Question

Write$\mathbf{\nabla }\mathit{U}\mathbf{,}\mathbf{}\mathbf{\nabla }\mathit{V}\mathbf{,}\mathbf{}{\mathbf{\nabla }}^{\mathbf{2}}\mathit{U}$

Short answer

$$\begin{gathered} Therequiredvaluesaregivenbelow. \\ \nabla U=\frac{\partial U}{\partial u}{\hat{e}}_u+\frac{\sqrt{2v-v^2}}{u}\frac{\partial U}{\partial v}{\hat{e}}_v \\ \nabla ·V=\frac{1}{u}\frac{\partial }{\partial u}\left(u{V}_u\right)+\frac{\sqrt{2v-v^2}}{u}\frac{\partial }{\partial v}{V}_v \\ {\nabla }^{2}U=\frac{1}{u}\frac{\partial }{\partial u}\left(u\frac{\partial U}{\partial u}+\frac{\partial }{h_u{h}_v}\frac{h_v}{\partial u}\frac{\partial U}{\partial u}\right)+\frac{1}{h_u{h}_v}\frac{\partial }{\partial v}\left(\frac{h_u}{h_v}\frac{\partial U}{\partial v}\left(\frac{\sqrt{2v-v^2}}{u}\frac{\partial U}{\partial v}\right)\right) \end{gathered}$$

Step-by-step solution

Given Information

The given values are mentioned below.

Definition of a covariant vector

Components of a covariant vector or cotangent vector (commonly abbreviated as co vector) co-vary as the basis changes.

Find the values

The given values are mentioned below.

$$\begin{gathered} x=u\left(1-v\right) \\ y=u\sqrt{2v-v^2} \end{gathered}$$

The scale factors are given below.

$$\begin{gathered} h_u=1 \\ {h}_v=\frac{u}{\sqrt{2v-v^2}} \end{gathered}$$

Find the value of $\nabla U$.

$$\begin{gathered} \nabla U=\frac{1}{h_u}\frac{\partial U}{\partial u}{\hat{e}}_u+\frac{1}{h_v}\frac{\partial U}{\partial v}{\hat{e}}_v \\ \nabla U=\frac{\partial U}{\partial u}{\hat{e}}_u+\frac{\sqrt{2v-v^2}}{u}\frac{\partial U}{\partial v}{\hat{e}}_v \end{gathered}$$

Find the value of $\nabla .V$

$$\begin{gathered} \nabla .V=\frac{1}{h_u{h}_v}\frac{\partial }{\partial u}\left(h_v{V}_u\right)+\frac{1}{h_u{h}_v}\frac{\partial }{\partial V}\left(h_u{V}_v\right) \\ \nabla .V=\frac{1}{u}\frac{\partial }{\partial u}\left(u{V}_{uv}\right)+\frac{\sqrt{2v-v^2}}{u}\frac{\partial }{\partial v}\left(V_v\right) \end{gathered}$$

Find the value of ${\nabla }^{2}U$.

${\nabla }^{2}U=\left(\frac{1}{u}\frac{\partial }{\partial u}\left(u\frac{\partial U}{\partial u}\right)+\frac{\partial }{h_u{h}_v}\frac{h_v}{\partial u}\frac{\partial U}{\partial u}\right)+\frac{1}{h_u{h}_v}\frac{\partial }{\partial v}\left(\frac{h_u}{h_v}\frac{\partial U}{\partial v}\left(\frac{\sqrt{2v-v^2}}{u}\frac{\partial U}{\partial v}\right)\right)$

The required values are given below.

$$\begin{gathered} \nabla U=\frac{\partial U}{\partial u}{\hat{e}}_u+\frac{\sqrt{2v-v^2}}{u}\frac{\partial U}{\partial v}{\hat{e}}_v \\ \nabla ·V=\frac{1}{u}\frac{\partial }{\partial u}\left(u{V}_u\right)+\frac{\sqrt{2v-v^2}}{u}\frac{\partial }{\partial v}{V}_v \\ {\nabla }^{2}U=\frac{1}{u}\frac{\partial }{\partial u}\left(u\frac{\partial U}{\partial u}\right)+\frac{\partial }{h_u{h}_v}\frac{h_v}{\partial u}\frac{\partial U}{\partial u}+\frac{1}{h_u{h}_v}\frac{\partial }{\partial v}\left(\frac{h_u}{h_v}\frac{\partial U}{\partial v}\left(\frac{\sqrt{2v-v^2}}{u}\frac{\partial U}{\partial v}\right)\right) \end{gathered}$$