4 Tensors
TABLE OF CONTENTS
1. Rotation and vectors
2. The tensor notation
2.1 Einstein summation convention
3. Matrix representation
4. Rotations and tensors
4.1 Pseudo-tensors
5. Playing with indices
5.1 The Kronecker delta and permutation symbol
6. Vector identities
7. The Kronecker delta and permutation symbol as tensors
7.1 Isotropic tensors
7.2 Physical significance of isotropic tensor of rank two
LEARNING OBJECTIVES
1. In this module a new perspective is given to the idea of vectors which are related to properties of physical quantities under rotation.
2. The tensor notation is introduced which is a big help in this study of rotational properties.
3. Next the Einstein summation convention which simplifies the formalism is introduced.
4. Vectors are defined from the perspective of their properties under rotation in the three dimensional physical space. The idea is then generalized to tensors of second and higher order ranks.
5. Further classification of physical quantities on the basis of their behaviour under inversion of the coordinate axes is also discussed.
6. In this tensor (or index) notation the indices play a very important role and it is demonstrated how playing around with indices leads to many significant results.
7. Two very special tensors, the Kronecker delta and the permutation tensor are introduced and their properties discussed in detail.
8. Next certain vector identities are written down and it is demonstrated how the tensor methods greatly simplify the derivation of such identities.
9. Next it is proved that Kronecker delta and the permutation tensor are indeed tensors of rank two and three respectively.
10. The idea of isotropic tensors is briefly introduced and isotropic tensors of rank up to four described.
T e n s o r s
1. Rotations and Vectors
So far we have studied two kinds of physical quantities; scalars which need only a magnitude to describe them, and vectors which, in addition, need a direction to specify them completely. There are many physical entities that do not fall in either of these two categories. They require more than one direction to specify them, so to say. We now wish to generalize our ideas to include such objects in our study as well. For that purpose we first reintroduce vectors from another point of view, viz., through their transformation properties under rotations in the physical three dimensional space. This is a more satisfactory way to introduce vectors than simply as objects having both magnitude and direction. This definition can then be easily generalized to include other physical entities that we have in mind.
2. The tensor notation
Together these equations can be written as
Hence
On comparing the coefficients of various terms we have
2.1 Einstein summation convention
The notation and writing can be further simplified by using what is called the Einstein summation convention. Under this convention
- An index (subscript) can appear only once or twice. Each index take values (1, 2, 3)
- An index that appears once is a free index. Every free index represents a set of three equations for three values of the index.
- An index that appears twice is a dummy index. The index is understood to be summed over, so that every such index represents a sum of three terms.
- One dummy index can be freely replaced by another. Thus
- The free indices must match on the two sides of an equation and on each term of an expression.
3. Matrix representation
Many of the equations above can be written in an even simpler form in terms of matrices. If we put the coordinates of a point in the form of a column matrix.
The symbol x is being used for the set of three coordinates and not for the abstract vector which it represents. The superscript (T) refers to the transpose of a matrix and I refers to the unit matrix. The matrix a is such that
A matrix with this property is called an orthogonal matrix. Thus rotation is represented by an orthogonal matrix. This in fact is true not only in two or three dimensions but for rotations in any dimensions.
Thus the inverse transformation is given by the transpose of the coefficients.4. Rotations and tensors
So far we have studied the properties of the position vector under rotations. We now extend the same ideas to other physical quantities and thereby define tensors of second and higher ranks as physical quantities having more complicated transformation properties under rotations.
We first redefine a vector in terms of its transformation properties under rotations. The position vector transforms according to the equation
Every physical quantity cannot be categorized as belonging to the set of scalars and vectors. There are others that transform in a more complicated way under rotations. Any set of nine physical quantities which transforms in the following way under rotations are said to form a tensor of rank two:
Moment of inertia, stress and strain, and Maxwell’s stress tensor are all examples of tensors of rank two. Vectors can be regarded as tensors of rank one and scalars as tensors of rank zero.
4.1 Pseudo Tensors
Certain further properties of the transformation coefficients are interesting. Let us take the determinant of both sides of equation (16)
Now for the identity transformation, det(a)=1. Since rotation is a continuous transformation, under any rotation det(a) will change continuously. Since det(a) is allowed only two values, +1 or -1, det(a)=1 under any continuous rotation. On the other hand for a reflection in a plane, say the z=0 plane, we have
This is represented by the transformation matrix
The same is true of inversion in three dimensions Based on this we have the following results
5. Playing with indices
5.1 The Kronecker delta and permutation symbol
6. Vector Identities
Many of the vector relations can be proved more conveniently by using the “tensor notation”. Let us, as an example, consider
To use the “tensor methods” to prove this relation, let us find the ith component of left hand side:
The differential operator as a tensor
The differential operators can obviously also be written in the tensor notation. Thus we have
7. The Kronecker delta and permutation symbol as tensors
We have so far defined Kronecker delta and permutation symbol as simply that, as objects having specific components. But these are indeed tensors in the sense that they have the required transformation properties under rotations. On using equations (13) and (22) we have
7.1 Isotropic tensors
An isotropic tensor is one that has the same components in all rotated frames of reference, i.e.,
We can classify isotropic tensors as follows:
7.2 Physical significance of isotropic tensor of rank two
An isotropic tensor behaves much like a scalar. Consider for example the conductivity tensor. In an anisotropic medium the relation between the electric field and the current density takes the form
SUMMARY
- In this module we develop a new perspective for the idea of vectors which are related to properties of physical quantities under rotation.
- We introduce the so called tensor notation which is a big help in this study of rotational properties.
- Next we introduce and explain with examples the Einstein summation convention which simplifies the formalism even further.
- We then define vectors from this new perspective of properties of physical quantities under rotations in the three dimensional physical space. We then generalize the idea to tensors of second and higher order ranks and explain how scalars and vectors are tensors of rank zero and one respectively.
- Next we further classify physical quantities on the basis of their behaviour under inversion of the coordinate axes and introduce true and pseudo tensors.
- In this tensor (or index) notation the indices play a very important role and we demonstrate how playing around with indices leads to many significant results.
- Next we formally introduce two very special tensors, the Kronecker delta and the permutation tensor and discuss their usefulness and properties.
- Next we write down certain vector identities and demonstrate how the tensor methods greatly simplify the derivation of such identities.
- Next we prove that Kronecker delta and the permutation tensor that were introduced are indeed tensors of rank two and three respectively.
- Finally we briefly discuss the idea of isotropic tensors and describe isotropic tensors of rank up to four.