# Dictionary Definition

curvilinear adj : characterized by or following a
curved line; "curvilinear tracery"; "curvilinear motion" [syn:
curvilineal]

# User Contributed Dictionary

## English

### Adjective

- Having bends.

# Extensive Definition

Curvilinear coordinates are a coordinate
system for the Euclidean
space based on some transformation that converts the standard
Cartesian coordinate system to a coordinate system with the same
number of coordinates in which the coordinate lines are curved. In
the two-dimensional case, instead of Cartesian
coordinates x and y, e.g., p and q are used: the level curves
of p and q in the xy-plane. Required is that the transformation is
locally invertible (a
one-to-one map) at each point. This means that one 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 to use than the Cartesian
coordinate system. For instance, a physical problem with spherical
symmetry defined in R3 (e.g., motion in the field of a point
mass/charge), is usually easier to solve in spherical
polar coordinates than in Cartesian coordinates. Also boundary
conditions may enforce symmetry. One would describe the motion of a
particle in a rectangular box in Cartesian coordinates, whereas one
would prefer spherical coordinates for a particle in a sphere. Many
of the concepts in vector
calculus, which are given in Cartesian or spherical
polar coordinates, can be formulated in arbitrary curvilinear
coordinates. This gives a certain economy of thought, as it is
possible to derive general expressions—valid for any
curvilinear coordinate system—for concepts as gradient,
divergence, curl, and the Laplacian. Well-known examples of
curvilinear systems are polar coordinates for R2, and
cylinder and spherical
polar coordinates for R3.

The name curvilinear coordinates, coined by the
French mathematician Lamé,
derives from the fact that the coordinate
surfaces of the curvilinear systems are curved . While a
Cartesian coordinate surface is a plane, e.g., z = 0 defines the
x-y plane, the coordinate surface r = 1 in spherical polar
coordinates is the surface of a unit sphere in R3—which
obviously is curved.

## General curvilinear coordinates

In Cartesian coordinates, the position of a point P(x,y,z) is determined by the intersection of three mutually perpendicular planes, x = const, y = const, z = const. The coordinates x, y and z are related to three new quantities q1,q2, and q3 by the equations:- x = x(q1,q2,q3) direct transformation
- y = y(q1,q2,q3) (curvilinear to Cartesian coordinates)
- z = z(q1,q2,q3)

- q1 = q1(x, y, z) inverse transformation
- q2 = q2(x, y, z) (Cartesian to curvilinear coordinates)
- q3 = q3(x, y, z)

- 1) They are smooth functions
- 2) The Jacobian determinant

A given point may be described by specifying
either x, y, z or q1, q2, q3 while each of the inverse equations
describes a surface in the new coordinates and the intersection of
three such surfaces locates the point in the three-dimensional
space (Fig. 1). The surfaces q1 = const, q2 = const, q3 = const are
called the coordinate surfaces; the space curves formed by their
intersection in pairs are called the coordinate lines. The
coordinate axes are determined by the tangents to the coordinate
lines at the intersection of three surfaces. They are not in
general fixed directions in space, as is true for simple Cartesian
coordinates. The quantities (q1, q2, q3 ) are the curvilinear
coordinates of a point P(q1, q2, q3 ).

In general, (q1, q2 ... qn ) are curvilinear
coordinates in n-dimensional space.

### Example: Spherical coordinates

Spherical coordinates are one of the most used curvilinear coordinate system in such fields as Earth sciences, cartography, and physics (quantum physics, relativity, etc.). The curvilinear coordinates (q1, q2, q3) in this system are, respectively, r (radial distance or polar radius, r ≥ 0), θ (azimuth or latitude, 0 ≤ θ ≤ 180°), and φ (zenith or longitude, 0 ≤ φ ≤ 360°). The relationship between Cartesian and spherical coordinates is given by:- x = r sin θ cos φ
- y = r sin θ sin φ
- z = r cos θ direct transformation (Cartesian to spherical coordinates)

Solving the above equation system for r, θ, and φ
gives the relations between spherical and Cartesian
coordinates:

- r=\sqrt
- =\arccos \left( \right) or \cos\theta=
- =\arctan \left( \right) or \tan\varphi= inverse transformation (spherical to Cartesian coordinates)

