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Chapter 10: Q11P (page 534)

In (10.18), show by raising and lowering indices that $a_{i} V^{i} = a^{i} V_{i}$ . Also, write (10.18) for an orthogonal coordinate system with $g_{i j}$ and $g^{i j}$ written in terms of the scale factors.

Short Answer

Expert verified

The proofs are complete in the solution below.

Step by step solution

01

Given information.

Co-variant and contra-variant components are defined.

02

Definition of an orthogonal coordinate system.

A curvilinear coordinate system in which each family of surfaces contacts the others at right angles is known as an orthogonal coordinate system.

03

Write vector with contravariant components.

Input the Kronecker Delta function and write the contravariant components of the vector V .

$a_{i} V^{i} = a_{i} 未_{j}^{i} V^{j} 未_{j}^{i} = g^{ik} g_{kj} a_{i} V^{i} = a_{i} g^{ik} g_{kj} V^{j}$

04

Use previous definitions and evaluate.

Use previous definitions to continue evaluation.

$a_{i} g^{ik} = a^{k} g_{kj} V^{j} = V_{k} a_{i} V^{i} = a^{i} V_{i}$

Continue evaluation.

$\begin{array}{l} a_{i} \frac{1}{h_{1}^{2}} = a^{i} \\ h_{j}^{2} V^{j} = V_{j} \end{array}$

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Most popular questions from this chapter

Verify equations(2.6).

Parabolic cylinder.

Write and prove in tensor notation:

(a) Chapter 6, Problem 3.13.

(b) Chapter 6, Problem 3.14.

(c) Lagrange鈥檚 identity: $(A 脳 B) 路 (C 脳 D) = (A 路 C) (B 路 D) - (A 路 D) (B 路 C)$ .

(d), role="math" localid="1659335462905" $(A 脳 B) (C 脳 D) = (ABD) C - (ABC) D$ where the symbol $(x y z)$ means the triple scalar product of the three vectors.

Write the tensor transformation equations for $蔚_{i j k} 蔚_{m n p}$ to show that this is a (rank 6) tensor (nota pseudo tensor). Hint:Write (6.1) for each $蔚$ and multiply them, being careful not to re-use a pair of summation indices.

Carry through the details of getting $(7 .4)$ from $(7 .2)$ and $(7 .3)$ . Hint: You need the dot product of $e'_{尾}$ and $e_{j}$ . This is the cosine of an angle between two axes since each eis a unit vector. Identify the result from matrixAin $(2 .10)$ .

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