Vector and Tensor Analysis  
Introduction  
Transport Phenomena is the subject which deals with the movement of different physical quantities in any chemical or mechanical process and describes the basic principles and laws of transport. It also describes the relations and similarities among different types of transport that may occur in any system. Transport in a chemical or mechanical process can be classified into three types:  


Three different types of physical quantities are used in transport phenomena: scalars (e.g. temperature, pressure and concentration), vectors (e.g. velocity, momentum and force) and second order tensors (e.g. stress or momentum flux and velocity gradient). It is essential to have a primary knowledge of the mathematical operations of scalar, vector and tensor quantities for solving the problems of transport phenomena. In fact, the use of the indicial notation in cartesian coordinates will enable us to express the long formulae encountered in transport phenomena in a concise and compact fashion. In addition, any equation written in vector tensor form is equally valid in any coordinate system.
In this course, we will using the following notations for scalar, vector and tensor quantities: 



Cartesian coordinates and unit vectors  
A xyz cartesian coordinate system may also be conventionally written as shown in Fig.1.1 below.  
Fig. 1.1 3dimensional cartesian coordinate system with unit vector


Here, and are the unit vectors in x, y and z direction respectively. 

Tensor quantities  
Most of us might have already encountered scalars and vectors in the study of highschool physics. It was pointed out that the vectors also have a direction associated with them along with a magnitude, whereas scalars only have a magnitude but no direction. Extending this definition, we can loosely define a 2nd order tensor as a physical quantity which has a magnitude and two different directions associated with it. To better understand, why we might need two different directions for specifying a particular physical quantity. Let us take the example of the stresses which may arise in a solid body, or in a fluid. Clearly, the stresses are associated with magnitude of forces, as well as with an area, whose direction is also need to be specified by the outward normal to the face of the area on which a particular force acting. Hence, we will require 3^{2}, i.e., 9 components to specify a stress completely in a 3 dimensional cartesian coordinate system. In general, an nth order tensor will be specified by 3^{n} components (in a 3dimensional system). However, the number of components alone cannot determine whether a physical quantity is a vector or a tensor. The additional requirement is that there should be some transformation rule for obtaining the corresponding tensors when we rotate the coordinate system about the origin. Thus, the tensor quantities can be defined by two essential conditions: 



Kronecker delta & Alternating Unit Tensor  
There are two quantities which are quite useful in conveniently and concisely expressing several mathematical operations on tensors. These are the Kronecker delta and the alternating unit tensor.  
Kronecker delta 

Kronecker delta or Kronecker’s delta is a function of two index variables, usually integer, which is 1 if they are equal and 0 otherwise.
It is expressed as a symbol δ_{ij} 

δ_{ij}=1, if i=j
δ_{ij}=0, if i≠j 

Thus, in three dimensions, we may also express the Kronecker delta in matrix form  
Alternating unit tensor  
The alternating unit tensor ε_{ijk} is useful when expressing certain results in a compact form in index notation. It may be noted that the alternating unit tensor has three index and therefore 27 possible combinations but it is a scalar quantity .  


Free indices and Dummy indices  
Free indices  
Free indices are the indices which occur only once in each tensor term. For example, i is the free index in following expression v_{ij} w_{j} 

In any tensorial equation, every term should have an equal number of free indices. For example, v_{ij} w_{j} =c_{j} d_{j} is not a valid tensorial expression since the number of free indices (index i) is not equal in both terms.  
Any free indices in a tensorial expression can be replaced by any other indices as long as this symbol has not already occurred in the expression. For example, A_{ij} B_{j}= C_{i}D_{j}E_{j} is equivalent to A_{kj} B_{j}= C_{k}D_{j}E_{j}.


The number of free indices in an equation gives the actual number of mathematical equations that will arise from it. For example, in equation A_{ij} B_{j}= C_{i}D_{j}E_{j} corresponds to 3^{1} = 3 equations since there is only one free indices i. It may be noted that each indices can take value i=1, 2 or 3.  
Dummy indices  
Dummy indices are the indices that occur twice in a tensor term. For example, j is the dummy index in A_{ij} B_{j}.  
Any dummy index implies the summation of all components of that tensor term associated with each coordinate axis. Thus, when we write A_{i}δ_{i}, we actually imply . 

Any dummy index in a tensor term can be replaced by any other symbol as long as this symbol has not already occurred in previous terms. For example, A_{ijk}δ_{j}δ_{k}= A_{ipq}δ_{p}δ_{q}.


Note: The dummy indices can be renamed in each term separately in a equations but free indices should be renamed for all terms in a tensor equations. For example, A_{ij} B_{j}= C_{i}D_{j}E_{j} can be replaced by A_{kp} B_{p}= C_{k}D_{j}E_{j}.


Here, i is the free index which has been replaced by k in both terms but j is a dummy index and can be replaced either in one term or both.  
Summation convention in vector and tensor analysis  
According to the summation convention rule, if k is a dummy index which repeats itself in a term then there should be a summation sign associated with it. Therefore, we can eliminate the implied summation sign and can write the expression in a more compact way. For example, using the summation convention  
can be simply written as ε_{ijk}ε_{ljk} . Since j and k are repeating, there is no need to write summation sign over these indices  