# 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 (a one-to-one map) 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 (**R**^{3}) are cylindrical and spherical polar 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 polar 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 (such as the gradient, divergence, curl, and Laplacian) 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 polar coordinates than in Cartesian coordinates; this is true of many physical problems with spherical symmetry defined in **R**^{3}. 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, the motion in a sphere is easier 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 3d space (or its position vector **r**) can be defined using Cartesian coordinates (*x*, *y*, *z*) [equivalently written (*x*^{1}, *x*^{2}, *x*^{3})], by , where **e**_{x}, **e**_{y}, **e**_{z} are the *standard basis vectors*.

It can also be defined by its **curvilinear coordinates** (*q*^{1}, *q*^{2}, *q*^{3}) 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 *q*^{1} = constant, *q*^{2} = constant, *q*^{3} = 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.

Note: for this article **e** is reserved for the standard basis (Cartesian) 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 (after Gabriel Lamé) by

and the curvilinear orthonormal basis vectors by

It is important to note that 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. (They technically form a basis for the tangent bundle of at *P*, and so are local to *P*.)

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 (with Einstein summation convention). 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*are the components of the metric tensor, which has only three non zero components in orthogonal coordinates:

_{ij}*g*=

_{11}*h*,

_{1}h_{1}*g*=

_{22}*h*,

_{2}h_{2}*g*=

_{33}*h*.

_{3}h_{3}## 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:
- basis vectors that are locally normal to the isosurface created by the other coordinates:

Consequently, a general curvilinear coordinate system has two sets of basis vectors for every point: {**b**_{1}, **b**_{2}, **b**_{3}} is the covariant basis, and {**b**^{1}, **b**^{2}, **b**^{3}} is the contravariant (a.k.a. reciprocal) 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.

Proof In the Cartesian coordinate system , we can write the dot product as:

Consider an infinitesimal displacement . Let dq

_{1}, dq_{2}and dq_{3}denote the corresponding infinitesimal changes in curvilinear coordinates q_{1}, q_{2}and q_{3}respectively.By the chain rule, dq

_{1}can be expressed as:If the displacement d

**r**is such that dq_{2}= dq_{3}= 0, i.e. the position vector**r**moves by an infinitesimal amount along the coordinate axis q_{2}=const and q_{3}=const, then:Dividing by dq

_{1}, and taking the limit dq_{1}→ 0:or equivalently:

Now if the displacement d

**r**is such that dq_{1}=dq_{3}=0, i.e. the position vector**r**moves by an infinitesimal amount along the coordinate axis q_{1}=const and q_{3}=const, then:Dividing by dq

_{2}, and taking the limit dq_{2}→ 0:or equivalently:

And so forth for the other dot products.

A vector **v** can be specified in terms either basis, i.e.,

Using the Einstein summation convention, the basis vectors relate to the components by[2]^{(pp30–32)}

and

where *g* is the metric tensor (see below).

A vector can be specified with covariant coordinates (lowered indices, written *v _{k}*) or contravariant coordinates (raised indices, written

*v*). 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.

^{k}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 (or contravariant manner) are paired with basis vectors that transform in a contravariant manner (or covariant 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 *q*^{1} is one of the curvilinear coordinates. The local (non-unit) basis vector is **b**_{1} (notated **h**_{1} above, with **b** reserved for unit vectors) and it is built on the *q*^{1} axis which is a tangent to that coordinate line at the point *P*. The axis *q*^{1} and thus the vector **b**_{1} form an angle with the Cartesian *x* axis and the Cartesian basis vector **e**_{1}.

It can be seen from triangle *PAB* that

where |**e**_{1}|, |**b**_{1}| are the magnitudes of the two basis vectors, i.e., the scalar intercepts *PB* and *PA*. Note that *PA* is also the projection of **b**_{1} 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*q*^{1}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 *q*^{1} 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 *q*^{1} axis almost coincides with *PE* measured on the *q*^{1} line. At the same time, the ratio *PD/PE* (*PD* being the projection of *PE* on the *x* axis) becomes almost exactly equal to .

