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 and , which we shall represent in short as just and respectively and always assume our index runs from 1 through 3. We shall assume that these coordinates systems are embedded in the three-dimensional euclidean space. Coordinates and 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 and are functions of each other
for
which can be written as
for
These three equations together are also called a coordinate transformation from to . Let us denote this transformation by . We will therefore represent the transformation from the coordinate system with coordinate variables to the coordinate system with coordinates as:
Similarly we can represent as a function of as follows:
for
and we can write the free equations more compactly as
for
These three equations together are also called a coordinate transformation from to . Let us denote this transformation by . We will represent the transformation from the coordinate system with coordinate variables to the coordinate system with coordinates as:
If the transformation is bijective then we call the image of the transformation, namely , a set of admissible coordinates for . If is linear the coordinate system will be called an affine coordinate system, otherwise is called a curvilinear coordinate system.
The Jacobian As we now see that the Coordinates and are functions of each other, we can take the derivative of the coordinate variable with respect to the coordinate variable .
Consider
for , these derivatives can be arranged in a matrix, say , in which is the element in the -th row and -th column
The resultant matrix is called the Jacobian matrix.
Vectors in curvilinear coordinates Let 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 can be expressed as
The components are the contravariant components of the vector .
The reciprocal basis is defined by the relation
where is the Kronecker delta.
The vector can also be expressed in terms of the reciprocal basis:
The components are the covariant components of the vector .
Second-order tensors in curvilinear coordinates A second-order tensor can be expressed as
The components are called the contravariant components, the mixed right-covariant components, the mixed left-covariant components, and the covariant components of the second-order tensor.
Metric tensor and relations between components The quantities , 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 defined by 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 is the permutation symbol and is a Cartesian basis vector. In curvilinear coordinates, the equivalent expression is:
where is the third-order alternating tensor. The cross product of two vectors is given by:
where is the permutation symbol and 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 defined by can be shown to be
Action of a second-order tensor on a vector The action can be expressed in curvilinear coordinates as
Inner product of two second-order tensors The inner product of two second-order tensors can be expressed in curvilinear coordinates as
Alternatively,
Determinant of a second-order tensor If is a second-order tensor, then the determinant is defined by the relation
where are arbitrary vectors and
Relations between curvilinear and Cartesian basis vectors Let be the usual Cartesian basis vectors for the Euclidean space of interest and let
where is a second-order transformation tensor that maps to . Then,
From this relation we can show that
Let 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 ,
Another interesting relation is derived below. Recall that
where is a, yet undetermined, constant. Then
This observation leads to the relations
In index notation,
where is the usual permutation symbol.
We have not identified an explicit expression for the transformation tensor 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
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 .
The coordinate curve represents a curve on which and are constant. Let 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 are called the curvilinear coordinate functions of the curvilinear coordinate system .
The coordinate curves are defined by the one-parameter family of functions given by
with , fixed.
Tangent vector to coordinate curves The tangent vector to the curve at the point (or to the coordinate curve at the point ) is
Gradient
Scalar field Let be a scalar field in space. Then
The gradient of the field is defined by
where is an arbitrary constant vector. If we define the components of are such that
then
If we set , then since , we have
which provides a means of extracting the contravariant component of a vector .
If is the covariant (or natural) basis at a point, and if 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 . The gradient is given by
If we consider the gradient of the position vector field , then we can show that
The vector field is tangent to the 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, . 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 .
Since is arbitrary, we can write
Note that the contravariant basis vector is perpendicular to the surface of constant and is given by
Christoffel symbols of the first kind The Christoffel symbols of the first kind are defined as
To express in terms of we note that
Since we have . 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 can be represented as
where are the covariant components of the field, 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 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 ,
we have
Recall that if is the matrix whose components are , then the inverse of the matrix is . The inverse of the matrix is given by
where is the cofactor matrix of the components . 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 is an arbitrary constant vector.
In curvilinear coordinates,
Laplacian
Scalar field The Laplacian of a scalar field 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 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 . As before, are covariant basis vectors and , are contravariant basis vectors. Also, let be a background, fixed, Cartesian basis. A list of orthogonal curvilinear coordinates is given below.
Metric tensor in orthogonal curvilinear coordinates