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 mechanics.
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 and physics 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.
Coordinate transformations.
Consider two coordinate systems with coordinate variables formula_1 and formula_2, which we shall represent in short as just formula_3 and formula_4 respectively and always assume our index formula_5 runs from 1 through 3. We shall assume that these coordinates systems are embedded in the three-dimensional euclidean space. Coordinates formula_3 and formula_4 may be used to explain each other, because as we move along the coordinate line in one coordinate system we can use the other to describe our position. In this way Coordinates formula_3 and formula_4 are functions of each other
which can be written as
These three equations together are also called a coordinate transformation from formula_4 to formula_3.Let us denote this transformation by formula_16. We will therefore represent the transformation from the coordinate system with coordinate variables formula_4 to the coordinate system with coordinates formula_3 as:
Similarly we can represent formula_4 as a function of formula_3 as follows:
similarly we can write the free equations more compactly as
These three equations together are also called a coordinate transformation from formula_3 to formula_4. Let us denote this transformation by formula_28. We will represent the transformation from the coordinate system with coordinate variables formula_3 to the coordinate system with coordinates formula_4 as:
If the transformation formula_16 is bijective then we call the image of the transformation,namely formula_3, a set of admissible coordinates for formula_4. If formula_16 is linear the coordinate system formula_3 will be called an affine coordinate system ,otherwise formula_3 is called a curvilinear coordinate system
The Jacobian.
As we now see that the Coordinates formula_3 and formula_4 are functions of each other, we can take the derivative of the coordinate variable formula_3 with respect to the coordinate variable formula_4
consider
The resultant matrix is called the Jacobian matrix.
Vectors in curvilinear coordinates.
Let (b1, b2, b3) be an arbitrary basis for three-dimensional Euclidean space. In general, the basis vectors are neither unit vectors nor mutually orthogonal. However, they are required to be linearly independent. Then a vector v can be expressed as
The components "vk" are the contravariant components of the vector v.
The reciprocal basis (b1, b2, b3) is defined by the relation 
where "δi j" is the Kronecker delta.
The vector v can also be expressed in terms of the reciprocal basis:
The components "vk" are the covariant components of the vector formula_58.
Second-order tensors in curvilinear coordinates.
A second-order tensor can be expressed as
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.
Metric tensor and relations between components.
The quantities "gij", "gij" are defined as
From the above equations we have
The components of a vector are related by
Also,
The components of the second-order tensor are related by
The alternating tensor.
In an orthonormal right-handed basis, the third-order alternating tensor is defined as
In a general curvilinear basis the same tensor may be expressed as
It can be shown that
Now,
Hence,
Similarly, we can show that
Vector operations.
Identity map.
The identity map I defined by formula_73 can be shown to be:
Scalar (dot) product.
The scalar product of two vectors in curvilinear coordinates is
Vector (cross) product.
The cross product of two vectors is given by:
where ε"ijk" is the permutation symbol and e"i" is a Cartesian basis vector. In curvilinear coordinates, the equivalent expression is:
where formula_78 is the third-order alternating tensor. The cross product of two vectors is given by:
where ε"ijk" is the permutation symbol and formula_80 is a Cartesian basis vector. Therefore,
and
Hence,
Returning to the vector product and using the relations:
gives us:
Tensor operations.
Identity map.
The identity map formula_86 defined by formula_87 can be shown to be
Action of a second-order tensor on a vector.
The action formula_89 can be expressed in curvilinear coordinates as
Inner product of two second-order tensors.
The inner product of two second-order tensors formula_91 can be expressed in curvilinear coordinates as
Alternatively,
Determinant of a second-order tensor.
If formula_94 is a second-order tensor, then the determinant is defined by the relation
where formula_96 are arbitrary vectors and
Relations between curvilinear and Cartesian basis vectors.
Let (e1, e2, e3) be the usual Cartesian basis vectors for the Euclidean space of interest and let
where "Fi" is a second-order transformation tensor that maps e"i" to b"i". Then,
From this relation we can show that
Let formula_101 be the Jacobian of the transformation. Then, from the definition of the determinant,
Since
we have
A number of interesting results can be derived using the above relations.
First, consider
Then
Similarly, we can show that
Therefore, using the fact that formula_108,
Another interesting relation is derived below. Recall that
where "A" is a, yet undetermined, constant. Then
This observation leads to the relations
In index notation,
where formula_114 is the usual permutation symbol.
