Question

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

Short answer

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 .

$$\begin{gathered} {\mathrm{a}}_{\mathrm{i}}{\mathrm{V}}^{\mathrm{i}}={\mathrm{a}}_{\mathrm{i}}{\mathrm{\delta }}_{\mathrm{j}}^{\mathrm{i}}{\mathrm{V}}^{\mathrm{j}} \\ {\mathrm{\delta }}_{\mathrm{j}}^{\mathrm{i}}={\mathrm{g}}^{\mathrm{ik}}{\mathrm{g}}_{\mathrm{kj}} \\ {\mathrm{a}}_{\mathrm{i}}{\mathrm{V}}^{\mathrm{i}}={\mathrm{a}}_{\mathrm{i}}{\mathrm{g}}^{\mathrm{ik}}{\mathrm{g}}_{\mathrm{kj}}{\mathrm{V}}^{\mathrm{j}} \end{gathered}$$

Use previous definitions and evaluate.

Use previous definitions to continue evaluation.

$$\begin{gathered} {\mathrm{a}}_{\mathrm{i}}{\mathrm{g}}^{\mathrm{ik}}={\mathrm{a}}^{\mathrm{k}} \\ {\mathrm{g}}_{\mathrm{kj}}{\mathrm{V}}^{\mathrm{j}}={\mathrm{V}}_{\mathrm{k}} \\ {\mathrm{a}}_{\mathrm{i}}{\mathrm{V}}^{\mathrm{i}}={\mathrm{a}}^{\mathrm{i}}{\mathrm{V}}_{\mathrm{i}} \end{gathered}$$

Continue evaluation.

$\begin{array}{l}{\mathrm{a}}_{\mathrm{i}}\frac{1}{{\mathrm{h}}_1^2}={\mathrm{a}}^{\mathrm{i}}\\ {\mathrm{h}}_{\mathrm{j}}^2{\mathrm{V}}^{\mathrm{j}}={\mathrm{V}}_{\mathrm{j}}\end{array}$