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

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

Given values are mentioned below.

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

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.

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

The formula states that

$$\begin{gathered} d{s}^2=h_i^{2}d{x}_i^2 \\ d{s}^2=\left(u^2+v^2\right)d{u}^2+\left(u^2+v^2\right)d{v}^2+d{z}^2 \end{gathered}$$

The following values are derived from the above equation.

$$\begin{gathered} a_u=ui+vj \\ {a}_v=-vi+uj \\ {a}_z=k \end{gathered}$$

More values are given below.

$$\begin{gathered} e_u=\frac{1}{\sqrt{u^2+v^2}}ui+vj \\ {e}_v=\frac{1}{\sqrt{u^2+v^2}}\left(ui+vj\right) \\ {e}_z=k \end{gathered}$$

More values are given below.

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

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

$$\begin{gathered} d{s}^2=h_i^{2}d{x}_i^2 \\ d{s}^2=\left(u^2+v^2\right)d{u}^2+\left(u^2+v^2\right)d{v}^2+d{z}^2 \end{gathered}$$

The value of dv is given below.

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

The value of $dS$is given below.

$ds=\left(ui+vj\right)du+\left(-vi+uj\right)dv+kdz$