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: Q8P (page 502)

Write the equations in (2.16) and so in (2.17) solved for the unprimed components in terms of the primed components.

Short Answer

Expert verified

Answer

The statement has been verified.

Step by step solution

01

Given Information

Vector U and vector V with components $U_{1}, U_{2}, U_{3}$ and $V_{1}, V_{2} V_{3}$ respectively.

02

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.

03

Prove the statement.

Vector U and vector V with components $U_{1}, U_{2}, U_{3}$ and $V_{1}, V_{2}, V_{3}$ respectively.

A is an orthogonal matrix hence $A A^{T} = δ_{i j}$

The formula for the cartesian vectors states that $V_{i} = \sum_{j = 1}^{3} a_{i j} V_{j}'$

$V_{i} = \sum_{j = 1}^{3} a_{i j} V_{j}' V_{i} (U_{i}) = \sum_{j = 1}^{3} a_{i j} V_{j}' (U_{j}') \sum_{k l} a_{k i} a_{i j} V_{l}' U_{k}' = \sum_{k} a_{k i} U_{k}' \sum_{i} a_{j i} V_{j}' \sum_{k l} a_{k i} a_{i j} V_{l}' U_{k}' = U_{i} V_{j}$

Hence, the statement has been proven.

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

Find the inertia tensor about the origin for a mass of uniform density =1, inside the part of the unit sphere where $x > 0, y > 0,$ and find the principal moments of inertia and the principal axes. Note that this is similar to Example 5 but the mass is both above and below the $(x, y)$ plane. Warning hint: This time don’t make the assumptions about symmetry that we did in Example 5.

Show that the sum of two $3^{r d}$ -rank tensors is a $3^{r d}$ -rank tensor. Hint: Write the transformation law for each tensor and then add your two equations. Divide out the factors to leave the result $T'_{α β γ} + S'_{α β γ} = a_{α i} a_{β j} a_{γ k} (T_{i j k} + S_{i j k})$ .

Show that the fourth expression in (3.1) is equal to $\frac{\partial u}{\partial x'_{i}}$ . By equations (2.6) and (2.10) , show that $\frac{\partial x_{j}}{\partial x'_{i}} = a$ , so $\frac{\partial u}{\partial x'_{i}} = \frac{\partial u}{\partial x_{j}} \frac{\partial x_{j}}{\partial x'_{i}} = a_{i j} \frac{\partial u}{\partial x_{j}}$

Compare this with equation (2.12) to show that $\nabla u$ is a Cartesian vector. Hint: Watch the summation indices carefully and if it helps, put back the summation signs or write sums out in detail as in (3.1) until you get used to summation convention.

$X_{i} A_{j} = B_{i j}$ .

Verify for a few representative cases that $(5 .6)$ gives the same results as a Laplace development. First note that if $α, β, γ = 1, 2, 3$ , then $(5 .6)$ is just $(5 .5)$ . Then try letting $α, β, γ =$ an even permutation of $1, 2, 3$ , and then try an odd permutation, to see that the signs work out correctly. Finally try a case when $ε_{α β γ} = 0$ (that is when two of the indices are equal) to see that the right hand side of $(5 .6)$ is zero because you are evaluating a determinant which has two identical rows.

See all solutions

Recommended explanations on Physics Textbooks

Measurements

Read Explanation

Famous Physicists

Read Explanation

Geometrical and Physical Optics

Read Explanation

Electromagnetism

Read Explanation

Turning Points in Physics

Read Explanation

Nuclear 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