate line at the point P. The axis q¹ and thus the vector b₁ form an angle with the Cartesian x axis and the Cartesian basis vector e₁.
It can be seen from triangle PAB that
where |e₁|, |b₁| are the magnitudes of the two basis vectors, i.e., the scalar intercepts PB and PA. PA is also the projection of b₁ on the x axis.
However, this method for basis vector transformations using directional cosines is inapplicable to curvilinear coordinates for the following reasons:

By increasing the distance from P, the angle between the curved line q¹ and Cartesian axis x increasingly deviates from.
At the distance PB the true angle is that which the tangent at point C forms with the x axis and the latter angle is clearly different from.

The angles that the q¹ line and that axis form with the x axis become closer in value the closer one moves towards point P and become exactly equal at P.
Let point E be located very close to P, so close that the distance PE is infinitesimally small. Then PE measured on the q¹ axis almost coincides with PE measured on the q¹ line. At the same time, the ratio PD/PE becomes almost exactly equal to.
Let the infinitesimally small intercepts PD and PE be labelled, respectively, as dx and dq¹. Then
Thus, the directional cosines can be substituted in transformations with the more exact ratios between infinitesimally small coordinate intercepts. It follows that the component of b₁ on the x axis is
If qⁱ = qⁱ and x_i = x_i are smooth functions the transformation ratios can be written as and. That is, those ratios are partial derivatives of coordinates belonging to one system with respect to coordinates belonging to the other system.