The respective spherical coordinate surfaces are
derived in terms of Cartesian coordinates by fixing the spherical
coordinates in the above inverse transformations to a constant
value. Thus (Fig.2), r = const are concentric spherical surfaces
centered at the origin, O, of the Cartesian coordinates, θ = const
are circular conical surfaces with apex in O and axis the Oz axis,
φ = const are half-planes bounded by the Oz axis and perpendicular
to the xOy Cartesian coordinate plane. Each spherical coordinate
line is formed at the pairwise intersection of the surfaces,
corresponding to the other two coordinates: r lines (radial
distance) are beams Or at the intersection of the cones θ = const
and the half-planes φ = const; θ lines (meridians) are semi-circles
formed by the intersection of the spheres r = const and the
half-planes φ = const ; and φ lines (parallels) are circles in
planes parallel to xOy at the intersection of the spheres r = const
and the cones θ = const. The location of a point P(r,θ,φ) is
determined by the point of intersection of the three coordinate
surfaces, or, alternatively, by the point of intersection of the
three coordinate lines. The θ and φ axes in P(r,θ,φ) are the
mutually perpendicular (orthogonal) tangents to the meridian and
parallel of this point, while the r axis is directed along the
radial distance and is orthogonal to both θ and φ axes.

The surfaces described by the inverse
transformations are smooth functions within their defined domains.
The Jacobian (functional determinant) of the inverse
transformations is:

- J^ =\frac

## Local basis

Coordinates are used to define location or distribution of physical quantities which are scalars, vectors, or tensors. Scalars are expressed as points and their location is defined by specifying their coordinates with the use of coordinate lines or coordinate surfaces. Vectors are objects that possess two characteristics: magnitude and direction. To define a vector in terms of coordinates, an additional coordinate-associated structure, called basis, is needed. A basis in three-dimensional space is a set of three linearly independent vectors , called basis vectors. Each basis vector is associated with a coordinate in the respective dimension. Any vector can be represented as a sum of vectors Anen formed by multiplication of a basis vector by a scalar coefficient, called component. Each vector, then, has exactly one component in each dimension and can be represented by the vector sum: A = A1e1 + A2e2 + A3e3, where An and en are the respective components and basis vectors. A requirement for the coordinate system and its basis is that A1e1 + A2e2 + A3e3 ≠ 0 when at least one of the An ≠ 0. This condition is called linear independence. Linear independence implies that there cannot exist bases with basis vectors of zero magnitude because the latter will give zero-magnitude vectors when multiplied by any component. Non-coplanar vectors are linearly independent, and any triple of non-coplanar vectors can serve as a basis in three dimensions.For the general curvilinear coordinates, basis
vectors and components vary from point to point. If vector A whose
origin is in point P (q1, q2, q3 ) is moved to point P' (q'1, q'2,
q'3 ) in such a way that its direction and orientation are
preserved, then the moved vector will be expressed by new
components A'n and basis vectors en. Therefore, the vector sum that
describes vector A in the new location is composed of different
vectors, although the sum itself remains the same. A coordinate
basis whose basis vectors change their direction and/or magnitude
from point to point is called local basis. All bases associated
with curvilinear coordinates are necessarily local. Global bases,
that is, bases composed of basis vectors that are the same in all
points can be associated only with linear coordinates. A more
exact, though seldom used, expression for such vector sums with
local basis vectors is \mathbf = \textstyle \sum_^n A_i(q_1\ldots
q_n)\mathbf_i(q_1\ldots q_n), where the dependence of both
components and basis vector on location is made explicit (n is the
number of dimensions). Local bases are composed of vectors with
arbitrary order, magnitude, and direction and magnitude/direction
vary in different points in space.

Basis vectors can be associated with a coordinate
system by two methods: they can be built along the coordinate axes
(colinear with axes) or they can be built to be perpendicular
(normal) to the coordinate surfaces. In the first case
(axis-colinear), basis vectors transform like
covariant vectors while in the second case (normal to
coordinate surfaces), basis vectors transform like
contravariant vectors. Those two types of basis vectors are
distinguished by the position of their indices: covariant vectors
are designated with lower indices while contravariant vectors are
designated with upper indices. Thus, depending on the method by
which they are built, for a general curvilinear coordinate system
there are two sets of basis vectors for every point: is the
covariant basis, and is the contravariant basis. A key property of
the vector and tensor representation 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), and these operations are inverse to one
another according to the transformation rules. This means that in
terms, in which an index occurs two times, one of the indices in
the pair must be upper and the other index must be lower. Thus in
the above vector sums, basis vectors with lower indices are
multiplied by components with upper indices or vice versa, so that
a given vector can be represented in two ways: A = A1e1 + A2e2 +
A3e3 = A1e1 + A2e2 + A3e3. Upon coordinate change, a vector
transforms in the same way as its components. Therefore, a vector
is covariant or contravariant if, respectively, its components are
covariant or contravariant. From the above vector sums, it can be
seen that contravariant vectors are represented with covariant
basis vectors, and covariant vectors are represented with
contravariant basis vectors. This is reflected in the Einstein
summation convention according to which in the vector sums
\textstyle \sum_^n A^i \mathbf_i and \textstyle \sum_^n A_i
\mathbf^i the basis vectors and the summation symbols are omitted,
leaving only Ai and Ai which represent, respectively, a
contravariant and a covariant vector.