Let the infinitesimally small intercepts *PD* and *PE* be labelled, respectively, as *dx* and d*q*^{1}. 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 (projection) of **b**_{1} on the *x* axis is

- .

If *q ^{i}* =

*q*(

^{i}*x*

_{1},

*x*

_{2},

*x*

_{3}) and

*x*=

_{i}*x*(

_{i}*q*

^{1},

*q*

^{2},

*q*

^{3}) are smooth (continuously differentiable) 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.

### Constructing a covariant basis in three dimensions

Doing the same for the coordinates in the other 2 dimensions, **b**_{1} can be expressed as:

Similar equations hold for **b**_{2} and **b**_{3} so that the standard basis {**e**_{1}, **e**_{2}, **e**_{3}} is transformed to a local (ordered and * normalised*) basis {

**b**

_{1},

**b**

_{2},

**b**

_{3}} by the following system of equations:

By analogous reasoning, one can obtain the inverse transformation from local basis to standard basis:

### Jacobian of the transformation

The above systems of linear equations can be written in matrix form using the Einstein summation convention as

- .

This coefficient matrix of the linear system is the Jacobian matrix (and its inverse) of the transformation. These are the equations that can be used to transform a Cartesian basis into a curvilinear basis, and vice versa.

In three dimensions, the expanded forms of these matrices are

In the inverse transformation (second equation system), the unknowns are the curvilinear basis vectors. For any specific location there can only exist one and only one set of basis vectors (else the basis is not well defined at that point). This condition is satisfied if and only if the equation system has a single solution, from linear algebra, a linear equation system has a single solution (non-trivial) only if the determinant of its system matrix is non-zero:

which shows the rationale behind the above requirement concerning the inverse Jacobian determinant.

## Generalization to *n* dimensions

*n*dimensions

The formalism extends to any finite dimension as follows.

Consider the real Euclidean *n*-dimensional space, that is **R**^{n} = **R** × **R** × ... × **R** (*n* times) where **R** is the set of real numbers and × denotes the Cartesian product, which is a vector space.

The coordinates of this space can be denoted by: **x** = (*x*_{1}, *x*_{2},...,*x _{n}*). Since this is a vector (an element of the vector space), it can be written as:

where **e**^{1} = (1,0,0...,0), **e**^{2} = (0,1,0...,0), **e**^{3} = (0,0,1...,0),...,**e**^{n} = (0,0,0...,1) is the *standard basis set of vectors* for the space **R**^{n}, and *i* = 1, 2,...*n* is an index labelling components. Each vector has exactly one component in each dimension (or "axis") and they are mutually orthogonal (perpendicular) and normalized (has unit magnitude).

More generally, we can define basis vectors **b**_{i} so that they depend on **q** = (*q*_{1}, *q*_{2},...,*q _{n}*), i.e. they change from point to point:

**b**

_{i}=

**b**

_{i}(

**q**). In which case to define the same point

**x**in terms of this alternative basis: the

*coordinates*with respect to this basis

*v*also necessarily depend on

_{i}**x**also, that is

*v*=

_{i}*v*(

_{i}**x**). Then a vector

**v**in this space, with respect to these alternative coordinates and basis vectors, can be expanded as a linear combination in this basis (which simply means to multiply each basis vector

**e**

_{i}by a number

*v*

_{i}– scalar multiplication):

The vector sum that describes **v** in the new basis is composed of different vectors, although the sum itself remains the same.

## Transformation of coordinates

From a more general and abstract perspective, a curvilinear coordinate system is simply a coordinate patch on the differentiable manifold **E**^{n} (n-dimensional Euclidean space) that is diffeomorphic to the Cartesian coordinate patch on the manifold.[3] Note that two diffeomorphic coordinate patches on a differential manifold need not overlap differentiably. With this simple definition of a curvilinear coordinate system, all the results that follow below are simply applications of standard theorems in differential topology.

