91Ó°ÊÓ

The given values are mentioned below.

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

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 $∇U$.

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

Find the value of $∇.V$

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

Find the value of ${∇}^{2}U$.

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

The required values are given below.

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