### Covariant basis

As stated above, contravariant vectors are vectors with contravariant components whose location is determined using covariant basis vectors that are built along the coordinate axes. In analogy to the other coordinate elements, transformation of the covariant basis of general curvilinear coordinates is described starting from the Cartesian coordinate system whose basis is called standard basis. The standard basis is a global basis that is composed of 3 mutually orthogonal vectors of unit length, that is, the magnitude of each basis vector equals 1. Regardless of the method of building the basis (axis-colinear or normal to coordinate surfaces), in the Cartesian system the result is a single set of basis vectors, namely, the standard basis. To avoid misunderstanding, in this section the standard basis will be thought of as built along the coordinate axes.In point P, taken as an origin, x is one of the
Cartesian coordinates, and q1 is one of the curvilinear coordinates
(Fig. 3). The local basis vector is e1 and it is built on the q1
axis which is a tangent to q1 coordinate line at the point P. The
axis q1 and thus the vector e1 form an angle α with the Cartesian x
axis and the Cartesian basis vector i. It can be seen from triangle
PAB that \cos \alpha = \tfrac where |e1| is the magnitude of the
basis vector e1 (the scalar intercept PB) and |i| is the magnitude
of the Cartesian basis vector i which is also the projection of e1
on the x axis (the scalar intercept PA). It follows, then, that
|\mathbf_1| = \tfrac and |\mathbf| = |\mathbf_1|\cos \alpha.
However, this method for basis vector transformations using
directional cosines is inapplicable to curvilinear coordinates for
the following reason. With increasing the distance from P, the
angle between the curved line q1 and Cartesian axis x increasingly
deviates from α. 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 α. The angles that the q1 line and q1
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 is
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 \tfrac (PD being the projection of PE on the x axis) becomes
almost exactly equal to cos α. Let the infinitesimally small
intercepts PD and PE be labelled, respectivelly, as dx and dq1.
Then \cos \alpha = \tfrac and \tfrac = \tfrac. Thus, the
directional cosines can be substituted in transformations with the
more exact ratios between infinitesimally small coordinate
intercepts. As pointed out
above, q_1 \equiv q_1(x,y,z) and x \equiv x(q_1,q_2,q_3) are
smooth (continuously differentiable) functions and, therefore, the
transformation ratios can be written as \tfrac = \tfrac = \tfrac
and \tfrac = \tfrac = \tfrac, that is, those ratios are partial
derivatives of coordinates belonging to one system with respect
to coordinates belonging to the other system.

From the foregoing discussion, it follows that
the component (projection) of e1 on the x axis is x =
\tfrac.|\mathbf_1| = \cos \alpha.|\mathbf_1| = \tfrac.|\mathbf_1|.
The projection of the normalized local basis vector (|e1| = 1) can
be made a vector directed along the x axis by multiplying it with
the standard basis vector i. Doing the same for the coordinates in
the other 2 dimensions, e1 can be expressed as: \mathbf_1 = \tfrac
\mathbf + \tfrac \mathbf + \tfrac \mathbf. Similar equations hold
for e2 and e2 so that the standard basis is transformed to local
(ordered and normalised) basis by the following system of
equations:

\begin \tfrac \mathbf + \tfrac \mathbf + \tfrac
\mathbf = \mathbf_1 \\ \tfrac \mathbf + \tfrac \mathbf + \tfrac
\mathbf = \mathbf_2 \\ \tfrac \mathbf + \tfrac \mathbf + \tfrac
\mathbf = \mathbf_3 \end

Vectors e1, e2, and e3 at the right hand side of
the above equation system are unit vectors (magnitude = 1) directed
along the 3 axes of the curvilinear coordinate system. However,
basis vectors in general curvilinear system are not required to be
of unit length: they can be of arbitrary magnitude and direction.
It can easily be shown that the condition |e1| = |e2| = |e3| = 1 is
a result of the above transformation, and not an a priori
requirement imposed on the curvilinear basis. Let the local basis
is not normalised, in effect, leaving the basis vectors with
arbitrary magnitudes. Then, instead of e1, e2, and e3 in the right
hand side, there will be \tfrac, \tfrac, and \tfrac which are again
unit vectors directed along the curvilinear coordinate axes.

