Chapter 10: Q6P (page 502)

Write the transformation equation for a $3 r d$ -rank tensor; for a $5 t h$ -rank tensor

Short Answer

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Answer

The transformations are written below.

$T'_{α' β' γ'} = \underset{α β γ}{\sum a_{α' α} a_{β' β} a_{γ' γ} T_{α β γ}}$

$T'_{α' β' γ' δ' ε'} = \sum_{α β γ δ ε} a_{α' α} a_{β' β} a_{γ' γ} a_{δ' δ} a_{ε' ε} T_{α β γ δ ε}$

Step by step solution

Given Information

A $3^{r d}$ -rank tensor and a $5^{t h}$ - 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 $T'_{k l} = \sum_{i = 1}^{3} \sum_{j = 1}^{2} a_{k l} a_{l j} T_{i j}$

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

role="math" localid="1654513090010" alt="" $T'_{α' β' γ'} = \sum_{α β γ} a_{α' α} a_{β' β} a_{γ' γ} T_{α β γ}$

The transformation A-rank tensor is mentioned below.

$T'_{α' β' γ' δ' ε'} = \sum_{α β γ δ ε} a_{α' α} a_{β' β} a_{γ' γ} a_{δ' δ} a_{ε' ε} T_{α β γ δ ε}$

Hence,the transformations are written below.

$T'_{α' β' γ'} = \sum_{α β γ} a_{α' α} a_{β' β} a_{γ' γ} T_{α β γ}$

$T'_{α' β' γ' δ' ε'} = \sum_{α β γ δ ε} a_{α' α} a_{β' β} a_{γ' γ} a_{δ' δ} a_{ε' ε} T_{α β γ δ ε}$

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Write the transformation equation for a $3 r d$ -rank tensor; for a $5 t h$ -rank tensor

Short Answer

Step by step solution

Given Information

Definition of a cartesian tensor.

Transformation of tensors.

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Write the transformation equation for a 3rd-rank tensor; for a 5th-rank tensor

Short Answer

Step by step solution

Given Information

Definition of a cartesian tensor.

Transformation of tensors.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Mechanics and Materials

Electricity

Further Mechanics and Thermal Physics

Radiation

Electrostatics

Electromagnetism

Study anywhere. Anytime. Across all devices.

Write the transformation equation for a $3 r d$ -rank tensor; for a $5 t h$ -rank tensor