91Ó°ÊÓ

A vector V .

The curl at a given place in the field is represented by a vector whose length and direction correspond to the greatest circulation's magnitude and axis.

Formula states that$∇u={∑}_i\frac{1}{h_i}\frac{∂u}{∂x_i}{\stackrel{Áåœ}{e}}_i$.

Let $u=x_k$,the equation becomes as follows.

$\begin{array}{l}∇x_k=\underset{i}{∑}\frac{1}{h}{\mathrm{δ}}_{ki}{\stackrel{Áåœ}{e}}_i\\ =\frac{1}{h_k}{\stackrel{Áåœ}{e}}_k\end{array}$

The curl of the gradient is zero, equation becomes as follows.

$∇×\left(\frac{1}{h_k}{\stackrel{Áåœ}{e}}_k\right)=0$

Evaluate curl of vector field V.

$\begin{array}{l}V=\underset{i}{∑}{h}_i{V}_i+\frac{{\stackrel{Áåœ}{e}}_i}{h_i}\\ ∇+V=\underset{i}{∑∇}×\left(h_i{V}_i\frac{{\stackrel{Áåœ}{e}}_i}{h_i}\right)\end{array}$

Vector field identity states that $∇×\left(\mathrm{ϕ}v\right)=\mathrm{ϕ}∇×v+∇\mathrm{ϕ}×v$

Apply the identity in the equation, the equation becomes as follows.

$$\begin{gathered} ∇×V=\underset{i}{∑}\left\{h_i{V}_i∇×\left(\frac{{\stackrel{Áåœ}{e}}_i}{h_i}\right)+∇\left(h_i{V}_i\right)×\frac{{\stackrel{Áåœ}{e}}_i}{h_i}\right\} \\ =\underset{i}{∑}∇\left(h_i{V}_i\right)×\left(\frac{{\stackrel{Áåœ}{e}}_i}{h_i}\right)+∇\left(h_i{V}_i\right)=\underset{i}{∑}\left(h_i{V}_i\right)\frac{{\stackrel{Áåœ}{e}}_j}{h_j} \\ =\underset{i}{∑}\frac{1}{h_j{h}_i}\frac{∂}{∂x_j}\left(h_i{V}_i\right){\stackrel{Áåœ}{e}}_j×{\stackrel{Áåœ}{e}}_i;{\stackrel{Áåœ}{e}}_j×{\stackrel{Áåœ}{e}}_i=\underset{k}{∑}{∈}_{jik}{\stackrel{Áåœ}{e}}_k \\ =\underset{jik}{∑}\frac{1}{h_j{h}_i{h}_k}{∈}_{jik}\left(h_i{V}_i\right)h_k{\stackrel{Áåœ}{e}}_k \end{gathered}$$

Hence, the value of V is$\underset{jik}{∑}\frac{1}{h_j{h}_i{h}_k}{∈}_{jik}\left(h_i{V}_i\right)h_k{\stackrel{Áåœ}{e}}_k$.