Chapter 10: Q5P (page 517)

Write the tensor transformation equations for $ε_{i j k} ε_{m n p}$ to show that this is a (rank 6) tensor (nota pseudo tensor). Hint:Write (6.1) for each $ε$ and multiply them, being careful not to re-use a pair of summation indices.

Short Answer

Expert verified

The tensor transformation equation for

${ε^{'}}_{α β γ} {ε^{'}}_{ζ n θ} = {(d e t A)}^{2} a_{α i} α_{β j} a_{γ k} a_{ζ m} a_{n m} a_{θ p} ε_{i j k} ε_{m n p} = a_{α i} α_{β j} a_{γ k} a_{ζ m} a_{n m} a_{θ p} ε_{i j k} ε_{m n p}$ .

Step by step solution

Given information.

Matrix definitions are given.

Definition of a rotation matrix.

The rotation matrix is defined in this way.

$[\begin{matrix} \cos ϕ & - \sin ϕ \\ \sin ϕ & \cos ϕ \end{matrix}]$

Define an orthogonal transformation.

It A is an orthogonal matrix, that defines an orthogonal transformation.

${ε^{'}}_{α β γ} = (d e t A) a_{α i} α_{β j} a_{γ k} ε_{i j k}$

Using the information $A = \pm 1$ make further simplifications.

${ε^{'}}_{α β γ} {ε^{'}}_{ζ n θ} = {(d e t A)}^{2} a_{α i} α_{β j} a_{γ k} a_{ζ m} a_{n m} a_{θ p} ε_{i j k} ε_{m n p} = a_{α i} α_{β j} a_{γ k} a_{ζ m} a_{n m} a_{θ p} ε_{i j k} ε_{m n p}$

Hence, the tensor transformation equation is shown below:

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Write the tensor transformation equations for $ε_{i j k} ε_{m n p}$ to show that this is a (rank 6) tensor (nota pseudo tensor). Hint:Write (6.1) for each $ε$ and multiply them, being careful not to re-use a pair of summation indices.

Short Answer

Step by step solution

Given information.

Definition of a rotation matrix.

Define an orthogonal transformation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Nuclear Physics

Physical Quantities And Units

Electromagnetism

Translational Dynamics

Turning Points in Physics

Atoms and Radioactivity

Study anywhere. Anytime. Across all devices.

Write the tensor transformation equations for εijkεmnpto show that this is a (rank 6) tensor (nota pseudo tensor). Hint:Write (6.1) for eachεand multiply them, being careful not to re-use a pair of summation indices.

Short Answer

Step by step solution

Given information.

Definition of a rotation matrix.

Define an orthogonal transformation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Nuclear Physics

Physical Quantities And Units

Electromagnetism

Translational Dynamics

Turning Points in Physics

Atoms and Radioactivity

Study anywhere. Anytime. Across all devices.

Write the tensor transformation equations for $ε_{i j k} ε_{m n p}$ to show that this is a (rank 6) tensor (nota pseudo tensor). Hint:Write (6.1) for each $ε$ and multiply them, being careful not to re-use a pair of summation indices.