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1Vector and tensor algebra in three-dimensional curvilinear coordinates
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2Vector and tensor calculus in three-dimensional curvilinear coordinates
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3Orthogonal curvilinear coordinates
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4Example: Cylindrical polar coordinates
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4.1Representing a physical vector field
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4.2Gradient of a scalar field
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4.3Gradient of a vector field
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4.4Divergence of a vector field
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4.5Laplacian of a scalar field
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4.6Representing a physical second-order tensor field
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4.7Gradient of a second-order tensor field
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4.8Divergence of a second-order tensor field
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5See also
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6References
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7External links
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.[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
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
which can be written as
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:
similarly we can write the free equations more compactly as
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
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[4]: 27
The reciprocal basis (b1, b2, b3) is defined by the relation [4]: 28–29
The vector v can also be expressed in terms of the reciprocal basis:
Second-order tensors in curvilinear coordinates
A second-order tensor can be expressed as
Metric tensor and relations between components
The quantities gij, gij are defined as[4]: 39
The components of a vector are related by[4]: 30–32
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
Vector operations
Identity map
The identity map I defined by can be shown to be:[4]: 39
Scalar (dot) product
The scalar product of two vectors in curvilinear coordinates is[4]: 32
Vector (cross) product
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
Identity map
The identity map defined by can be shown to be[4]: 39
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 (e1, e2, e3) be the usual Cartesian basis vectors for the Euclidean space of interest and let
First, consider
Another interesting relation is derived below. Recall that
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
Vector and tensor calculus in three-dimensional curvilinear coordinates
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
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 qi coordinate curves are defined by the one-parameter family of functions given by
Tangent vector to coordinate curves
The tangent vector to the curve xi at the point xi(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
If we set , then since , we have
If bi is the covariant (or natural) basis at a point, and if bi is the contravariant (or reciprocal) basis at that point, then
Vector field
A similar process can be used to arrive at the gradient of a vector field f(x). The gradient is given by
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
The Christoffel symbols of the first kind are defined as
Christoffel symbols of the second kind
The Christoffel symbols of the second kind are defined as
This implies that
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
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
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.,
Divergence
Vector field
The divergence of a vector field ()is defined as
An alternative equation for the divergence of a vector field is frequently used. To derive this relation recall that
Second-order tensor field
The divergence of a second-order tensor field is defined using
Laplacian
Scalar field
The Laplacian of a scalar field φ(x) is defined as
Curl of a vector field
The curl of a vector field v in covariant curvilinear coordinates can be written as
Orthogonal curvilinear coordinates
Assume, for the purposes of this section, that the curvilinear coordinate system is orthogonal, i.e.,
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 (q1, q2, q3) as
The symmetric quantity
Note also that
If we define the scale factors, hi, using
Example: Polar coordinates
If we consider polar coordinates for R2, note that
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
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
Normally in the calculation of line integrals we are interested in calculating
by the chain rule. And from the definition of the Lamé coefficients,
and thus
Now, since when , we have
Surface integrals
Likewise, if we are interested in a surface integral, the relevant calculation, with the parameterization of the surface in Cartesian coordinates is:
Therefore,
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
The curl of a vector field is given by
Example: Cylindrical polar coordinates
For cylindrical coordinates we have
Then the covariant and contravariant basis vectors are
Note that the components of the metric tensor are such that
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
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 . In cylindrical polar coordinates
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,
See also
- Covariance and contravariance
- Basic introduction to the mathematics of curved spacetime
- Orthogonal coordinates
- Frenet–Serret formulas
- Covariant derivative
- Tensor derivative (continuum mechanics)
- Curvilinear perspective
- Del in cylindrical and spherical coordinates
References
- Notes
- ^ a b c Green, A. E.; Zerna, W. (1968). Theoretical Elasticity. Oxford University Press. ISBN 0-19-853486-8.
- ^ a b c Ogden, R. W. (2000). Nonlinear elastic deformations. Dover.
- ^ Naghdi, P. M. (1972). "Theory of shells and plates". In S. Flügge (ed.). Handbook of Physics. Vol. VIa/2. pp. 425–640.
- ^ 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.
- ^ a b Basar, Y.; Weichert, D. (2000). Numerical continuum mechanics of solids: fundamental concepts and perspectives. Springer.
- ^ a b c Ciarlet, P. G. (2000). Theory of Shells. Vol. 1. Elsevier Science.
- ^ Einstein, A. (1915). "Contribution to the Theory of General Relativity". In Laczos, C. (ed.). The Einstein Decade. p. 213.
- ^ Misner, C. W.; Thorne, K. S.; Wheeler, J. A. (1973). Gravitation. W. H. Freeman and Co. ISBN 0-7167-0344-0.
- ^ 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.
- ^ 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.
- ^ "The divergence of a tensor field". Introduction to Elasticity/Tensors. Wikiversity. Retrieved 2010-11-26.
- Further reading