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

English

Adjective

1. Having bends.

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)

The above equation system can be solved for the arguments q1, q2, and q3 with solutions in the form:

q1 = q1(x, y, z)     inverse transformation

q2 = q2(x, y, z)     (Cartesian to curvilinear coordinates)

q3 = q3(x, y, z)

The transformation functions are such that there is 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:

1) They are smooth functions

2) The Jacobian determinant

=\begin \frac & \frac & \frac \\ \frac & \frac & \frac \\ \frac & \frac & \frac \end \neq 0 is not zero; that is, 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.

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

=\begin \sin\theta\cos\varphi & \sin\theta\sin\varphi & \cos\theta\\ \frac\cos\theta\cos\varphi & \frac\cos\theta\sin\varphi & -\frac\sin\theta \\ -\frac\frac & \frac\frac & 0 \end = \frac \neq 0

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_

where δij is the Kronecker 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) \,\!

which is vector too. Two vectors h = (x_1, x_2, x_3) and f=(y_l, y_2, y_3) from the same origin can be added and result is the vector with coordinates (x_l+ y_l, x_2 + y_2, x_3 + y_3). A vector can also be multiplied by any real number. The Euclidean scalar product of two (real) vectors is the number

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

The scalar product of the vector with itself give the square of the vector length. The square of the length of a line element in space with scalar product is called the metric of the space. The metric of Euclidean space is

\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' .

The symmetric tensor

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) \,\!

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

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

where x(t) parametrizes C in Cartesian coordinates. In curvilinear coordinates, the term

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

by the chain rule. But from the definition of the curvilinear coordinates,

= h_i \mathbf_

and thus

\left|\right| = \sqrt

and we can proceed normally.

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

Again, in curvilinear coordinates, the term

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

and we make use of the definition of curvilinear coordinates again to yield

= \sum h_ \mathbf_

and

= \sum h_ \mathbf_

where the cross product, in terms of curvilinear coordinates, will be:

\begin

\mathbf_ & \mathbf_ & \mathbf_ \\ && \\ h_1 & h_2 & h_3 \\ && \\ h_1 & h_2 & h_3 \end

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\times = \frac \sum_i \hat e_ \sum_ \epsilon_ h_i \frac \qquad (\hbox n=3)

\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

• Covariance and contravariance
• Basic introduction to the mathematics of curved spacetime
• Orthogonal coordinates
• Frenet-Serret formulas

External links

• Derivation of Unit Vectors in Curvilinear Coordinates
• MathWorld's page on Curvilinear Coordinates