In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible (a one-to-one map) at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The name curvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved. Well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space (R3) are cylindrical and spherical coordinates. A Cartesian coordinate surface in this space is a coordinate plane; for example z = 0 defines the x-y plane. In the same space, the coordinate surface r = 1 in spherical coordinates is the surface of a unit sphere, which is curved. The formalism of curvilinear coordinates provides a unified and general description of the standard coordinate systems. Curvilinear coordinates are often used to define the location or distribution of physical quantities which may be, for example, scalars, vectors, or tensors. Mathematical expressions involving these quantities in vector calculus and tensor analysis (such as the gradient, divergence, curl, and Laplacian) can be transformed from one coordinate system to another, according to transformation rules for scalars, vectors, and tensors. Such expressions then become valid for any curvilinear coordinate system. A curvilinear coordinate system may be simpler to use than the Cartesian coordinate system for some applications. The motion of particles under the influence of central forces is usually easier to solve in spherical coordinates than in Cartesian coordinates; this is true of many physical problems with spherical symmetry defined in R3. Equations with boundary conditions that follow coordinate surfaces for a particular curvilinear coordinate system may be easier to solve in that system. While one might describe the motion of a particle in a rectangular box using Cartesian coordinates, it is easier to describe the motion in a sphere with spherical coordinates. Spherical coordinates are the most common curvilinear coordinate systems and are used in Earth sciences, cartography, quantum mechanics, relativity, and engineering.
Orthogonal curvilinear coordinates in 3 dimensions
Coordinates, basis, and vectors
For now, consider 3-D space. A point P in 3-D space (or its position vector r) can be defined using Cartesian coordinates (x, y, z) [equivalently written (x1, x2, x3)], by r = x e x + y e y + z e z {\displaystyle \mathbf {r} =x\mathbf {e} _{x}+y\mathbf {e} _{y}+z\mathbf {e} _{z}} , where ex, ey, ez are the standard basis vectors. It can also be defined by its curvilinear coordinates (q1, q2, q3) 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:
x = f 1 ( q 1 , q 2 , q 3 ) , y = f 2 ( q 1 , q 2 , q 3 ) , z = f 3 ( q 1 , q 2 , q 3 ) {\displaystyle x=f^{1}(q^{1},q^{2},q^{3}),\,y=f^{2}(q^{1},q^{2},q^{3}),\,z=f^{3}(q^{1},q^{2},q^{3})}
q 1 = g 1 ( x , y , z ) , q 2 = g 2 ( x , y , z ) , q 3 = g 3 ( x , y , z ) {\displaystyle q^{1}=g^{1}(x,y,z),\,q^{2}=g^{2}(x,y,z),\,q^{3}=g^{3}(x,y,z)}
The surfaces q1 = constant, q2 = constant, q3 = constant are called the coordinate surfaces; and the space curves formed by their intersection in pairs are called the coordinate curves. The coordinate axes are determined by the tangents to the coordinate curves at the intersection of three surfaces. They are not in general fixed directions in space, which happens to be the case for simple Cartesian coordinates, and thus there is generally no natural global basis for curvilinear coordinates. In the Cartesian system, the standard basis vectors can be derived from the derivative of the location of point P with respect to the local coordinate
e x = ∂ r ∂ x ; e y = ∂ r ∂ y ; e z = ∂ r ∂ z . {\displaystyle \mathbf {e} _{x}={\dfrac {\partial \mathbf {r} }{\partial x}};\;\mathbf {e} _{y}={\dfrac {\partial \mathbf {r} }{\partial y}};\;\mathbf {e} _{z}={\dfrac {\partial \mathbf {r} }{\partial z}}.}
Applying the same derivatives to the curvilinear system locally at point P defines the natural basis vectors:
h 1 = ∂ r ∂ q 1 ; h 2 = ∂ r ∂ q 2 ; h 3 = ∂ r ∂ q 3 . {\displaystyle \mathbf {h} _{1}={\dfrac {\partial \mathbf {r} }{\partial q^{1}}};\;\mathbf {h} _{2}={\dfrac {\partial \mathbf {r} }{\partial q^{2}}};\;\mathbf {h} _{3}={\dfrac {\partial \mathbf {r} }{\partial q^{3}}}.}
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 (Cartesian) and h or b is for the curvilinear basis. These may not have unit length, and may also not be orthogonal. In the case that they are orthogonal at all points where the derivatives are well-defined, we define the Lamé coefficients (after Gabriel Lamé) by
h 1 = | h 1 | ; h 2 = | h 2 | ; h 3 = | h 3 | {\displaystyle h_{1}=|\mathbf {h} _{1}|;\;h_{2}=|\mathbf {h} _{2}|;\;h_{3}=|\mathbf {h} _{3}|}
and the curvilinear orthonormal basis vectors by
b 1 = h 1 h 1 ; b 2 = h 2 h 2 ; b 3 = h 3 h 3 . {\displaystyle \mathbf {b} _{1}={\dfrac {\mathbf {h} _{1}}{h_{1}}};\;\mathbf {b} _{2}={\dfrac {\mathbf {h} _{2}}{h_{2}}};\;\mathbf {b} _{3}={\dfrac {\mathbf {h} _{3}}{h_{3}}}.}
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 space of R 3 {\displaystyle \mathbb {R} ^{3}} at P, and so are local to P.) In general, curvilinear coordinates allow the natural basis vectors hi not all mutually perpendicular to each other, and not required to be of unit length: they can be of arbitrary magnitude and direction. The use of an orthogonal basis makes vector manipulations simpler than for non-orthogonal. However, some areas of physics and engineering, particularly fluid mechanics and continuum mechanics, require non-orthogonal bases to describe deformations and fluid transport to account for complicated directional dependences of physical quantities. A discussion of the general case appears later on this page.
