Introduction to Tensor Analysis

Tensor:

Laws of nature are represented by anequation in physics, Physical equation gives the variation of physical quantity with respect to variation of other physical quantities. If the Physical equation truly represent the law of nature then it's form shouldn't be changeunder co-ordinate transformation i.e. Physical equation should be co-variant ( not changed)

Tensor is the set of quantities representing rule of co-ordinate transformation preserving the covariant nature of physical equation. All physical quantities are tensor. E.g, Scalar are tensor of rank 0, vector are tensor of rank one, and so on.

Let S and J be two co-ordinate systems having co-ordinate s\(\to (x',x^2,x^3,\dotsm,x^N\)) and \(\bar s\to (\bar x, \bar x^2,\dotsm,\bar x^N\)). Then we can find the transformation relation between s and \(\bar s\).

$$\bar x'= \bar x'(x^1, x^2, \dotsm, x^N)$$

$$\bar x^2= \bar x^2(x^1x^2,\dotsm, x^N)$$

$$\vdots$$

$$\bar x^N=\bar x^N (x^1, x^2,\dotsm, x^N)$$

and

$$x'=x'(\bar x^1 , \bar x^2, \dotsm \bar x^N)$$

$$x^2=x^2(\bar x^1, \bar x^2, \dotsm \bar x^N)$$

$$\vdots$$

$$x^N= x^N(\bar x^1 , \bar x^2 , \dotsm \bar x^N)$$

The transformation relation are,

$$d\bar x'=\frac{\partial \bar x}{\partial x'}dx'+\frac{\partial x'}{\partial x^2}\partial x^2+\dotsm+\frac{\partial \bar x'}{\partial x^N}dx^N$$

In general,

$$d\bar x^i= \frac{\partial \bar x ^i}{\partial x'}dx'+\frac{\partial \bar x^i}{\partial x^2}dx^2+\dotsm+\frac{\partial \bar x^i}{\partial x^N}dx^N$$

$$=\sum_{\alpha=1}^N\frac{\partial \bar x^i}{\partial x^\alpha }dx^{\alpha}$$

Now,

$$dx^i=\sum_{\alpha =1}^N\frac{\partial x^i}{\partial \bar x^\alpha} d\bar x^{-\alpha}$$

Rank of Tensor:

The number of mode of change of physical quantity during co-ordinate transformation is represented by rank. Rank of a tensor is equal to number of indices present in sub script and super script of the symbol representing tensor. Example,

\(A^j=\) Rank 1 ( contravariant )

\(A^{ij}=\) Rank 2 ( contravariant )

\(A_{ijk}=\) Rank 3 ( covariant )

\(A^{ijk}_{lmn}= \)Rank 6 ( Mixed )

Einstein summation convention:

The sum \(a_1 x^1+ a_2 x^2+\dotsm + a_N x^N\) can be written as \(\sum_{i=1}^n a_i x^i\) . The further short form representation of above summation is \(a_i x^i\) by considering that summation is done over repeated index.

Contravariant and covariant tensor of rank one:

The N quantities (\( A^1, A^2,\dotsm A^N\)) in the co-ordinate system (\( x^1,x^2,\dotsm, x^N\)) are said to be contravariant tensor of rank one if these quantities are related with N quantities (\(\bar A^1, \bar A^2, \bar A^3,\dotsm, \bar A^N\)) in co-ordinate system (\( \bar x^1,\bar x^2,\dotsm , \bar x^N\)) by following relation.

$$\bar A^\alpha =\frac{\partial \bar x^\alpha}{\partial x^i}A^i$$

The N quantities \(B_1, B_2,\dotsm , B_N)\) in the co-ordinate system (\( x^1, x^2, \dotsm x^N)\) are said to be co-varient tensor of rank one if these quantities are related with N component (\(\bar B_1, \bar B_2, \dotsm \bar B_N\)) in co-ordinate system \(\bar x^1, \bar x^2, \dotsm , \bar x^N\)) by following relation

$$\bar B_\alpha=\frac{\partial x^i}{\partial x^\alpha} B_i$$

Mixed tensor :

The tensor is said to be mixed tensor if it is both covariant and contravariant. For example, \(c_k^{ij}\) is mixed tensor of rank 3 if it satisfy the following relation,

$$\bar c_{\gamma}^{\alpha \beta} = \frac{\partial \bar x^\alpha}{\partial x^i}\frac{\partial \bar x^\beta}{\partial x^j}\frac{\partial x^k}{\partial \bar x^\gamma } c_k^{ij}$$

Symmetric and Anti symmetric tensor:

A tensor is said to be symmetric in two co-variant and contravariant indices the tensor remain unchanged during exchange of two indices. Example

\(C_k^{ij}=C_k^{ji} \) (symmetric n i and j)

\(C_{lm}^{\alpha\beta\gamma}= C_{lm}^{\gamma\alpha\beta}\) (symmetric in \(\alpha \) and \(\gamma\)

A tensor is said to be antisymmetric in two co-variant or contravariant indices if the sign of the tensor changes during change of two indices. Example

\( C_k^{ij}=-C_k^{ji}\) ( Antisym. in i and j)

\( C_{lm}^{\alpha\beta\gamma}=C_{lm}^{\gamma\beta\alpha}\) (antisym. in \(\alpha\) and \(\gamma\))

Any tensor can be written as in terms of sum of symmetric and antisymmetric tensor

Le t \(A^{ij}\) be any tensor then it can be written as

$$A^{ij}=\frac12 (A^{ij}+A^{ji})+\frac12 (A^{ij}-A^{ji})$$

$$= C^{ij}+ D^{ij}$$

Which is symmetric. and

$$D_{ij}= \frac12(A^{ij}-A^{ji})$$

$$= \frac12 (A^{ji}-A^{ij})$$

$$=D^{ji}$$

Which is antisymmetric.

Show that velocity of vector is tensor of rank one:

Let us consider a barred and unbarred co-ordinate system then component of velocity in barred and unbarred co-ordinate system is,

$$\bar v^\alpha=\frac{\partial \bar x^\alpha}{\partial t}\dotsm(1)$$

$$v^i=\frac{\partial x^i}{\partial t}\dotsm(2)$$

Now,

$$\bar v^\alpha=\frac{\partial \bar x^\alpha}{\partial t}$$

$$=\frac{\partial \bar x^\alpha}{\partial x^i}\cdot \frac{\partial x^i}{\partial t}$$

$$=\frac{\partial \bar x^\alpha}{\partial x^i}v_i$$

$$\bar v^\alpha=\frac{\partial \bar x^\alpha}{\partial x^i}v_i$$

This shows that velocity is the cotravariant tensor of rank one.

Tensor of rank one:

The physical quantities which remains invariant under co-ordinate transformation are called tensor of rank zero this type of tensor is scalar.

Higher order tensor:.The higher order tensor are tensor of rank 2 or more than 2 . A quantities is said to be contravariant or covarient tensor of rank two if it satisfy the following transformation relation.

$$\bar C_\gamma ^{\alpha \beta} = \frac{\partial \bar x^\alpha}{\partial x^i}{\frac{\partial\bar x^\beta}{\partial x^j}\frac{\partial x^k}{\partial \bar ^\gamm} C_k^{ij}$$

Reference:

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5. Adhikari, Pitri Bhakta, and Dya Nidhi Chhatkuli.A Textbook of Physics. Third Revised Edition ed. Vol. III. Kathmandu: SUKUNDA PUSTAK BHAWAN, 2072.