Tensors in curvilinear coordinates

Summary

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 edit

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.[6]

Coordinate transformations edit

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  

similarly 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 edit

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 edit

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[4]: 27 

 
The components vk are the contravariant components of the vector v.

The reciprocal basis (b1, b2, b3) is defined by the relation [4]: 28–29 

 
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  .

Second-order tensors in curvilinear coordinates edit

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 edit

The quantities gij, gij are defined as[4]: 39 

 
From the above equations we have
 

The components of a vector are related by[4]: 30–32 

 
 
Also,
 
 

The components of the second-order tensor are related by

 

The alternating tensor edit

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 edit

Identity map edit

The identity map I defined by   can be shown to be:[4]: 39 

 

Scalar (dot) product edit

The scalar product of two vectors in curvilinear coordinates is[4]: 32 

 

Vector (cross) product edit

The cross product of two vectors is given by:[4]: 32–34 

 

where εijk is the permutation symbol and ei 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 εijk 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 edit

Identity map edit

The identity map   defined by   can be shown to be[4]: 39 

 

Action of a second-order tensor on a vector edit

The action   can be expressed in curvilinear coordinates as

 

Inner product of two second-order tensors edit

The inner product of two second-order tensors   can be expressed in curvilinear coordinates as

 

Alternatively,

 

Determinant of a second-order tensor edit

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 edit

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 ei to bi. 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 A 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 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 edit

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.[6]

Basic definitions edit

Let the position of a point in space be characterized by three coordinate variables  .

The coordinate curve q1 represents a curve on which q2, q3 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 [2]: 55 

 
The fields ψi(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 edit

The tangent vector to the curve xi at the point xi(α) (or to the coordinate curve qi at the point x) is

 

Gradient edit

Scalar field edit

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  , then since  , we have

 
which provides a means of extracting the contravariant component of a vector c.

If bi is the covariant (or natural) basis at a point, and if bi 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 edit

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 bi 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, bi. 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 bi is perpendicular to the surface of constant ψi and is given by

 

Christoffel symbols of the first kind edit

The Christoffel symbols of the first kind are defined as

 
To express Γijk in terms of gij we note that
 
Since bi,j = bj,i we have Γijk = Γjik. Using these to rearrange the above relations gives
 

Christoffel symbols of the second kind edit

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 edit

The following expressions for the gradient of a vector field in curvilinear coordinates are quite useful.

 

Representing a physical vector field edit

The vector field v 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 edit

The gradient of a second order tensor field can similarly be expressed as

 

Explicit expressions for the gradient edit

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 edit

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 edit

Vector field edit

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 [gij] is the matrix whose components are gij, then the inverse of the matrix is  . 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 edit

The divergence of a second-order tensor field is defined using

 
where a is an arbitrary constant vector. [11] In curvilinear coordinates,
 

Laplacian edit

Scalar field edit

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 edit

The curl of a vector field v in covariant curvilinear coordinates can be written as

 
where
 

Orthogonal curvilinear coordinates edit

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 bi, bj 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 edit

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 (q1, q2, q3) 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 edit

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 br = (cos θ, sin θ), bθ = (−r sin θ, r cos θ). The normalized basis vectors are er = (cos θ, sin θ), eθ = (−sin θ, cos θ) and the scale factors are hr = 1 and hθ= r. The fundamental tensor is g11 =1, g22 =r2, g12 = g21 =0.

Line and surface integrals edit

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  -dimensional problems though there are some additional terms in the expressions when the coordinate system is not orthogonal.

Line integrals edit

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   when  , we have

 
and we can proceed normally.

Surface integrals edit

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   is the permutation symbol.

In determinant form, the cross product in terms of curvilinear coordinates will be:

 

Grad, curl, div, Laplacian edit

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 εijk is the Levi-Civita symbol.

Example: Cylindrical polar coordinates edit

For cylindrical coordinates we have

 
and
 
where
 

Then the covariant and contravariant basis vectors are

 
where   are the unit vectors in the   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 edit

The normalized contravariant basis vectors in cylindrical polar coordinates are

 
and the physical components of a vector v are
 

Gradient of a scalar field edit

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 edit

Similarly, the gradient of a vector field, v(x), in cylindrical coordinates can be shown to be

 

Divergence of a vector field edit

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 edit

The Laplacian is more easily computed by noting that  . In cylindrical polar coordinates

 
Hence,
 

Representing a physical second-order tensor field edit

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 edit

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 edit

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,

 

See also edit

References edit

Notes
  1. ^ a b c Green, A. E.; Zerna, W. (1968). Theoretical Elasticity. Oxford University Press. ISBN 0-19-853486-8.
  2. ^ a b c Ogden, R. W. (2000). Nonlinear elastic deformations. Dover.
  3. ^ Naghdi, P. M. (1972). "Theory of shells and plates". In S. Flügge (ed.). Handbook of Physics. Vol. VIa/2. pp. 425–640.
  4. ^ a b c d e f g h i j k Simmonds, J. G. (1994). A brief on tensor analysis. Springer. ISBN 0-387-90639-8.
  5. ^ a b Basar, Y.; Weichert, D. (2000). Numerical continuum mechanics of solids: fundamental concepts and perspectives. Springer.
  6. ^ a b c Ciarlet, P. G. (2000). Theory of Shells. Vol. 1. Elsevier Science.
  7. ^ Einstein, A. (1915). "Contribution to the Theory of General Relativity". In Laczos, C. (ed.). The Einstein Decade. p. 213.
  8. ^ Misner, C. W.; Thorne, K. S.; Wheeler, J. A. (1973). Gravitation. W. H. Freeman and Co. ISBN 0-7167-0344-0.
  9. ^ Greenleaf, A.; Lassas, M.; Uhlmann, G. (2003). "Anisotropic conductivities that cannot be detected by EIT". Physiological Measurement. 24 (2): 413–419. doi:10.1088/0967-3334/24/2/353. PMID 12812426. S2CID 250813768.
  10. ^ Leonhardt, U.; Philbin, T. G. (2006). "General relativity in electrical engineering". New Journal of Physics. 8 (10): 247. arXiv:cond-mat/0607418. Bibcode:2006NJPh....8..247L. doi:10.1088/1367-2630/8/10/247. S2CID 12100599.
  11. ^ "The divergence of a tensor field". Introduction to Elasticity/Tensors. Wikiversity. Retrieved 2010-11-26.
Further reading
  • Spiegel, M. R. (1959). Vector Analysis. New York: Schaum's Outline Series. ISBN 0-07-084378-3.
  • Arfken, George (1995). Mathematical Methods for Physicists. Academic Press. ISBN 0-12-059877-9.

External links edit

  • Derivation of Unit Vectors in Curvilinear Coordinates
  • MathWorld's page on Curvilinear Coordinates
  • Prof. R. Brannon's E-Book on Curvilinear Coordinates