International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 |
International Tables for Crystallography (2013). Vol. D, ch. 1.1, pp. 7-10
Section 1.1.3. Mathematical notion of tensor^{a}Institut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France |
For the mathematical definition of tensors, the reader may consult, for instance, Lichnerowicz (1947), Schwartz (1975) or Sands (1995).
A linear form in the space is written where is independent of the chosen basis and the 's are the coordinates of T in the dual basis. Let us consider now a bilinear form in the product space of two vector spaces with n and p dimensions, respectively:
The np quantities 's are, by definition, the components of a tensor of rank 2 and the form is invariant if one changes the basis in the space . The tensor is said to be twice covariant. It is also possible to construct a bilinear form by replacing the spaces and by their respective conjugates and . Thus, one writes where is the doubly contravariant form of the tensor, whereas and are mixed, once covariant and once contravariant.
We can generalize by defining in the same way tensors of rank 3 or higher by using trilinear or multilinear forms. A vector is a tensor of rank 1, and a scalar is a tensor of rank 0.
Let us consider two vector spaces, with n dimensions and with p dimensions, and let there be two linear forms, in and in . We shall associate with these forms a bilinear form called a tensor product which belongs to the product space with np dimensions, :
This correspondence possesses the following properties:
The analytical expression of the tensor product is then One deduces from this that
It is a tensor of rank 2. One can equally well envisage the tensor product of more than two spaces, for example, in npq dimensions. We shall limit ourselves in this study to the case of affine tensors, which are defined in a space constructed from the product of the space with itself or with its conjugate . Thus, a tensor product of rank 3 will have components. The tensor product can be generalized as the product of multilinear forms. One can write, for example,
A multilinear form is, by definition, invariant under a change of basis. Let us consider, for example, the trilinear form (1.1.3.1). If we change the system of coordinates, the components of vectors x, y, z become
Let us put these expressions into the trilinear form (1.1.3.1):
Now we can equally well make the components of the tensor appear in the new basis:
As the decomposition is unique, one obtains
One thus deduces the rule for transforming the components of a tensor q times covariant and r times contravariant: they transform like the product of q covariant components and r contravariant components.
This transformation rule can be taken inversely as the definition of the components of a tensor of rank .
Example. The operator O representing a symmetry operation has the character of a tensor. In fact, under a change of basis, O transforms into O′: so that Now the matrices A and B are inverses of one another: The symmetry operator is a tensor of rank 2, once covariant and once contravariant.
It is necessary that the tensors are of the same nature (same rank and same variance).
Here we are concerned with an operation that only exists in the case of tensors and that is very important because of its applications in physics. In practice, it is almost always the case that tensors enter into physics through the intermediary of a contracted product.
Let us first consider the dielectric constant. In the introduction, we remarked that for an isotropic medium
If the medium is anisotropic, we have, for one of the components, This relation and the equivalent ones for the other components can also be written using the Einstein convention.
The scalar product of D by an arbitrary vector x is
The right-hand member of this relation is a bilinear form that is invariant under a change of basis. The set of nine quantities constitutes therefore the set of components of a tensor of rank 2. Expression (1.1.3.3) is the contracted product of by .
A similar demonstration may be used to show the tensor nature of the various physical properties described in Section 1.1.1, whatever the rank of the tensor. Let us for instance consider the piezoelectric effect (see Section 1.1.4.4.3). The components of the electric polarization, , which appear in a medium submitted to a stress represented by the second-rank tensor are where the tensor nature of will be shown in Section 1.3.2 . If we take the contracted product of both sides of this equation by any vector of covariant components , we obtain a linear form on the left-hand side, and a trilinear form on the right-hand side, which shows that the coefficients are the components of a third-rank tensor. Let us now consider the piezo-optic (or photoelastic) effect (see Sections 1.1.4.10.5 and 1.6.7 ). The components of the variation of the dielectric impermeability due to an applied stress are
In a similar fashion, consider the contracted product of both sides of this relation by two vectors of covariant components and , respectively. We obtain a bilinear form on the left-hand side, and a quadrilinear form on the right-hand side, showing that the coefficients are the components of a fourth-rank tensor.
Let us consider a tensor represented in an orthonormal frame where variance is not important. The value of component in an arbitrary direction is given by where the , are the direction cosines of that direction with respect to the axes of the orthonormal frame.
The representation surface of the tensor is the polar plot of .
The representation surfaces of second-rank tensors are quadrics. The directions of their principal axes are obtained as follows. Let be a second-rank tensor and let be a vector with coordinates . The doubly contracted product, , is a scalar. The locus of points M such that is a quadric. Its principal axes are along the directions of the eigenvectors of the matrix with elements . They are solutions of the set of equations where the associated quantities λ are the eigenvalues.