We have not identified an explicit expression for the transformation tensor F because an alternative form of the mapping between curvilinear and Cartesian bases is more useful. Assuming a sufficient degree of smoothness in the mapping (and a bit of abuse of notation), we have
Similarly,
From these results we have
and
Vector and tensor calculus in three-dimensional curvilinear coordinates.
Simmonds, in his book on tensor analysis, quotes Albert Einstein 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 manifolds in general relativity, in the mechanics of curved shells, in examining the invariance properties of Maxwell's equations which has been of interest in metamaterials 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.
Basic definitions.
Let the position of a point in space be characterized by three coordinate variables formula_119.
The coordinate curve "q"1 represents a curve on which "q"2, "q"3 are constant. Let x be the position vector of the point relative to some origin. Then, assuming that such a mapping and its inverse exist and are continuous, we can write 
The fields ψ"i"(x) are called the curvilinear coordinate functions of the curvilinear coordinate system ψ(x) = φ−1(x).
The "qi" coordinate curves are defined by the one-parameter family of functions given by
with "qj, qk" fixed.
Tangent vector to coordinate curves.
The tangent vector to the curve x"i" at the point x"i"(α) (or to the coordinate curve "qi" at the point x) is
Gradient.
Scalar field.
Let "f"(x) be a scalar field in space. Then
The gradient of the field "f" is defined by
where c is an arbitrary constant vector. If we define the components "ci" of c are such that
then
If we set formula_127, then since formula_128, we have
which provides a means of extracting the contravariant component of a vector c.
If b"i" is the covariant (or natural) basis at a point, and if b"i" is the contravariant (or reciprocal) basis at that point, then
A brief rationale for this choice of basis is given in the next section.
Vector field.
A similar process can be used to arrive at the gradient of a vector field f(x). The gradient is given by
If we consider the gradient of the position vector field r(x) = x, then we can show that
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.
Since c is arbitrary, we can write
Note that the contravariant basis vector b"i" is perpendicular to the surface of constant ψ"i" and is given by
Christoffel symbols of the first kind.
The Christoffel symbols of the first kind are defined as
To express Γ"ijk" in terms of "gij" we note that
Since b"i,j" = b"j,i" we have Γ"ijk" = Γ"jik". Using these to rearrange the above relations gives
Christoffel symbols of the second kind.
The Christoffel symbols of the second kind are defined as
in which
This implies that
Other relations that follow are
Another particularly useful relation, which shows that the Christoffel symbol depends only on the metric tensor and its derivatives, is
Explicit expression for the gradient of a vector field.
The following expressions for the gradient of a vector field in curvilinear coordinates are quite useful.
Representing a physical vector field.
The vector field v can be represented as
where formula_145 are the covariant components of the field, formula_146 are the physical components, and (no summation)
is the normalized contravariant basis vector.
Second-order tensor field.
The gradient of a second order tensor field can similarly be expressed as
Explicit expressions for the gradient.
If we consider the expression for the tensor in terms of a contravariant basis, then
We may also write
Representing a physical second-order tensor field.
The physical components of a second-order tensor field can be obtained by using a normalized contravariant basis, i.e.,
where the hatted basis vectors have been normalized. This implies that (again no summation)
Divergence.
Vector field.
The divergence of a vector field (formula_58)is defined as
In terms of components with respect to a curvilinear basis
An alternative equation for the divergence of a vector field is frequently used. To derive this relation recall that
Now,
Noting that, due to the symmetry of formula_158,
we have
Recall that if ["gij"] is the matrix whose components are "gij", then the inverse of the matrix is formula_161. The inverse of the matrix is given by
where "Aij" are the Cofactor matrix of the components "gij". From matrix algebra we have
Hence,
Plugging this relation into the expression for the divergence gives
A little manipulation leads to the more compact form
Second-order tensor field.
The divergence of a second-order tensor field is defined using
where a is an arbitrary constant vector.
In curvilinear coordinates,
Laplacian.
Scalar field.
The Laplacian of a scalar field φ(x) is defined as
Using the alternative expression for the divergence of a vector field gives us
Now
Therefore,
Curl of a vector field.
The curl of a vector field v in covariant curvilinear coordinates can be written as
where
Orthogonal curvilinear coordinates.
Assume, for the purposes of this section, that the curvilinear coordinate system is orthogonal, i.e.,
or equivalently,
where formula_177. As before, formula_178 are covariant basis vectors and b"i", b"j" are contravariant basis vectors. Also, let (e1, e2, e3) be a background, fixed, Cartesian basis. A list of orthogonal curvilinear coordinates is given below.
