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

Given information.

Co-variant and contra-variant components are defined.

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.

Write vector with contravariant components.

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

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

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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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

Step by step solution

Given information.

Definition of an orthogonal coordinate system.

Write vector with contravariant components.

Use previous definitions and evaluate.

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In (10.18), show by raising and lowering indices that aiVi=aiVi . Also, write (10.18) for an orthogonal coordinate system with gijandgijwritten in terms of the scale factors.

Short Answer

Step by step solution

Given information.

Definition of an orthogonal coordinate system.

Write vector with contravariant components.

Use previous definitions and evaluate.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Quantum Physics

Work Energy and Power

Radiation

Physical Quantities And Units

Dynamics

Scientific Method Physics

Study anywhere. Anytime. Across all devices.

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.