Question

As we did in (3.3) , show that the contracted tensor ${\mathit{T}}_{\mathbf{i}\mathbf{i}\mathbf{j}}$is a first-rank tensor, that is, a vector.

Short answer

Answer

The equation has been proven.

Step-by-step solution

Given Information 

The contracted tensor$T_{iij}$.

Definition of a cartesian tensor.

The first rank tensor is just a vector. A tensor of second rank has nine components (in three dimensions) in every rectangular coordinate system.

Prove the statement.

The contracted tensor$T_{iij}$.

A is an orthogonal matrix.

$A{A}^T={\delta }_{ij}$

$$\begin{gathered} T{\text{'}}_{iij}=a_{ik}{a}_{ij}{a}_{jr}{T}_{klr} \\ T{\text{'}}_{iij}={\delta }_{kl}{a}_{jr}{T}_{klr} \\ T{\text{'}}_{iij}=a_{jr}{T}_{llr} \\ T{\text{'}}_{iij}=a_{jr}{T}_{iir} \end{gathered}$$

Hence, $T_{iij}$is a vector.