By analogous reasoning, but this time projecting
the standard basis on the curvilinear axes ( |i| = |j| = |k| = 1
according to the definition of standard
basis), one can obtain the inverse transformation from local
basis to standard basis:

\begin \tfrac \mathbf_1 + \tfrac \mathbf_2 +
\tfrac \mathbf_3 = \mathbf \\ \tfrac \mathbf_1 + \tfrac \mathbf_2 +
\tfrac \mathbf_3 = \mathbf \\ \tfrac \mathbf_1 + \tfrac \mathbf_2 +
\tfrac \mathbf_3 = \mathbf \end

The above
systems of linear equations can be written in matrix form as
\tfrac \mathbf_i = \mathbf_k and \tfrac \mathbf_i = \mathbf_k where
xi (i = 1,2,3) are the Cartesian coordinates x, y, z and ii are the
standard basis vectors i, j, k. The system matrices (that is,
matrices composed of the coefficients in front of the unknowns)
are, respectively, \tfrac and \tfrac. At the same time, those two
matrices are the Jacobian matrices
Jik and J-1ik of the transformations of basis vectors from
curvilinear to Cartesian coordinates and vice versa. In the second
equation system (the inverse transformation), the unknowns are the
curvilinear basis vectors which are subject to the condition that
in each point of the curvilinear coordinate system there must exist
one and only one set of basis vectors. This condition is satisfied
iff (if and only if) the equation system has a single solution.
From linear
algebra, it is known that a linear equation system has a single
solution only if the determinant of its system matrix is non-zero.
For the second equation system, the determinant of the system
matrix is \det = J^ = \tfrac \neq 0 which shows the rationale
behind the
above requirement concerning the inverse Jacobian
determinant.

Another, very important, feature of the above
transformations is the nature of the derivatives: in front of the
Cartesian basis vectors stand derivatives of Cartesian coordinates
while in front of the curvilinear basis vectors stand derivatives
of curvililear coordinates. In general, the following definition
holds: This definition is so general that it applies to
covariance in the very abstract sense, and includes not only
basis vectors, but also all vectors, components, tensors,
pseudovectors, and pseudotensors (in the last two there is an
additional sign flip). It also serves to define tensors in one of
their most usual
treatment.

The partial derivative coefficients through which
vector transformation is achieved are called also scale factors or
Lamé coefficients (named after Gabriel
Lamé): h_ = \tfrac. However, the hik designation is very rarely
used, being largely replaced with √gik, the components of the
metric
tensor.

### Contravariant basis

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 defined to be orthogonal when- \mathbf_\cdot\mathbf_ = \delta_

Cartesian coordinates x_1,x_2,x_3 which have the
scalar product, are called Euclidean coordinates. It is often
convenient to associate the points of Euclidean space with vectors,
for example, with each point P we associate the vector (or arrow)
with its tail at the origin of coordinates and its tip at P. This
vector is called the radius vector with components (x_1, x_2, x_3).
At any point P of Euclidean space we can construct the small line
element

- d \bold = (dx_1,dx_2,dx_3) \,\!

- \lang f,h \rang =\sum_ x_ y_ \,\!.

- \lang d\mathbf,d\mathbf \rang = dx_1^2+dx_2^2+dx_3^2 .

The same Euclidean metric in curvilinear
coordinates is

- \lang d\mathbf,d\mathbf \rang = \sum_^3 \frac \frac dx_i' dx_j' .

- g_(x_i',x_j')= \sum_^3 \frac \frac

are called the fundamental (or metric) tensor of
the Euclidean space in curvilinear coordinates. Connection between
fundamental tensor and Lamé coefficients is g_(x_i',x_j')=
h_i^2.

### Example

If we consider polar coordinates for R2, note that- (x, y)=(r \cos \theta, r \sin \theta) \,\!

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. The
fundamental tensor is g1,1 =1, g2,2 =r2, g1,2 = g2,1 =0.

## Line and surface 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- \int_C f \,ds = \int_a^b f(\mathbf(t))\left|\right|\; dt

- \left|\right| = \left| \sum \right|

- = h_i \mathbf_

- \left|\right| = \sqrt

### Surface integrals

Likewise, if we are interested in a surface integral, the relevant calculation, with the parameterization of the surface in Cartesian coordinates is:- \int_S f \,ds = \iint_T f(\mathbf(s, t)) \left|\times \right| ds dt

- \left|\times \right| = \left| \times \right|

- = \sum h_ \mathbf_

- = \sum h_ \mathbf_

- \begin

## Grad, curl, div, Laplacian

In orthogonal curvilinear coordinates, one can express the gradient, curl, divergence, and Laplacian of a function or vector field as follows:- \nabla f = \sum_i \hat e_

- \nabla\cdot = \sum_i \left ( \right )

\nabla^2 f = \frac \sum_i \frac \frac
\frac,

where \Omega is the product of all h_i and
\epsilon_ is the Levi-Civita
symbol.

## References

- M. R. Spiegel, Vector Analysis, Schaum's Outline Series, New York, (1959).
- Mathematical Methods for Physicists

## See also

## External links

curvilinear in Czech: Křivočará soustava
souřadnic

curvilinear in French: Système de coordonnées
curvilignes

curvilinear in Italian: Coordinate
curvilinee