Question

Show that ${\mathit{T}}_{\mathbf{i}\mathbf{j}\mathbf{k}\mathbf{l}\mathbf{m}}{\mathit{S}}_{\mathbf{l}\mathbf{m}}$is a tensor and find its rank (assuming that T and S are tensors of the rank indicated by the indices).

Short answer

Answer

The equation has been proven

Step-by-step solution

Given Information

The tensor $T_{\mathrm{ijklm}}{S}_{\mathrm{lm}}$
.

Definition of a cartesian tensor.

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

Prove the statement.

The tensor $T_{ijklm}{S}_{Im}$

The number of dummy index pairs in $T_{ijklm}{S}_{Im}$is 2, double contraction.

Each contraction reduces the rank of tensor by 2.

So, the rank of the tensor be $5+2-2\times 2=3$.

Prove that the rank is 3.

$$\begin{gathered} (T_{i\text{'}j\text{'}l\text{'}m\text{'}}{S}_{l\text{'}m\text{'}}=a_{i\text{'}i\text{'}}{a}_{j\text{'}j\text{'}}{a}_{k\text{'}k\text{'}}{a}_{l\text{'}l\text{'}}{a}_{m\text{'}m\text{'}}{a}_{l\text{'}l\text{'}}{a}_{m\text{'}m\text{'}}{T}_{ijklm}{S}_{lm} \\ (T_{i\text{'}j\text{'}l\text{'}m\text{'}}{S}_{l\text{'}m\text{'}}=a_{i\text{'}i\text{'}}{a}_{j\text{'}j\text{'}}{a}_{k\text{'}k\text{'}}{\delta }_{l\text{'}l\text{'}}{\delta }_{m\text{'}m\text{'}}{a}_{l\text{'}l\text{'}}{a}_{m\text{'}m\text{'}}{T}_{ijklm}{S}_{lm} \\ (T_{i\text{'}j\text{'}l\text{'}m\text{'}}{S}_{l\text{'}m\text{'}}=a_{i\text{'}i\text{'}}{a}_{j\text{'}j\text{'}}{a}_{k\text{'}k\text{'}}{T}_{ijklm}{S}_{lm{j}_1...j_n} \end{gathered}$$

Hence, the rank of the tensor is 3.