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.
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.
[1] 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,[2] Naghdi,[3] Simmonds,[4] Green and Zerna,[1] Basar and Weichert,[5] and Ciarlet.
In an orthonormal right-handed basis, the third-order alternating tensor is defined as
where εijk is the permutation symbol and ei is a Cartesian basis vector.
Let (e1, e2, e3) be the usual Cartesian basis vectors for the Euclidean space of interest and let
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
Simmonds,[4] in his book on tensor analysis, quotes Albert Einstein saying[7] 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,[8] in the mechanics of curved shells,[6] in examining the invariance properties of Maxwell's equations which has been of interest in metamaterials[9][10] 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,[2] Simmonds,[4] Green and Zerna,[1] Basar and Weichert,[5] and Ciarlet.
Note that the contravariant basis vector bi is perpendicular to the surface of constant ψi and is given by
Another particularly useful relation, which shows that the Christoffel symbol depends only on the metric tensor and its derivatives, is
The following expressions for the gradient of a vector field in curvilinear coordinates are quite useful.
The physical components of a second-order tensor field can be obtained by using a normalized contravariant basis, i.e.,
The curl of a vector field v in covariant curvilinear coordinates can be written as
Assume, for the purposes of this section, that the curvilinear coordinate system is orthogonal, i.e.,
Let r(x) be the position vector of the point x with respect to the origin of the coordinate system.
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 (q1, q2, q3) as
is called the fundamental (or metric) tensor of the Euclidean space in curvilinear coordinates.
For simplicity, we again restrict the discussion to three dimensions and orthogonal curvilinear coordinates.
-dimensional problems though there are some additional terms in the expressions when the coordinate system is not orthogonal.
In determinant form, the cross product in terms of curvilinear coordinates will be:
The normalized contravariant basis vectors in cylindrical polar coordinates are
Similarly, the gradient of a vector field, v(x), in cylindrical coordinates can be shown to be
Using the above definitions we can show that the gradient of a second-order tensor field in cylindrical polar coordinates can be expressed as
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.