Vector calculus
Differential elements In orthogonal curvilinear coordinates, since the total differential change in r is
d r = ∂ r ∂ q 1 d q 1 + ∂ r ∂ q 2 d q 2 + ∂ r ∂ q 3 d q 3 = h 1 d q 1 b 1 + h 2 d q 2 b 2 + h 3 d q 3 b 3 {\displaystyle d\mathbf {r} ={\dfrac {\partial \mathbf {r} }{\partial q^{1}}}dq^{1}+{\dfrac {\partial \mathbf {r} }{\partial q^{2}}}dq^{2}+{\dfrac {\partial \mathbf {r} }{\partial q^{3}}}dq^{3}=h_{1}dq^{1}\mathbf {b} _{1}+h_{2}dq^{2}\mathbf {b} _{2}+h_{3}dq^{3}\mathbf {b} _{3}}
so scale factors are h i = | ∂ r ∂ q i | {\displaystyle h_{i}=\left|{\frac {\partial \mathbf {r} }{\partial q^{i}}}\right|}
In non-orthogonal coordinates the length of d r = d q 1 h 1 + d q 2 h 2 + d q 3 h 3 {\displaystyle d\mathbf {r} =dq^{1}\mathbf {h} _{1}+dq^{2}\mathbf {h} _{2}+dq^{3}\mathbf {h} _{3}} is the positive square root of d r ⋅ d r = d q i d q j h i ⋅ h j {\displaystyle d\mathbf {r} \cdot d\mathbf {r} =dq^{i}dq^{j}\mathbf {h} _{i}\cdot \mathbf {h} _{j}} (with Einstein summation convention). The six independent scalar products gij=hi.hj of the natural basis vectors generalize the three scale factors defined above for orthogonal coordinates. The nine gij are the components of the metric tensor, which has only three non zero components in orthogonal coordinates: g11=h1h1, g22=h2h2, g33=h3h3.
Covariant and contravariant bases
Spatial gradients, distances, time derivatives and scale factors are interrelated within a coordinate system by two groups of basis vectors:
basis vectors that are locally tangent to their associated coordinate pathline: b i = ∂ r ∂ q i {\displaystyle \mathbf {b} _{i}={\dfrac {\partial \mathbf {r} }{\partial q^{i}}}} are contravariant vectors (denoted by lowered indices), and basis vectors that are locally normal to the isosurface created by the other coordinates: b i = ∇ q i {\displaystyle \mathbf {b} ^{i}=\nabla q^{i}} are covariant vectors (denoted by raised indices), ∇ is the del operator. Note that, because of Einstein's summation convention, the position of the indices of the vectors is the opposite of that of the coordinates. Consequently, a general curvilinear coordinate system has two sets of basis vectors for every point: {b1, b2, b3} is the contravariant basis, and {b1, b2, b3} 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:
b i ⋅ b j = δ j i {\displaystyle \mathbf {b} ^{i}\cdot \mathbf {b} _{j}=\delta _{j}^{i}}
wherein δ j i {\displaystyle \delta _{j}^{i}} denotes the generalized Kronecker delta.
