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Chapter 10: Q3P (page 513)

Show that δijϵklmis an isotropic tensor of rank 5. Hint: Combine equations (5.4) and (5.7).

Short Answer

Expert verified

The expressionδij∈klm is an isotropic tensor.

Step by step solution

01

Given Information

An isotropic tensor of rank 5is .δij∈klm

02

Definition of Isomorphic tensor

Isotropic tensor is defined as a tensor with components that remain unaltered when the coordinate system is rotated arbitrarily.

03

Simplify the isotropic tensor

Use the result given in previous question.

ϵρλσ'=aρiaλjaσkϵijk=ϵρλσdetA=ϵρλσ

It is known thatδαβ'=δαβ

Use the above result to getδαβ'∈ρλσ'=δαβ∈ρλσ

Hence proved thatδij∈klm is an isotropic tensor.

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Most popular questions from this chapter

Show that the nine quantities Tij=(∂Vi)/(∂xJ) (which are the Cartesian components of ∇V where V is a vector) satisfy the transformation equations (2.14)for a Cartesian 2nd -rank tensor. Show that they do not satisfy the general tensor transformation equations as in (10.12) . Hint: Differentiate (10.9)or(10.10)partially with respect to, say,x'k. You should get the expected terms [as in(10.12) ] plus some extra terms; these extraneous terms show that(∂Vi)/(∂xJ) is not a tensor under general transformations. Comment: It is possible to express the components of∇V correctly in general coordinate systems by taking into account the variation of the basis vectors in length and direction.

For Example 1, write out the components of U,V, and U×Vin the original right-handed coordinate system and in the left-handed coordinate system S' with the axis reflected. Show that each component ofU×VinS'has the “wrong” sign to obey the vector transformation laws.

Show that, in polar coordinates, theθcontravariant component of dsis which is unitless, the dθphysical component of ds is rdθwhich has units of length, and theθcovariant component of ds isr2dθwhich has units role="math" localid="1659265070715" (length)2 .

Parabolic.

Any rotation of axes in three dimensions can be described by giving the nine direction cosines of the angle between the (x,y,z)axes and the (x',y',z')axes. Show that the matrix A of these direction cosines in (2.7)or (2.10)is an orthogonal matrix. Hint: See Chapter 3, Section 9. Find AATand use Problem 3.
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