Chapter 10: Q5P (page 513)

Let $T_{j k m n}$ be the tensor in $(5 .8)$ . This is a $4^{t h}$ -rank tensor and so has $3^{4} = 81$ components. Most of the components are zero. Find the nonzero components and their values. Hint: See discussion after $(5 .8)$ .

Short Answer

Expert verified

The non-zero components are mentioned below:

$\begin{matrix} T_{j k j k} = 1 \\ T_{j k k j} = - 1 \end{matrix}$

Step by step solution

Given Information

The $4^{t h}$ -rank tensor $T_{j k m n}$ .

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.

Find the nonzero components

Formula states that $T_{j k m n} = δ_{j m} δ_{k n} - δ_{j r} δ_{k m}$ .

If $k = j$ ,then the formula becomes as follows.

$\begin{matrix} T_{j k m n} = δ_{j m} δ_{k n} - δ_{j r} δ_{k m} \\ = 0 \end{matrix}$

If ,then the formula becomes as follows.

$\begin{matrix} T_{j k m n} = δ_{j m} δ_{k n} - δ_{j r} δ_{k m} \\ = 0 \end{matrix}$

$T_{j k m n} \neq 0$ for the following conditions mentioned below.

$\begin{matrix} 1) j = m \\ k = n \\ 2) j = n \\ k = m \end{matrix}$

Or $j \neq k$ .

There are six pairs, in which exactly two pairs satisfy the equation.

Hence the non-zero components are mentioned above

$\begin{matrix} T_{j k j k} = 1 \\ T_{j k k j} = - 1 \end{matrix}$

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.

Let $T_{j k m n}$ be the tensor in $(5 .8)$ . This is a $4^{t h}$ -rank tensor and so has $3^{4} = 81$ components. Most of the components are zero. Find the nonzero components and their values. Hint: See discussion after $(5 .8)$ .

Short Answer

Step by step solution

Given Information

Definition of a Cartesian tensor

Find the nonzero components

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Nuclear Physics

Mechanics and Materials

Medical Physics

Space Physics

Translational Dynamics

Particle Model of Matter

Study anywhere. Anytime. Across all devices.

LetTjkmn be the tensor in (5.8). This is a4th -rank tensor and so has 34=81 components. Most of the components are zero. Find the nonzero components and their values. Hint: See discussion after(5.8) .

Short Answer

Step by step solution

Given Information

Definition of a Cartesian tensor

Find the nonzero components

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Nuclear Physics

Mechanics and Materials

Medical Physics

Space Physics

Translational Dynamics

Particle Model of Matter

Study anywhere. Anytime. Across all devices.

Let $T_{j k m n}$ be the tensor in $(5 .8)$ . This is a $4^{t h}$ -rank tensor and so has $3^{4} = 81$ components. Most of the components are zero. Find the nonzero components and their values. Hint: See discussion after $(5 .8)$ .