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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.

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Most popular questions from this chapter

Parabolic cylinder.

Mass of uniform density=1, bounded by the coordinate planes and the plane x +y +x=1 .

Let $T_{j k m n}$ be the tensor in $(5 .8)$ . This is a $4^{t h}$ -rank tensor and so has $3^{4} = 81$ components. Most of the components are zero. Find the nonzero components and their values. Hint: See discussion after $(5 .8)$ .

In $(7 .9)$ we have written the first row of elements in the inertia matrix. Write the formulas for the other6elements and compare with Section 4.

Show by the quotient rule (Section 3 ) that $C_{ijkm}$ in $(7.5)$ is a $4^{th}$ -rank tensor.

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