Find study content
Learning Materials

Discover learning materials by subject, university or textbook.

Explanations

Textbooks

Exams
All Subjects

Anthropology

Archaeology

Architecture

Art and Design

Bengali

Biology

Business Studies

Greek

Chemistry

Chinese

Combined Science

Computer Science

Economics

Engineering

English

English Literature

Environmental Science

French

Geography

German

History

Hospitality and Tourism

Human Geography

Italian

Japanese

Law

Macroeconomics

Marketing

Math

Media Studies

Medicine

Microeconomics

Music

Nursing

Nutrition and Food Science

Physics

Polish

Politics

Psychology

Religious Studies

Sociology

Spanish

Sports Sciences

Translation

All Subjects

Biology

Business Studies

Chemistry

Combined Science

Computer Science

Economics

English

Environmental Science

Geography

History

Math

Physics

Psychology

Sociology

EXAM TYPES

GCSE

IGCSE

AS

A Level

International A Level

University Admissions Tests

GCSE SUBJECTS

GCSE 1

GCSE 2

GCSE 3

GCSE 4

GCSE 5

SOME-TEXT

IGCSE SUBJECTS

IGCSE 1

AS SUBJECTS

AS 1

A Level SUBJECTS

A Level 1

International A Level SUBJECTS

International A Level 1

University Admissions Tests SUBJECTS

University Admissions Tests 1
Features
Features

Discover all of these amazing features with a free account.

Flashcards

Vaia AI

Notes

Study Plans

Study Sets
What’s new?

Flashcards
Study your flashcards with three learning modes.

Study Sets
All of your learning materials stored in one place.

Notes
Create and edit notes or documents.

Study Plans
Organise your studies and prepare for exams.
Resources
Discover

All the hacks around your studies and career - in one place.

Find a job

Find a degree

Student Deals

Magazine

Mobile App
Featured

Magazine
Trusted advice for anyone who wants to ace their studies & career.

Job Board
The largest student job board with the most exciting opportunities.

Vaia Deals
Verified student deals from top brands.

Our App
Discover our mobile app to take your studies anywhere.

Go to App

Learning Materials

Features

Discover

Chapter 10: Q13P (page 505)

Show that the first parenthesis in (3.5) is a symmetric tensor and the second parenthesis is antisymmetric.

Short Answer

Expert verified

Answer

$T$ is a summation of an antisymmetric tensor and a symmetric tensor.

Step by step solution

01

Given information.

Physics definitions are given.

02

Definition of a tensor.

Tensors, like scalars and vectors, are mathematical constructs that can be used to describe physical qualities. Tensors are just a combination of scalars and vectors, with a scalar being a zero rank tensor and a vector being a first rank tensor.

03

Write the equation mentioned in the question.

Write the given equation.

$T_{i j} = \frac{1}{2} (T_{i j} + T_{j i}) + \frac{1}{2} (T_{i j} - T_{j i})$

Define new tensors.

$S_{i j} = \frac{1}{2} (T_{i j} + T_{j i}) A_{i j} = \frac{1}{2} (T_{i j} - T_{j i})$

Add the above equations.

$T_{i j} = S_{i j} + A_{i j}$

04

Invert the indices of the tensors.

Invert the indices of the tensors.

$S_{j i} = \frac{1}{2} (T_{j i} + T_{i j}) = S_{i j} A_{j i} = \frac{1}{2} (T_{j i} - T_{i j}) = - A_{i j}$

This implies that $T$ is a summation of an antisymmetric tensor and a symmetric tensor.

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!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Show that $δ_{i j} ϵ_{k l m}$ is an isotropic tensor of rank 5. Hint: Combine equations (5.4) and (5.7).

Show that, in polar coordinates, the $θ$ contravariant component of dsis which is unitless, the $d θ$ physical component of ds is $r d θ$ which has units of length, and the $θ$ covariant component of ds is $r^{2} d θ$ which has units role="math" localid="1659265070715" ${(l e n g t h)}^{2}$ .

In equation $(10.13),$ let the variables be rectangular coordinates x, y, z, and let $x_{1}, x_{2}, x_{3}$ , be general curvilinear coordinates, orthogonal or not (see end of Section 8 ). Show that $J^{T} J$ is the $g_{ij}$ matrix in $(8.13)$ [or in $(8.16)$ for an orthogonal system]. Thus show that the volume element in a general coordinate system is $dV = \sqrt{g} {dx}_{1} {dx}_{2} {dx}_{3}$ where $g = {detg}_{ij}$ , and that for an orthogonal system, this becomes [by $(8.16)$ or $(10.19)$ ], $dV = h_{1} h_{2} h_{3} {dx}_{1} {dx}_{2} {dx}_{3}$ . Hint: To evaluate the products of partial derivatives in $J^{T} J$ , observe that the same expressions arise as in finding ${ds}^{2}$ . In fact, from $(8.11)$ and $(8.12)$ , you can show that row i times column j in $J^{T} J$ is just $a_{i} . a_{j} = g_{ij}$ in equations $(8.11)$ to $(8.14)$ .

Show that $T_{i j k l m} S_{l m}$ is a tensor and find its rank (assuming that T and S are tensors of the rank indicated by the indices).

Write equations (2.12) out in detail and solve the three simultaneous equations (say by determinants) for $x_{1}, x_{2}, x_{3}$ in terms of $x_{1}', x_{2}', x_{3}'$ to verify equations (2.13) . Use your results in Problem 4.

See all solutions

Recommended explanations on Physics Textbooks

Fluids

Read Explanation

Electricity

Read Explanation

Work Energy and Power

Read Explanation

Particle Model of Matter

Read Explanation

Circular Motion and Gravitation

Read Explanation

Turning Points in Physics

Read Explanation

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free