91Ó°ÊÓ

The given values are mentioned below.

$$\begin{gathered} x=uv\cos \mathrm{Ï•} \\ y=uv\sin \mathrm{Ï•} \\ z=\frac{1}{2}\left(u^2-v^2\right) \end{gathered}$$

The coordinate system primarily utilized in three-dimensional systems is the cylindrical coordinates of the system. The cylindrical coordinate system is used to find the surface area in three-dimensional space.

The given values are mentioned below.

$$\begin{gathered} x=uv\cos \mathrm{Ï•} \\ y=uv\sin \mathrm{Ï•} \\ z=\frac{1}{2}\left(u^2-v^2\right) \end{gathered}$$

The formula states the equation mentioned below.

$\left[\begin{array}{c}dx\\ dy\\ dz\end{array}\right]=\left(\begin{array}{ccc}\cos \mathrm{Ï•}& u\cos \mathrm{Ï•}& -uv\sin \mathrm{Ï•}\\ v\sin \mathrm{Ï•}& u\sin \mathrm{Ï•}& uv\cos \mathrm{Ï•}\\ u& -v& 0\end{array}\right)\left[\begin{array}{c}du\\ dv\\ d\mathrm{Ï•}\end{array}\right]$.

Since the matrix is orthogonal. Each column is proportional to the unit vector.

$$\begin{gathered} a_u=v\cos \mathrm{Ï•}{e}_x+v\sin \mathrm{Ï•}{e}_y+u{e}_z \\ {a}_v=u\cos \mathrm{Ï•}{e}_x+u\sin \mathrm{Ï•}{e}_y+u{e}_z \\ {a}_u=uv\cos \mathrm{Ï•}{e}_x+uv\sin \mathrm{Ï•}{e}_y \end{gathered}$$

More values are given below.

$$\begin{gathered} e_u=\frac{v\cos {\mathrm{Ï•}}^{âˆ\S }}{\sqrt{v^2+u^2}}{e}_x+\frac{v\sin {\mathrm{Ï•}}^{âˆ\S }}{\sqrt{v^2+u^2}}{e}_y+\frac{v{}^{âˆ\S }}{\sqrt{v^2+u^2}}e \\ {e}_v=\frac{u\cos {\mathrm{Ï•}}^{âˆ\S }}{\sqrt{v^2+u^2}}{e}_x+\frac{u\sin {\mathrm{Ï•}}^{âˆ\S }}{\sqrt{v^2+u^2}}{e}_y+\frac{v{}^{âˆ\S }}{\sqrt{v^2+u^2}}{e}_z \\ {e}_{\mathrm{Ï•}}=\sin {\mathrm{Ï•}}^{âˆ\S }{e}_x+\cos \stackrel{âˆ\S }{\mathrm{Ï•}}{e}_y \end{gathered}$$

More values are given below.

$$\begin{gathered} h_u=\sqrt{v^2+u^2} \\ {h}_v=\sqrt{v^2+u^2} \\ {h}_{\mathrm{Ï•}}=uv \end{gathered}$$

The value of $d{s}^2$is given below.

$d{s}^2=\left(u^2+v^2\right)d{u}^2\left(u^2+v^2\right)d{v}^2+u^2{v}^{2}d{\mathrm{Ï•}}^2$

The value of $dv$is given below.

$$\begin{gathered} dV=h_u{h}_v{h}_{\mathrm{Ï•}}dudvd\mathrm{Ï•} \\ dV=uv\left(u^2+v^2\right)dudvd\mathrm{Ï•} \end{gathered}$$

The value of $dS$is given below.

$ds=a_{i}d{q}_i$