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Chapter 10: Q17P (page 496)

Repeat Problems 8.15and 10.16above for the (u,v)coordinate system if x=2u-v , y=u-2v.

Short Answer

Expert verified

The value of matrix is .

Step by step solution

01

Given Information:

The equations are mentioned below.

x=2u

y=u-2v

02

Definition of covariant basis vector:

A covariant vector, also known as a cotangent vector, has components that co-vary when the basis changes. That is, the components must be converted using the same matrix as the base matrix change.

03

Find the value:

The equations are mentioned below.

x=2u

y=u-2v

Position vector is,

Covariant basis vectors are mentioned below.

The metric tensor and inverse of metric tensor is given below.

Hence, the value of matrix is.

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Most popular questions from this chapter

∂B∂t=-∇×E

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.

Use equations (9.2), (9.8), and (9.11) to evaluate the following expressions.In cylindrical coordinates∇.er,∇.eθ,∇×er,∇×eθ. .

Prove (9.4) in the following way. Using (9.2) with, show that. Similarly, show thatand ∇. Letin that order form a right-handed triad (so that, etc.) and show that. Take the divergence of this equation and, using the vector identities (h) and (b) in the table at the end of Chapter 6, show that. The other parts of (9.4) are proved similar.

XijklAkl=Bij.
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