91Ó°ÊÓ

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

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

Calculate the gradient.

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

Continue the evaluation.

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

$\begin{array}{c}∇â‹\dots V=\frac{1}{\sqrt{g}}\frac{∂}{∂x_i}\left\{\sqrt{g}{V}^i\right\}\\ \sqrt{g}=h_1{h}_2{h}_3\\ ∇â‹\dots V=\frac{1}{h_1{h}_2{h}_3}\frac{∂}{∂x_i}\left\{h_1{h}_2{h}_3{V}^i\right\}\end{array}$

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

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

$∇â‹\dots V=\frac{1}{h_1{h}_2{h}_3}\{\frac{∂}{∂x_1}\{h_2{h}_{3}Vâ‹\dots {\widehat{e}}_1\}+\frac{∂}{∂x_2}\{h_1{h}_{3}Vâ‹\dots {\widehat{e}}_2\}+\frac{∂}{∂x_3}\{h_1{h}_{2}Vâ‹\dots {\widehat{e}}_3\}\}$