Introduction

The principal target of tensor calculus is to investigate the relations that remain the same when we change from one coordinate system to any other. The laws of physics are independent of the frame of references in which physicists describe physical phenomena by means of laws. Therefore, it is useful to exploit tensor calculus as the mathematical tool in which such laws can be formulated.

Transformation of Coordinates

Let there be two reference systems, S with coordinates (x1, x2,…xn) and with coordinates. (Fig. 1). The new system S depends on the old system S as

Here ϕi are single-valued continuous differentiable functions of (x1, x2,…xn) and further the Jacobian

Differentiation of Eq. (1.1) yields

Now and onward, we use the Einstein summation convention, i.e., omit the summation symbol Σ and write the above equations as

The repeated index r or m is known as dummy index. The index i is not dummy and is known as free index.

The transformation matrices are inverse to each other

The symbol is Kronecker delta, is defined as

Obviously vectors in system are linked with (S) system

Covariant and Contravariant Vector and Tensor

Usually one can describe the tensors by means of their properties of transformation under coordinate transformation. There are two possible ways of transformations from one coordinate system .xi) to the other coordinate system .

Let us consider a set of n functions Ai of the coordinates xi. The functions Ai are said to be the components of covariant vector if these components transform according to the following rule

Also, one can find by multiplying and using and

Gradient of a scalar is a covariant vector.

Here, Ai is known as the covariant tensor of first order or of the type (0, 1).

The functions Ai are said to be the components of the contravariant vector if these components transform according to the following rule

Also, one can find by multiplying both sides with and using

Here, Ai is known as the contravariant tensor of first order or of the type (1, 0).