Let us take as axes the principal axes. The equation of the quadric reduces to
If the eigenvalues are all of the same sign, the quadric is an ellipsoid; if two are positive and one is negative, the quadric is a hyperboloid with one sheet; if one is positive and two are negative, the quadric is a hyperboloid with two sheets (see Section 1.3.1 ).
Associated quadrics are very useful for the geometric representation of physical properties characterized by a tensor of rank 2, as shown by the following examples:
Equation (1.1.2.17) describing the behaviour of the quantities under a change of basis shows that they are the components of a tensor of rank 2, the metric tensor. In the same way, equation (1.1.2.19) shows that the 's transform under a change of basis like the product of two contravariant coordinates. The coefficients and are the components of a unique tensor, in one case doubly contravariant, in the other case doubly covariant. In a general way, the Euclidean tensors (constructed in a space where one has defined the scalar product) are geometrical entities that can have covariant, contravariant or mixed components.
Let us take a tensor product We know that It follows that is a tensor product of two vectors expressed in the dual space:
One can thus pass from the doubly covariant form to the doubly contravariant form of the tensor by means of the relation
This result is general: to change the variance of a tensor (in practice, to raise or lower an index), it is necessary to make the contracted product of this tensor using or , according to the case. For instance,
Let us consider, for example, the force, F, which is a tensor quantity (tensor of rank 1). One can define it:
The tensor defined by is called the outer product of vectors x and y. (Note: The symbol is different from the symbol for the vector product.) The analytical expression of this tensor of rank 2 is
The components of this tensor satisfy the properties It is an antisymmetric tensor of rank 2.
Consider the so-called permutation tensor of rank 3 (it is actually an axial tensor – see Section 1.1.4.5.3) defined by and let us form the contracted product It is easy to check that
One recognizes the coordinates of the vector product.
Expression (1.1.3.4) of the vector product shows that it is of a covariant nature. This is indeed correct, and it is well known that the vector product of two vectors of the direct lattice is a vector of the reciprocal lattice [see Section 1.1.4 of Volume B of International Tables for Crystallography (2008)].
The vector product is a very particular vector which it is better not to call a vector: sometimes it is called a pseudovector or an axial vector in contrast to normal vectors or polar vectors. The components of the vector product are the independent components of the antisymmetric tensor . In the space of n dimensions, one would write
The number of independent components of is equal to or 3 in the space of three dimensions and 6 in the space of four dimensions, and the independent components of are not the components of a vector in the space of four dimensions.
Let us also consider the behaviour of the vector product under the change of axes represented by the matrix
This is a symmetry with respect to a point that transforms a right-handed set of axes into a left-handed set and reciprocally. In such a change, the components of a normal vector change sign. Those of the vector product, on the contrary, remain unchanged, indicating – as one well knows – that the orientation of the vector product has changed and that it is not, therefore, a vector in the normal sense, i.e. independent of the system of axes.
We have under a change of axes: This shows that the new components, , can be considered linear functions of the old components, , and one can write It should be noted that the covariance has been increased.
Consider a field of tensors that are functions of space variables. In a change of coordinate system, one has Differentiate with respect to : It can be seen that the partial derivatives behave under a change of axes like a tensor of rank 3 whose covariance has been increased by 1 with respect to that of the tensor . It is therefore possible to introduce a tensor of rank 1, (nabla), of which the components are the operators given by the partial derivatives .
If one applies the operator nabla to a scalar ϕ, one obtains This is a covariant vector in reciprocal space.
Now let us form the tensor product of by a vector v of variable components. We then have
The quantities form a tensor of rank 2. If we contract it, we obtain the divergence of v: Taking the vector product, we get The curl is then an axial vector.
Let be a vector function. Its development as a Taylor series is writtenThe coefficients of the expansion, , are tensors of rank .
An example is given by the relation between displacement and electric field: (see Sections 1.6.2 and 1.7.2 ).
We see that the linear relation usually employed is in reality a development that is arrested at the first term. The second term corresponds to nonlinear optics. In general, it is very small but is not negligible in ferroelectric crystals in the neighbourhood of the ferroelectric–paraelectric transition. Nonlinear optics are studied in Chapter 1.7 .
References
International Tables for Crystallography (2008). Vol. B, Reciprocal Space, edited by U. Shmueli. Heidelberg: Springer.Lichnerowicz, A. (1947). Algèbre et analyse linéaires. Paris: Masson.
Sands, D. E. (1995). Vectors and Tensors in Crystallography. New York: Dover.
Schwartz, L. (1975). Les tenseurs. Paris: Hermann.