/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q8P Using (10.15) show that聽gij聽is... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Q8P (page 534)

Using (10.15) show thatgijis a2ndV-rank covariant tensor. Hint:Write the transformationequation for eachdx, and set the scalards'2=ds2to find the transformationequation forgij.

Short Answer

Expert verified

The results are proved in the solution.

Step by step solution

01

Given information.

The scalards'2=ds2

02

Definition of a covariance and contravariance.

The components of a vector relative to a tangent bundle basis are covariant in differential geometry if they change with the same linear transformation as the basis. If they change as a result of the inverse transformation, they are contravariant

03

Write the general form of a differential.

Length is an invariant quantity. This implies that ds'2=ds2. Use10.14,10.15, and continue evaluation.

ds2=gijdxidxjds2=gij鈭�xi鈭�xk'鈭�xj鈭�xl'dxk'dxl'ds'2=gkl'dxk'dxl'

04

Compare the equations and write the result.

Compare the deduced equations and write the result.

gkl'=鈭�xi鈭�xk'鈭�xj鈭�xl'gij

Thus, it can be concluded that gijis a second order 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.

Over 30 million students worldwide already upgrade their learning with 91影视!

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

As in Problem 2, complete Example 5.

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鈥檛 make the assumptions about symmetry that we did in Example 5.

Show by the quotient rule (Section 3 ) that Cijkmin (7.5)is a 4th-rank tensor.

Parabolic cylinder.

Interpret the elements of the matrices in Chapter 3, Problems 11.18 to11.21, as components of stress tensors. In each case diagonalize the matrix and so find the principal axes of the stress (along which the stress is pure tension or compression). Describe the stress relative to these axes. (See Example 1.)
See all solutions

Recommended explanations on Physics Textbooks

Physical Quantities And Units

Read Explanation

Wave Optics

Read Explanation

Kinematics Physics

Read Explanation

Force

Read Explanation

Fluids

Read Explanation

Rotational Dynamics

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.