The transformation functions are such that there's a one-to-one relationship between points in the "old" and "new" coordinates, that is, those functions are bijections, and fulfil the following requirements within their domains:

- They are smooth functions: q
^{i}= q^{i}(**x**) - The inverse Jacobian determinant
is not zero; meaning the transformation is invertible:

according to the inverse function theorem. The condition that the Jacobian determinant is not zero reflects the fact that three surfaces from different families intersect in one and only one point and thus determine the position of this point in a unique way.[4]*x*(_{i}**q**).

## Vector and tensor algebra in three-dimensional curvilinear coordinates

*Note: the Einstein summation convention of summing on repeated indices is used below.*

Elementary vector and tensor algebra in curvilinear coordinates is used in some of the older scientific literature in mechanics and physics and can be indispensable to understanding work from the early and mid-1900s, for example the text by Green and Zerna.[5] Some useful relations in the algebra of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden,[6] Naghdi,[7] Simmonds,[2] Green and Zerna,[5] Basar and Weichert,[8] and Ciarlet.[9]

## Tensors in curvilinear coordinates

A second-order tensor can be expressed as

where denotes the tensor product. The components *S ^{ij}* are called the

**contravariant**components,

*S*the

^{i}_{j}**mixed right-covariant**components,

*S*the

_{i}^{j}**mixed left-covariant**components, and

*S*the

_{ij}**covariant**components of the second-order tensor. The components of the second-order tensor are related by

### The metric tensor in orthogonal curvilinear coordinates

At each point, one can construct a small line element d**x**, so the square of the length of the line element is the scalar product d**x** • d**x** and is called the metric of the space, given by:

- .

The following portion of the above equation

is a *symmetric* tensor called the **fundamental (or metric) tensor** of the Euclidean space in curvilinear coordinates.

Indices can be raised and lowered by the metric:

#### Relation to Lamé coefficients

Defining the scale factors *h _{i}* by

gives a relation between the metric tensor and the Lamé coefficients. Note also that

where *h _{ij}* are the Lamé coefficients. For an orthogonal basis we also have:

#### Example: Polar coordinates

If we consider polar coordinates for **R**^{2}, note that

(r, θ) are the curvilinear coordinates, and the Jacobian determinant of the transformation (*r*,θ) → (*r* cos θ, *r* sin θ) is *r*.

The orthogonal basis vectors are **b**_{r} = (cos θ, sin θ), **b**_{θ} = (−sin θ, cos θ). The scale factors are *h*_{r} = 1 and *h*_{θ}= *r*. The fundamental tensor is *g*_{11} =1, *g*_{22} =*r*^{2}, *g*_{12} = *g*_{21} =0.

### The alternating tensor

In an orthonormal right-handed basis, the third-order alternating tensor is defined as

In a general curvilinear basis the same tensor may be expressed as

It can also be shown that

### Christoffel symbols

- Christoffel symbols of the first kind

where the comma denotes a partial derivative (see Ricci calculus). To express Γ_{ijk} in terms of *g _{ij}* we note that

Since

using these to rearrange the above relations gives

- Christoffel symbols of the second kind

This implies that

Other relations that follow are

### Vector operations

**Dot product:**The scalar product of two vectors in curvilinear coordinates is[2]

^{(p32)}**Cross product:**The cross product of two vectors is given by[2]

^{(pp32–34)}where is the permutation symbol and is a Cartesian basis vector. In curvilinear coordinates, the equivalent expression is

## Vector and tensor calculus in three-dimensional curvilinear coordinates

*Note: the Einstein summation convention of summing on repeated indices is used below.*

Adjustments need to be made in the calculation of line, surface and volume integrals. For simplicity, the following restricts to three dimensions and orthogonal curvilinear coordinates. However, the same arguments apply for *n*-dimensional spaces. When the coordinate system is not orthogonal, there are some additional terms in the expressions.

