MATHEMATICAL PHYSICS UNIT – 7 Tensor Algebra PRESENTED BY: DR. RAJESH MATHPAL ACADEMIC CONSULTANT SCHOOL OF SCIENCES U.O.U. TEENPANI, HALDWANI UTTRAKHAND MOB:9758417736,7983713112 STRUCTURE OF UNIT 7.1. INTRODUCTION 7.2. n-DIMENSIONAL SPACE 7.3. CO-ORDINATE TRANSFORMATIONS 7.4. INDICAL AND SUMMATION CONVENTIONS 7.5. DUMMY AND REAL INDICES 7.6. KEONECKER DELTA SYMBOL 7.7. SCALARS, CONTRAVARIANT VECTORS AND COVARIANT VECTORS 7.8. TENSORS OF HIGHER RANKS 7.9. SYMMETRIC AND ANTISYMMETRIC TENSORS 7.10. ALGEBRAIC OPERATIONS ON TENSORS 7.1. INTRODUCTION Tensors are mathematical objects that generalize scalars, vectors and matrices to higher dimensions. If you are familiar with basic linear algebra, you should have no trouble understanding what tensors are. In short, a single-dimensional tensor can be represented as a vector. The term rank of a tensor extends the notion of the rank of a matrix in linear algebra, although the term is also often used to mean the order (or degree) of a tensor. The rank of a matrix is the minimum number of column vectors needed to span the range of the matrix. A tensor is a vector or matrix of n-dimensions that represents all types of data. All values in a tensor hold identical data type with a known (or partially known) shape. The shape of the data is the dimensionality of the matrix or array. A tensor can be originated from the input data or the result of a computation. Tensors are simply mathematical objects that can be used to describe physical properties, just like scalars and vectors. In fact tensors are merely a generalisation of scalars and vectors; a scalar is a zero rank tensor, and a vector is a first rank tensor. Pressure itself is looked upon as a scalar quantity; the related tensor quantity often talked about is the stress tensor. that is, pressure is the negative one-third of the sum of the diagonal components of the stress tensor (Einstein summation convention is implied here in which repeated indices imply a sum). Tensors are to multilinear functions as linear maps are to single variable functions. If you want to apply techniques in linear algebra to problems depending on more than one variable linearly (usually something like problems that are more than one-dimensional), the objects you are studying are tensors. A tensor field has a tensor corresponding to each point space. An example is the stress on a material, such as a construction beam in a bridge. Other examples of tensors include the strain tensor, the conductivity tensor, and the inertia tensor. A tensor is a generalization of vectors and matrices and is easily understood as a multidimensional array. ... It is a term and set of techniques known in machine learning in the training and operation of deep learning models can be described in terms of tensors. The tensor is a more generalized form of scalar and vector. Or, the scalar, vector are the special cases of tensor. If a tensor has only magnitude and no direction (i.e., rank 0 tensor), then it is called scalar. If a tensor has magnitude and one direction (i.e., rank 1 tensor), then it is called vector. Tensors are a type of data structure used in linear algebra, and like vectors and matrices, you can calculate arithmetic operations with tensors. After completing this tutorial, you will know: That tensors are a generalization of matrices and are represented using n-dimensional arrays. A tensor is a container which can house data in N dimensions. Often and erroneously used interchangeably with the matrix (which is specifically a 2-dimensional tensor), tensors are generalizations of matrices to N-dimensional space. Mathematically speaking, tensors are more than simply a data container, however. Stress is a tensor because it describes things happening in two directions simultaneously. ... Pressure is part of the stress tensor. The diagonal elements form the pressure. For example, σxx measures how much x-force pushes in the x-direction. Think of your hand pressing against the wall, i.e. applying pressure. 