![Curvilinear](https://upload.wikimedia.org/wikipedia/commons/0/0c/Curvilinear.svg)
In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, curvilinear coordinates are a
coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
for
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
in which the
coordinate line
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
s may be curved. These coordinates may be derived from a set of
Cartesian coordinate
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
s 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
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
of the curvilinear systems are curved.
Well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space (R
3) are
cylindrical
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infini ...
and
spherical
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
coordinates. A Cartesian coordinate surface in this space is a
coordinate plane
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
; 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
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, 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,
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
s,
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
s, or
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
s. Mathematical expressions involving these quantities in
vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...
and
tensor analysis
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
(such as the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
,
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
,
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
, and
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
) 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 force
In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force.
: \vec = \mathbf(\mathbf) = \left\vert F( \mathbf ) \right\vert \hat
where \vec F is the force, F is a vecto ...
s is usually easier to solve in
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
than in Cartesian coordinates; this is true of many physical problems with
spherical symmetry
In geometry, circular symmetry is a type of continuous symmetry for a planar object that can be rotated by any arbitrary angle and map onto itself.
Rotational circular symmetry is isomorphic with the circle group in the complex plane, or the ...
defined in R
3. Equations with
boundary conditions
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
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's 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
Earth science or geoscience includes all fields of natural science related to the planet Earth. This is a branch of science dealing with the physical, chemical, and biological complex constitutions and synergistic linkages of Earth's four sphere ...
,
cartography
Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an im ...
,
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
,
relativity, and
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
.
Orthogonal curvilinear coordinates in 3 dimensions
Coordinates, basis, and vectors
![Spherical coordinate elements](https://upload.wikimedia.org/wikipedia/commons/9/96/Spherical_coordinate_elements.svg)
For now, consider
3-D space. A point ''P'' in 3d space (or its
position vector
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or s ...
r) can be defined using Cartesian coordinates (''x'', ''y'', ''z'')
1, ''x''2, ''x''3)">quivalently written (''x''1, ''x''2, ''x''3) by
, where e
''x'', e
''y'', e
''z'' are the ''
standard basis
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the c ...
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
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
. The coordinate axes are determined by the
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
s 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
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the c ...
(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
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 inv ...
(after
Gabriel Lamé
Gabriel Lamé (22 July 1795 – 1 May 1870) was a French mathematician who contributed to the theory of partial differential equations by the use of curvilinear coordinates, and the mathematical theory of elasticity (for which linear elasticity ...
) 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. (They technically form a basis for the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
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
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
, particularly
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...
and
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
, 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
In calculus, the differential represents the principal part of the change in a function ''y'' = ''f''(''x'') with respect to changes in the independent variable. The differential ''dy'' is defined by
:dy = f'(x)\,dx,
where f'(x) is the ...
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
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of i ...
). 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
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
, which has only three non zero components in orthogonal coordinates: ''g''
11=''h''
1''h''
1, ''g''
22=''h''
2''h''
2, ''g''
33=''h''
3''h''
3.
Covariant and contravariant bases
![Vector 1-form](https://upload.wikimedia.org/wikipedia/commons/6/6b/Vector_1-form.svg)
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 (denoted by lowered indices), and
# basis vectors that are locally normal to the isosurface created by the other coordinates:
are
covariant vectors (denoted by raised indices), ∇ is the
del
Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes th ...
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 (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.
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 (see below).
A vector can be specified with covariant coordinates (lowered indices, written ''vk'') or contravariant coordinates (raised indices, written ''vk''). 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 (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
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
, ''x'' is one of the Cartesian coordinates, and ''q''1 is one of the curvilinear coordinates. The local (non-unit) basis vector is b1 (notated h1 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 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 ''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 b1 on the ''x'' axis is
:.
If ''qi'' = ''qi''(''x''1, ''x''2, ''x''3) and ''xi'' = ''xi''(''q''1, ''q''2, ''q''3) are smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebraic ...
(continuously differentiable) functions the transformation ratios can be written as and . That is, those ratios are partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
s 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, b1 can be expressed as:
:
Similar equations hold for b2 and b3 so that the standard basis is transformed to a local (ordered and ''normalised'') basis 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
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables.
For example,
:\begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of three equations in the ...
can be written in matrix form using the Einstein summation convention as
:.
This coefficient matrix In linear algebra, a coefficient matrix is a matrix consisting of the coefficients of the variables in a set of linear equations. The matrix is used in solving systems of linear equations.
Coefficient matrix
In general, a system with ''m'' linear e ...
of the linear system is the Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as ...
(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. In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
, 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
The formalism extends to any finite dimension as follows.
Consider the real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
Euclidean ''n''-dimensional space, that is R''n'' = R × R × ... × R (''n'' times) where R is the set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
and × denotes the Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ti ...
, which is a vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
.
The coordinates
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
of this space can be denoted by: x = (''x''1, ''x''2,...,''xn''). Since this is a vector (an element of the vector space), it can be written as:
:
where e1 = (1,0,0...,0), e2 = (0,1,0...,0), e3 = (0,0,1...,0),...,e''n'' = (0,0,0...,1) is the ''standard basis
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the c ...
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
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
(perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
) and normalized (has unit magnitude).
More generally, we can define basis vectors b''i'' so that they depend on q = (''q''1, ''q''2,...,''qn''), 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
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
'' with respect to this basis ''vi'' also necessarily depend on x also, that is ''vi'' = ''vi''(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
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
e''i'' by a number ''v''''i'' – scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector by ...