Simmonds,[2] in his book on tensor analysis, quotes Albert Einstein saying[10]

The magic of this theory will hardly fail to impose itself on anybody who has truly understood it; it represents a genuine triumph of the method of absolute differential calculus, founded by Gauss, Riemann, Ricci, and Levi-Civita.

Vector and tensor calculus in general curvilinear coordinates is used in tensor analysis on four-dimensional curvilinear manifolds in general relativity,[11] in the mechanics of curved shells,[9] in examining the invariance properties of Maxwell's equations which has been of interest in metamaterials[12][13] and in many other fields.

Some useful relations in the calculus of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden,[14] Simmonds,[2] Green and Zerna,[5] Basar and Weichert,[8] and Ciarlet.[9]

Let φ = φ(**x**) be a well defined scalar field and **v** = **v**(**x**) a well-defined vector field, and *λ*_{1}, *λ*_{2}... be parameters of the coordinates

### Geometric elements

**Tangent vector:**If**x**(*λ*) parametrizes a curve*C*in Cartesian coordinates, thenis a tangent vector to

*C*in curvilinear coordinates (using the chain rule). Using the definition of the Lamé coefficients, and that for the metric*g*= 0 when_{ij}*i*≠*j*, the magnitude is:**Tangent plane element:**If**x**(*λ*_{1},*λ*_{2}) parametrizes a surface*S*in Cartesian coordinates, then the following cross product of tangent vectors is a normal vector to*S*with the magnitude of infinitesimal plane element, in curvilinear coordinates. Using the above result,where is the permutation symbol. In determinant form:

### Integration

Operator Scalar field Vector field Line integral Surface integral Volume integral

### Differentiation

The expressions for the gradient, divergence, and Laplacian can be directly extended to *n*-dimensions, however the curl is only defined in 3d.

The vector field **b**_{i} is tangent to the *q ^{i}* coordinate curve and forms a

**natural basis**at each point on the curve. This basis, as discussed at the beginning of this article, is also called the

**covariant**curvilinear basis. We can also define a

**reciprocal basis**, or

**contravariant**curvilinear basis,

**b**

^{i}. All the algebraic relations between the basis vectors, as discussed in the section on tensor algebra, apply for the natural basis and its reciprocal at each point

**x**.

Operator Scalar field Vector field 2nd order tensor field Gradient Divergence N/A where

**a**is an arbitrary constant vector. In curvilinear coordinates,Laplacian Curl N/A For vector fields in 3d only, where is the Levi-Civita symbol.

See Curl of a tensor field

## Fictitious forces in general curvilinear coordinates

By definition, if a particle with no forces acting on it has its position expressed in an inertial coordinate system, (*x*_{1}, *x*_{2}, *x*_{3}, *t*), then there it will have no acceleration (d^{2}*x*_{j}/d*t*^{2} = 0).[15] In this context, a coordinate system can fail to be “inertial” either due to non-straight time axis or non-straight space axes (or both). In other words, the basis vectors of the coordinates may vary in time at fixed positions, or they may vary with position at fixed times, or both. When equations of motion are expressed in terms of any non-inertial coordinate system (in this sense), extra terms appear, called Christoffel symbols. Strictly speaking, these terms represent components of the absolute acceleration (in classical mechanics), but we may also choose to continue to regard d^{2}*x*_{j}/d*t*^{2} as the acceleration (as if the coordinates were inertial) and treat the extra terms as if they were forces, in which case they are called fictitious forces.[16] The component of any such fictitious force normal to the path of the particle and in the plane of the path’s curvature is then called centrifugal force.[17]