7.2. n-DIMENSIONAL SPACE In three dimensional space a point is determined by a set of three numbers called the co-ordinates of that point in particular system. For example (x, y, z) are the co-ordinates of a point in rectangular Cartesian co-ordinate system. By analogy if a point is respected by an ordered set of n real variables (x1, x2, x3,……xi …..xn) or more conveniently (x1, x2, x3, ....xi …..xn) Hence the suffixes 1, 2, 3, …, i, ….., n denote variables and not the powers of the variables involved], then all the points corresponding to all values of co-ordinates (i.e., variables) are said to form an n- dimensional space, denoted by Vn. A curve in n-dimensional space (Vn) is defined as the collection of points which satisfy the n-equations. xi = xi (u), (i =1, 2, 3, …, n) where u is a parameter and xi (u) are n functions of u, which satisfy certain continuity conditions. A sub-space Vm (m < n) of Vn is defined as the collection of points which satisfy n-equations xi = xi (u1, u2, …, um) (i = 1, 2, …, n) where u1, u2, …, um are m parameters and xi (u1, u2, …, um) are n functions of u1, u2, …, um which satify certain continuity conditions. 7.3. CO-ORDINATE TRANSFORMATIONS Tensor analysis is intimately connected with the subject of co-ordinate transformations. Consider two sets of variables (x1, x2, x3, …, xn) and 푥1, 푥2, 푥3, … 푥푛 which determine the co-ordinates of point in an n-dimensional space in two different frames of reference. Let the two sets of variables be related to each other by the transformation equations 푥1 = 휙1 (푥1, 푥2, 푥3, … 푥푛) 푥2 = 휙2 (푥1, 푥2, 푥3, … 푥푛) … … … … … … … … 푥푛 = 휙푛 (푥1, 푥2, 푥3, … 푥푛) or briefly 푥휇 = 휙휇 (푥1, 푥2, 푥3, … , 푥푖, … , 푥푛) …(7.1) (i = 1, 2, 3, …, n) where function 휙휇 are single valued, continuous differentiable functions of co-ordinates. Iit sis essential that the n-function 휙휇 be independent. Equations (7.1) can be solved for co-ordinates xi as functions of 푥휇 to yield 푖 푖 1 2 3 휇 푛 푥 = 휓 (푥 , 푥 , 푥 , … , 푥 , … , 푥 ) …(7.2) Equations (4.1) and (4.2) are said to define co-ordinate transformations. 휇 From equations (4.1) the differentials 푑푥 are transformed as 휇 휇 휇 휇 휕푥 1 휕푥 2 휕푥 푛 푑푥 = 푑푥 + 푑푥 + ⋯ + 푑푥 휕푥1 휕푥2 휕푥푛 n x i = i dx , (휇 = 1, 2, 3, …, n) …(7.3) i1 x 7.4.INDICAL AND SUMMATION CONVENTIONS Let us now introduce the following two conventions : (1) Indicial convention. Any index, used either as subscript or superscript will take all values from 1 to n unless the contrary is specified. Thus equations (4.1) can be briefly written as 푥휇 = 휙휇 푥푖 …(7.4) The convention reminds us that there are n equations with 휇 = 1, 2, …n and 휙휇 are the functions of n-co-ordinates with (i = 1, 2, …, n). (2) Einstein’s summation convention. If any index is repeated in a term, then a summation with respected to that index over the range 1, 2, 3, …, n is implied. This convention instead is called Einstein’s summation convention. n axj According to this conversation instead of expression , j i1 i we merely write ai x . Using above tow conversation eqn. (7.3) is written as 휇 휇 휕푥 푖 푑푥 = 푑푥 …(7.5a) 휕푥푖 Thus the summation convention means the drop of sigma sign for the index appearing twice in a given term. In other words the summation convention implies the sum of the term for the index appearing twice in that term over defined range. 7.5. DUMMY AND REAL INDICES Any index which is repeated in a given term, so that the summation convention implies, is called a dummy index and it may be replaced freely by any other 휇 푖 index not already used in the term. For example I is a dummy index in 푎푖 푥 . Also I is a dummy index in eqn. (4.5a), so that equation (4.5a) is equally written as 휇 휇 휇 휕푥 휕푥 푑푥 = 푑푥푘 = 푑푥휆. …(7.5b) 휕푥푘 휕푥휆 Also two or more dummy indices can be interchanged. In order to avoid confusion the same index must not be used more than twice in any single team. i i i j For example will not be written as aix ai x but rather aiajx x . Any index which is not repeated in a given term is called a real index. For 휇 푖 example 휇 is a real index in푎푖 푥 . A real index cannot be replaced by another real index, e.g. 휇 푖 푣 푖 푎푖 푥 ≠ 푎푖 푥 7.6.KEONECKER DELTA SYMBOL The symbol kronecker delta 1 푖푓 푗 = 푘 훿푗 = ቊ …(7.6) 푘 0 푖푓 푗 ≠ 푘 Some properties of kronecker delta (i) If x1, x2, x3, …xn are independent variables, then 휕푥푗 = 훿푗 …(7.7) 휕푥푘 푘 (ii) An obvious property of kronecker delta symbol is 푗 푗 푘 훿푘 퐴 = 퐴 . …(7.8) Since by summation convention in the left hand side of this equation the summation is with respect to j and by definition of kronecker delta, the only surviving term is that for which j = k. (iii) If we are dealing with n dimensions, then 푗 푘 훿푗 = 훿푘 = 푛 …(7.9) By summation convention 푗 1 2 3 푛 훿푗 = 훿1 + 훿2 + 훿3 + ⋯ + 훿푛 = 1 + 1 + 1 + ⋯ + 1 = 푛 푗 푗 푖 (iv) 훿푗 훿푘 = 훿푘.