Question

Write the transformation equation for a $\mathbf{3}\mathit{r}\mathit{d}$-rank tensor; for a $\mathbf{5}\mathit{t}\mathit{h}$-rank tensor

Short answer

Answer

The transformations are written below.

$T{\text{'}}_{\alpha \text{'}\beta \text{'}\gamma \text{'}}=\underset{\alpha \beta \gamma }{\sum a_{\alpha \text{'}\alpha }{a}_{\beta \text{'}\beta }{a}_{\gamma \text{'}\gamma }{T}_{\alpha \beta \gamma }}$

$T{\text{'}}_{\alpha \text{'}\beta \text{'}\gamma \text{'}\delta \text{'}\epsilon \text{'}}=\sum _{\alpha \beta \gamma \delta \epsilon }{a}_{\alpha \text{'}\alpha }{a}_{\beta \text{'}\beta }{a}_{\gamma \text{'}\gamma }{a}_{\delta \text{'}\delta }{a}_{\epsilon \text{'}\epsilon }{T}_{\alpha \beta \gamma \delta \epsilon }$

Step-by-step solution

Given Information

A $3^{rd}$-rank tensor and a$5^{th}$- rank tensor.

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.

Transformation of tensors.

The formula for the cartesian tensor is $T{\text{'}}_{kl}=\sum _{i=1}^3\sum _{j=1}^2{a}_{kl}{a}_{lj}{T}_{ij}$

The transformation A $3^{rd}$-rank tensor is mentioned below.

role="math" localid="1654513090010" alt="" $T{\text{'}}_{\alpha \text{'}\beta \text{'}\gamma \text{'}}=\sum _{\alpha \beta \gamma }{a}_{\alpha \text{'}\alpha }{a}_{\beta \text{'}\beta }{a}_{\gamma \text{'}\gamma }{T}_{\alpha \beta \gamma }$

The transformation A-rank tensor is mentioned below.

$T{\text{'}}_{\alpha \text{'}\beta \text{'}\gamma \text{'}\delta \text{'}\epsilon \text{'}}=\sum _{\alpha \beta \gamma \delta \epsilon }{a}_{\alpha \text{'}\alpha }{a}_{\beta \text{'}\beta }{a}_{\gamma \text{'}\gamma }{a}_{\delta \text{'}\delta }{a}_{\epsilon \text{'}\epsilon }{T}_{\alpha \beta \gamma \delta \epsilon }$

Hence,the transformations are written below.

$T{\text{'}}_{\alpha \text{'}\beta \text{'}\gamma \text{'}}=\sum _{\alpha \beta \gamma }{a}_{\alpha \text{'}\alpha }{a}_{\beta \text{'}\beta }{a}_{\gamma \text{'}\gamma }{T}_{\alpha \beta \gamma }$

$T{\text{'}}_{\alpha \text{'}\beta \text{'}\gamma \text{'}\delta \text{'}\epsilon \text{'}}=\sum _{\alpha \beta \gamma \delta \epsilon }{a}_{\alpha \text{'}\alpha }{a}_{\beta \text{'}\beta }{a}_{\gamma \text{'}\gamma }{a}_{\delta \text{'}\delta }{a}_{\epsilon \text{'}\epsilon }{T}_{\alpha \beta \gamma \delta \epsilon }$