This more general context makes clear the correspondence between the concepts of centrifugal force in rotating coordinate systems and in stationary curvilinear coordinate systems. (Both of these concepts appear frequently in the literature.[18][19][20]) For a simple example, consider a particle of mass *m* moving in a circle of radius *r* with angular speed *w* relative to a system of polar coordinates rotating with angular speed *W*. The radial equation of motion is *mr*” = *F*_{r} + *mr*(*w* + *W*)^{2}. Thus the centrifugal force is *mr* times the square of the absolute rotational speed *A* = *w* + *W* of the particle. If we choose a coordinate system rotating at the speed of the particle, then *W* = *A* and *w* = 0, in which case the centrifugal force is *mrA*^{2}, whereas if we choose a stationary coordinate system we have *W* = 0 and *w* = *A*, in which case the centrifugal force is again *mrA*^{2}. The reason for this equality of results is that in both cases the basis vectors at the particle’s location are changing in time in exactly the same way. Hence these are really just two different ways of describing exactly the same thing, one description being in terms of rotating coordinates and the other being in terms of stationary curvilinear coordinates, both of which are non-inertial according to the more abstract meaning of that term.

When describing general motion, the actual forces acting on a particle are often referred to the instantaneous osculating circle tangent to the path of motion, and this circle in the general case is not centered at a fixed location, and so the decomposition into centrifugal and Coriolis components is constantly changing. This is true regardless of whether the motion is described in terms of stationary or rotating coordinates.

## See also

## References

- J.A. Wheeler; C. Misner; K.S. Thorne (1973).
*Gravitation*. W.H. Freeman & Co. ISBN 0-7167-0344-0. - Simmonds, J. G. (1994).
*A brief on tensor analysis*. Springer. ISBN 0-387-90639-8. - Boothby, W. M. (2002).
*An Introduction to Differential Manifolds and Riemannian Geometry*(revised ed.). New York, NY: Academic Press. - McConnell, A. J. (1957).
*Application of Tensor Analysis*. New York, NY: Dover Publications, Inc. Ch. 9, sec. 1. ISBN 0-486-60373-3. - Green, A. E.; Zerna, W. (1968).
*Theoretical Elasticity*. Oxford University Press. ISBN 0-19-853486-8. - Ogden, R. W. (2000).
*Nonlinear elastic deformations*. Dover. - Naghdi, P. M. (1972). "Theory of shells and plates". In S. Flügge (ed.).
*Handbook of Physics*. VIa/2. pp. 425–640. - Basar, Y.; Weichert, D. (2000).
*Numerical continuum mechanics of solids: fundamental concepts and perspectives*. Springer. - Ciarlet, P. G. (2000).
*Theory of Shells*.**1**. Elsevier Science. - Einstein, A. (1915). "Contribution to the Theory of General Relativity". In Laczos, C. (ed.).
*The Einstein Decade*. p. 213. ISBN 0-521-38105-3. - Misner, C. W.; Thorne, K. S.; Wheeler, J. A. (1973).
*Gravitation*. W. H. Freeman and Co. ISBN 0-7167-0344-0. - Greenleaf, A.; Lassas, M.; Uhlmann, G. (2003). "Anisotropic conductivities that cannot be detected by EIT".
*Physiological measurement*.**24**(2): 413–419. doi:10.1088/0967-3334/24/2/353. PMID 12812426. - Leonhardt, U.; Philbin, T.G. (2006). "General relativity in electrical engineering".
*New Journal of Physics*.**8**(10): 247. arXiv:cond-mat/0607418. doi:10.1088/1367-2630/8/10/247. - Ogden
- Friedman, Michael (1989).
*The Foundations of Space–Time Theories*. Princeton University Press. ISBN 0-691-07239-6. - Stommel, Henry M.; Moore, Dennis W. (1989).
*An Introduction to the Coriolis Force*. Columbia University Press. ISBN 0-231-06636-8. - Beer; Johnston (1972).
*Statics and Dynamics*(2nd ed.). McGraw–Hill. p. 485. ISBN 0-07-736650-6. - Hildebrand, Francis B. (1992).
*Methods of Applied Mathematics*. Dover. p. 156. ISBN 0-13-579201-0. - McQuarrie, Donald Allan (2000).
*Statistical Mechanics*. University Science Books. ISBN 0-06-044366-9. - Weber, Hans-Jurgen; Arfken, George Brown (2004).
*Essential Mathematical Methods for Physicists*. Academic Press. p. 843. ISBN 0-12-059877-9.