Chapter 10: Q3P (page 505)

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

Short Answer

Expert verified

Answer

The equation has been proven.

Step by step solution

Given Information

The contracted tensor $T_{i i j}$ .

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 $T_{i i j}$ .

A is an orthogonal matrix.

$A A^{T} = δ_{i j}$

$T'_{i i j} = a_{i k} a_{i j} a_{j r} T_{k l r} T'_{i i j} = δ_{k l} a_{j r} T_{k l r} T'_{i i j} = a_{j r} T_{l l r} T'_{i i j} = a_{j r} T_{i i r}$

Hence, $T_{i i j}$ is a vector.

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.

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

Short Answer

Step by step solution

Given Information

Definition of a cartesian tensor.

Prove the statement.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Classical Mechanics

Linear Momentum

Medical Physics

Work Energy and Power

Translational Dynamics

Atoms and Radioactivity

Study anywhere. Anytime. Across all devices.

As we did in (3.3) , show that the contracted tensor Tiijis a first-rank tensor, that is, a vector.

Short Answer

Step by step solution

Given Information

Definition of a cartesian tensor.

Prove the statement.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Classical Mechanics

Linear Momentum

Medical Physics

Work Energy and Power

Translational Dynamics

Atoms and Radioactivity

Study anywhere. Anytime. Across all devices.

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