Question

Do Problem 14 for an orthogonal coordinate system with scale factors${\mathit{h}}_{\mathbf{1}}\mathbf{,}{\mathit{h}}_{\mathbf{2}}\mathbf{,}{\mathit{h}}_{\mathbf{3}}$,and compare with the Section 9 formulas

Short answer

The gradient, divergence and curl are obtained.

$\begin{array}{c}\nabla u=\frac{1}{h_i}\frac{\partial u}{\partial x_i}{\widehat{e}}_i\\ \nabla \cdot V=\frac{1}{h_1{h}_2{h}_3}\frac{\partial }{\partial x_i}\left\{h_1{h}_2{h}_3{V}^i\right\}\\ {\nabla }^{2}u=\frac{1}{h_1{h}_2{h}_3}\frac{\partial }{\partial x_i}\left\{\frac{h_1{h}_2{h}_3}{h_i^2}\frac{\partial u}{\partial x_i}\right\}\end{array}$

Step-by-step solution

Given information.

An orthogonal coordinate system is given with scale factors$h_1,h_2,h_3$ .

Definition of a covariance and contravariance.

The components of a vector relative to a tangent bundle basis are covariant in differential geometry if they change with the same linear transformation as the basis. If they change as a result of the inverse transformation, they are contravariant

Begin by calculating the gradient.

Calculate the gradient.

$\begin{array}{c}\nabla u=\frac{\partial u}{\partial x_i}{a}^i\\ a^i=g^{ij}{a}_j\\ =\frac{1}{h_i^2}{a}_i\\ \nabla u=\frac{\partial u}{\partial x_i}\frac{a_i}{h_i^2}\end{array}$

Continue the evaluation.

$\begin{array}{c}{a}_i=h_i{\widehat{e}}_i\\ \nabla u=\frac{1}{h_i}\frac{\partial u}{\partial x_i}{\widehat{e}}_i\end{array}$

Calculate the divergence.

$\begin{array}{c}\nabla \cdot V=\frac{1}{\sqrt{g}}\frac{\partial }{\partial x_i}\left\{\sqrt{g}{V}^i\right\}\\ \sqrt{g}=h_1{h}_2{h}_3\\ \nabla \cdot V=\frac{1}{h_1{h}_2{h}_3}\frac{\partial }{\partial x_i}\left\{h_1{h}_2{h}_3{V}^i\right\}\end{array}$

Calculate the Laplacian.

${\nabla }^{2}u=\frac{1}{h_1{h}_2{h}_3}\frac{\partial }{\partial x_i}\left\{\frac{h_1{h}_2{h}_3}{h_i^2}\frac{\partial u}{\partial x_i}\right\}$

Recognize the relation between physical components and contravariance components and substitute.

The$V\cdot {\widehat{e}}_i=h_i{V}^i$describes the relation between physical components and contravariance components. Substitute this in earlier equations.

$\nabla \cdot V=\frac{1}{h_1{h}_2{h}_3}\{\frac{\partial }{\partial x_1}\{h_2{h}_{3}V\cdot {\widehat{e}}_1\}+\frac{\partial }{\partial x_2}\{h_1{h}_{3}V\cdot {\widehat{e}}_2\}+\frac{\partial }{\partial x_3}\{h_1{h}_{2}V\cdot {\widehat{e}}_3\}\}$