Parabolic cylinder coordinates $\mathit{u}\mathbf{,}\mathit{v}\mathbf{,}\mathit{z}\mathbf{.}$

$$\begin{gathered} \mathit{x}\mathbf{=}\mathit{u}\mathit{v}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{\varphi } \\ \mathit{y}\mathbf{=}\mathit{u}\mathit{v}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathit{\varphi } \\ \mathit{z}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\left(u^2-v^2\right) \end{gathered}$$

The given values are mentioned below.

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

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.

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

The formula states the equation mentioned below.

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

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

$$\begin{gathered} a_u=v\cos \varphi e_x+v\sin \varphi e_y+u{e}_z \\ {a}_v=u\cos \varphi e_x+u\sin \varphi e_y+u{e}_z \\ {a}_u=uv\cos \varphi e_x+uv\sin \varphi e_y \end{gathered}$$

More values are given below.

$$\begin{gathered} e_u=\frac{v\cos {\varphi }^{\wedge }}{\sqrt{v^2+u^2}}{e}_x+\frac{v\sin {\varphi }^{\wedge }}{\sqrt{v^2+u^2}}{e}_y+\frac{v{}^{\wedge }}{\sqrt{v^2+u^2}}e \\ {e}_v=\frac{u\cos {\varphi }^{\wedge }}{\sqrt{v^2+u^2}}{e}_x+\frac{u\sin {\varphi }^{\wedge }}{\sqrt{v^2+u^2}}{e}_y+\frac{v{}^{\wedge }}{\sqrt{v^2+u^2}}{e}_z \\ {e}_{\varphi }=\sin {\varphi }^{\wedge }{e}_x+\cos \stackrel{\wedge }{\varphi }{e}_y \end{gathered}$$

More values are given below.

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

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

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

The value of $dv$is given below.

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

The value of $dS$is given below.

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