Metric tensor in orthogonal curvilinear coordinates.
Let r(x) be the position vector of the point x with respect to the origin of the coordinate system. The notation can be simplified by noting that x = r(x). At each point we can construct a small line element dx. The square of the length of the line element is the scalar product dx • dx and is called the metric of the space. Recall that the space of interest is assumed to be Euclidean when we talk of curvilinear coordinates. Let us express the position vector in terms of the background, fixed, Cartesian basis, i.e.,
Using the chain rule, we can then express dx in terms of three-dimensional orthogonal curvilinear coordinates ("q"1, "q"2, "q"3) as
Therefore, the metric is given by
The symmetric quantity
is called the fundamental (or metric) tensor of the Euclidean space in curvilinear coordinates.
Note also that
where "hij" are the Lamé coefficients.
If we define the scale factors, "hi", using
we get a relation between the fundamental tensor and the Lamé coefficients.
Example: Polar coordinates.
If we consider polar coordinates for R2, note that
(r, θ) are the curvilinear coordinates, and the Jacobian determinant of the transformation ("r",θ) → ("r" cos θ, "r" sin θ) is "r".
The orthogonal basis vectors are b"r" = (cos θ, sin θ), bθ = (−"r" sin θ, "r" cos θ). The normalized basis vectors are e"r" = (cos θ, sin θ), eθ = (−sin θ, cos θ) and 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.
Line and surface integrals.
If we wish to use curvilinear coordinates for vector calculus calculations, adjustments need to be made in the calculation of line, surface and volume integrals. For simplicity, we again restrict the discussion to three dimensions and orthogonal curvilinear coordinates. However, the same arguments apply for formula_186-dimensional problems though there are some additional terms in the expressions when the coordinate system is not orthogonal.
Line integrals.
Normally in the calculation of line integrals we are interested in calculating
where x("t") parametrizes C in Cartesian coordinates.
In curvilinear coordinates, the term
by the chain rule. And from the definition of the Lamé coefficients,
and thus
Now, since formula_191 when formula_192, we have
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:
Again, in curvilinear coordinates, we have
and we make use of the definition of curvilinear coordinates again to yield
Therefore,
where formula_198 is the permutation symbol.
In determinant form, the cross product in terms of curvilinear coordinates will be:
Grad, curl, div, Laplacian.
In orthogonal curvilinear coordinates of 3 dimensions, where
one can express the gradient of a scalar or vector field as
For an orthogonal basis
The divergence of a vector field can then be written as
Also,
Therefore,
We can get an expression for the Laplacian in a similar manner by noting that
Then we have
The expressions for the gradient, divergence, and Laplacian can be directly extended to "n"-dimensions.
The curl of a vector field is given by
where ε"ijk" is the Levi-Civita symbol.
Example: Cylindrical polar coordinates.
For cylindrical coordinates we have
and
where
Then the covariant and contravariant basis vectors are
where formula_213 are the unit vectors in the formula_214 directions.
Note that the components of the metric tensor are such that
which shows that the basis is orthogonal.
The non-zero components of the Christoffel symbol of the second kind are
Representing a physical vector field.
The normalized contravariant basis vectors in cylindrical polar coordinates are
and the physical components of a vector v are
Gradient of a scalar field.
The gradient of a scalar field, "f"(x), in cylindrical coordinates can now be computed from the general expression in curvilinear coordinates and has the form
Gradient of a vector field.
Similarly, the gradient of a vector field, v(x), in cylindrical coordinates can be shown to be
Divergence of a vector field.
Using the equation for the divergence of a vector field in curvilinear coordinates, the divergence in cylindrical coordinates can be shown to be
Laplacian of a scalar field.
The Laplacian is more easily computed by noting that formula_222. In cylindrical polar coordinates
Hence,
Representing a physical second-order tensor field.
The physical components of a second-order tensor field are those obtained when the tensor is expressed in terms of a normalized contravariant basis. In cylindrical polar coordinates these components are:
Gradient of a second-order tensor field.
Using the above definitions we can show that the gradient of a second-order tensor field in cylindrical polar coordinates can be expressed as
Divergence of a second-order tensor field.
The divergence of a second-order tensor field in cylindrical polar coordinates can be obtained from the expression for the gradient by collecting terms where the scalar product of the two outer vectors in the dyadic products is nonzero. Therefore,