):
:
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
In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an a ...
on the differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
En (n-dimensional Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
) that is diffeomorphic
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...
to the Cartesian Cartesian means of or relating to the French philosopher René Descartes—from his Latinized name ''Cartesius''. It may refer to:
Mathematics
*Cartesian closed category, a closed category in category theory
*Cartesian coordinate system, modern ...
coordinate patch on the manifold. 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
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
.
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 bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
s, and fulfil the following requirements within their domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
s:
Vector and tensor algebra in three-dimensional curvilinear coordinates
Elementary vector and tensor algebra in curvilinear coordinates is used in some of the older scientific literature in mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
and physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and can be indispensable to understanding work from the early and mid-1900s, for example the text by Green and Zerna. 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, Naghdi, Simmonds, Green and Zerna,[ Basar and Weichert,] and Ciarlet.
Tensors in curvilinear coordinates
A second-order tensor can be expressed as
:
where denotes the tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
. The components ''Sij'' are called the contravariant components, ''Si j'' the mixed right-covariant components, ''Si j'' the mixed left-covariant components, and ''Sij'' the 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 , so the square of the length of the line element is the scalar product dx • dx and is called the metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathema ...
of the space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
, given by:
:.
The following portion of the above equation
:
is a ''symmetric'' tensor called the fundamental (or metric) tensor of the Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
in curvilinear coordinates.
Indices can be raised and lowered by the metric:
:
Relation to Lamé coefficients
Defining the scale factors ''hi'' by
:
gives a relation between the metric tensor and the Lamé coefficients, and
:
where ''hij'' are the Lamé coefficients. For an orthogonal basis we also have:
:
Example: Polar coordinates
If we consider polar coordinates for R2,
:
(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
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
basis vectors are b''r'' = (cos θ, sin θ), bθ = (−r sin θ, r 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 In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. section §7. The index subset must generally either be all ...
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
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distance ...
of the first kind :
:
where the comma denotes a partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
(see Ricci calculus
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be cal ...
). To express Γ''kij'' in terms of ''gij'',
:
Since
:
using these to rearrange the above relations gives
:
;Christoffel symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distance ...
of the second kind :
:
This implies that
: since .
Other relations that follow are
:
Vector operations
Vector and tensor calculus in three-dimensional curvilinear coordinates
Adjustments need to be made in the calculation of line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
, surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
and volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
integrals
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with di ...
. 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,[ in his book on ]tensor analysis
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
, quotes Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
saying
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 manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s in general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, in the mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
of curved shells,[ in examining the invariance properties of ]Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
which has been of interest in metamaterials
A metamaterial (from the Ancient Greek, Greek word Meta, μετά ''meta'', meaning "beyond" or "after", and the Latin word ''materia'', meaning "matter" or "material") is any material engineered to have a property that is not found in naturally ...
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, Simmonds, Green and Zerna,[ Basar and Weichert,][ and Ciarlet.][
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
Integration
:
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 ''qi'' 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.
:
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 (d2''x''''j''/d''t''2 = 0). 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 d2''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. 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
In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
.
This more general context makes clear the correspondence between the concepts of centrifugal force in rotating coordinate system
A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth. (This article considers onl ...
s and in stationary curvilinear coordinate systems. (Both of these concepts appear frequently in the literature.) 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
* Covariance and contravariance
* Introduction to the mathematics of general relativity
The mathematics of general relativity is complex. In Newton's theories of motion, an object's length and the rate at which time passes remain constant while the object accelerates, meaning that many problems in Newtonian mechanics may be solve ...
* Special cases:
** Orthogonal coordinates In mathematics, orthogonal coordinates are defined as a set of ''d'' coordinates q = (''q''1, ''q''2, ..., ''q'd'') in which the coordinate hypersurfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate su ...
** Skew coordinates A system of skew coordinates is a curvilinear coordinate system where the coordinate surfaces are not orthogonal, in contrast to orthogonal coordinates.
Skew coordinates tend to be more complicated to work with compared to orthogonal coordinates ...
* Tensors in curvilinear coordinates Curvilinear coordinates can be formulated in tensor calculus, with important applications in physics and engineering, particularly for describing transportation of physical quantities and deformation of matter in fluid mechanics and continuum mecha ...
* Frenet–Serret formulas
In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space \mathbb^, or the geometric properties of the curve itself irrespective ...
* Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a different ...
* Tensor derivative (continuum mechanics)
The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in ...
* Curvilinear perspective
Curvilinear perspective, also five-point perspective, is a graphical projection used to draw 3D objects on 2D surfaces. It was formally codified in 1968 by the artists and art historians André Barre and Albert Flocon in the book ''La Perspective c ...
* Del in cylindrical and spherical coordinates
This is a list of some vector calculus formulae for working with common curvilinear coordinates, curvilinear coordinate systems.
Notes
* This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11#Coordinate systems, ISO 31- ...
References
Further reading
*
*
External links
Planetmath.org Derivation of Unit vectors in curvilinear coordinates
Prof. R. Brannon's E-Book on Curvilinear Coordinates
* Wikiversity:Introduction to Elasticity/Tensors#The divergence of a tensor field – Wikiversity
Wikiversity is a Wikimedia Foundation project that supports learning communities, their learning materials, and resulting activities. It differs from Wikipedia in that it offers tutorials and other materials for the fostering of learning, rather ...
, Introduction to Elasticity/Tensors.
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Coordinate systems
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