A vector v can be specified in terms of either basis, i.e.,
v = v 1 b 1 + v 2 b 2 + v 3 b 3 = v 1 b 1 + v 2 b 2 + v 3 b 3 {\displaystyle \mathbf {v} =v^{1}\mathbf {b} _{1}+v^{2}\mathbf {b} _{2}+v^{3}\mathbf {b} _{3}=v_{1}\mathbf {b} ^{1}+v_{2}\mathbf {b} ^{2}+v_{3}\mathbf {b} ^{3}}
Using the Einstein summation convention, the basis vectors relate to the components by
v ⋅ b i = v k b k ⋅ b i = v k δ k i = v i {\displaystyle \mathbf {v} \cdot \mathbf {b} ^{i}=v^{k}\mathbf {b} _{k}\cdot \mathbf {b} ^{i}=v^{k}\delta _{k}^{i}=v^{i}}
v ⋅ b i = v k b k ⋅ b i = v k δ i k = v i {\displaystyle \mathbf {v} \cdot \mathbf {b} _{i}=v_{k}\mathbf {b} ^{k}\cdot \mathbf {b} _{i}=v_{k}\delta _{i}^{k}=v_{i}}
and
v ⋅ b i = v k b k ⋅ b i = g k i v k {\displaystyle \mathbf {v} \cdot \mathbf {b} _{i}=v^{k}\mathbf {b} _{k}\cdot \mathbf {b} _{i}=g_{ki}v^{k}}
v ⋅ b i = v k b k ⋅ b i = g k i v k {\displaystyle \mathbf {v} \cdot \mathbf {b} ^{i}=v_{k}\mathbf {b} ^{k}\cdot \mathbf {b} ^{i}=g^{ki}v_{k}}
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, x is one of the Cartesian coordinates, and q1 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 q1 axis which is a tangent to that coordinate line at the point P. The axis q1 and thus the vector b1 form an angle α {\displaystyle \alpha } with the Cartesian x axis and the Cartesian basis vector e1. It can be seen from triangle PAB that
cos α = | e 1 | | b 1 | ⇒ | e 1 | = | b 1 | cos α {\displaystyle \cos \alpha ={\cfrac {|\mathbf {e} _{1}|}{|\mathbf {b} _{1}|}}\quad \Rightarrow \quad |\mathbf {e} _{1}|=|\mathbf {b} _{1}|\cos \alpha }
where |e1|, |b1| are the magnitudes of the two basis vectors, i.e., the scalar intercepts PB and PA. PA is also the projection of b1 on the x axis. However, this method for basis vector transformations using directional cosines is inapplicable to curvilinear coordinates for the following reasons:
By increasing the distance from P, the angle between the curved line q1 and Cartesian axis x increasingly deviates from α {\displaystyle \alpha } . 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 {\displaystyle \alpha } . The angles that the q1 line and that axis form with the x axis become closer in value the closer one moves towards point P and become exactly equal at P. Let point E be located very close to P, so close that the distance PE is infinitesimally small. Then PE measured on the q1 axis almost coincides with PE measured on the q1 line. At the same time, the ratio PD/PE (PD being the projection of PE on the x axis) becomes almost exactly equal to cos α {\displaystyle \cos \alpha } . Let the infinitesimally small intercepts PD and PE be labelled, respectively, as dx and dq1. Then
cos α = d x d q 1 = | e 1 | | b 1 | {\displaystyle \cos \alpha ={\cfrac {dx}{dq^{1}}}={\frac {|\mathbf {e} _{1}|}{|\mathbf {b} _{1}|}}} . 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
p 1 = b 1 ⋅ e 1 | e 1 | = | b 1 | | e 1 | | e 1 | cos α = | b 1 | d x d q 1 ⇒ p 1 | b 1 | = d x d q 1 {\displaystyle p^{1}=\mathbf {b} _{1}\cdot {\cfrac {\mathbf {e} _{1}}{|\mathbf {e} _{1}|}}=|\mathbf {b} _{1}|{\cfrac {|\mathbf {e} _{1}|}{|\mathbf {e} _{1}|}}\cos \alpha =|\mathbf {b} _{1}|{\cfrac {dx}{dq^{1}}}\quad \Rightarrow \quad {\cfrac {p^{1}}{|\mathbf {b} _{1}|}}={\cfrac {dx}{dq^{1}}}} . If qi = qi(x1, x2, x3) and xi = xi(q1, q2, q3) are smooth (continuously differentiable) functions the transformation ratios can be written as ∂ q i ∂ x j {\displaystyle {\cfrac {\partial q^{i}}{\partial x_{j}}}} and ∂ x i ∂ q j {\displaystyle {\cfrac {\partial x_{i}}{\partial q^{j}}}} . That is, those ratios are partial derivatives of coordinates belonging to one system with respect to coordinates belonging to the other system.
Constructing a covariant basis in three dimensions Doing the same for the coordinates in the other 2 dimensions, b1 can be expressed as:
b 1 = p 1 e 1 + p 2 e 2 + p 3 e 3 = ∂ x 1 ∂ q 1 e 1 + ∂ x 2 ∂ q 1 e 2 + ∂ x 3 ∂ q 1 e 3 {\displaystyle \mathbf {b} _{1}=p^{1}\mathbf {e} _{1}+p^{2}\mathbf {e} _{2}+p^{3}\mathbf {e} _{3}={\cfrac {\partial x_{1}}{\partial q^{1}}}\mathbf {e} _{1}+{\cfrac {\partial x_{2}}{\partial q^{1}}}\mathbf {e} _{2}+{\cfrac {\partial x_{3}}{\partial q^{1}}}\mathbf {e} _{3}}
Similar equations hold for b2 and b3 so that the standard basis {e1, e2, e3} is transformed to a local (ordered and normalised) basis {b1, b2, b3} by the following system of equations:
b 1 =