Question

Show that the first parenthesis in (3.5) is a symmetric tensor and the second parenthesis is antisymmetric.

Short answer

Answer

$T$is a summation of an antisymmetric tensor and a symmetric tensor.

Step-by-step solution

Given information.

Physics definitions are given.

Definition of a tensor.

Tensors, like scalars and vectors, are mathematical constructs that can be used to describe physical qualities. Tensors are just a combination of scalars and vectors, with a scalar being a zero rank tensor and a vector being a first rank tensor.

Write the equation mentioned in the question.

Write the given equation.

$T_{ij}=\frac{1}{2}\left(T_{ij}+T_{ji}\right)+\frac{1}{2}\left(T_{ij}-T_{ji}\right)$

Define new tensors.

$$\begin{gathered} S_{ij}=\frac{1}{2}\left(T_{ij}+T_{ji}\right) \\ {A}_{ij}=\frac{1}{2}\left(T_{ij}-T_{ji}\right) \end{gathered}$$

Add the above equations.

$T_{ij}=S_{ij}+A_{ij}$

Invert the indices of the tensors.

Invert the indices of the tensors.

$$\begin{gathered} S_{ji}=\frac{1}{2}\left(T_{ji}+T_{ij}\right) \\ =S_{ij} \\ {A}_{ji}=\frac{1}{2}\left(T_{ji}-T_{ij}\right) \\ =-A_{ij} \end{gathered}$$

This implies that $T$is a summation of an antisymmetric tensor and a symmetric tensor.