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

$$\begin{gathered} \mathit{x}\mathbf{=}\mathit{a}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{h}\mathit{u}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{v} \\ \mathit{y}\mathbf{=}\mathbf{}\mathit{a}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\mathbf{}\mathit{u}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{v} \\ \mathit{z}\mathbf{=}\mathit{z} \end{gathered}$$

The given values are mentioned below.

$$\begin{gathered} x=a\cos hu\cos v \\ y=a\sin hu\sin v \\ z=z \end{gathered}$$

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.

$$\begin{gathered} x=a\cos hu\cos v \\ y=a\sin hu\sin v \\ z=z \end{gathered}$$

The formula states that $ds=idx+jdy+kdz$.

$$\begin{gathered} ds=idx+jdy+kdz \\ ds=i\left(a\sin hu\cos vdu-a\cos hu\sin vdu\right)+j\left(a\cos hu\sin u\cos vdv\right)+kdz \\ ds=\left(ia\sin hu\cos v+ja\cos hu\sin v\right)du+\left(-ia\cos hu\sin v+ja\sin hu\cos v\right)dv+kdz \end{gathered}$$

The following values are derived from the above equation.

$$\begin{gathered} a_u=ia\sin hu\cos v+ja\cos hu\sin v \\ {a}_v=-ia\cos hu\sin v+ja\sin hu\cos v \\ {a}_z=k \end{gathered}$$

More values are given below.

$$\begin{gathered} e_u=\frac{1}{\sqrt{\sin h^{2}u+{\sin }^{2}v}}\left(i\sin hu\cos v+j\cos hu\sin v\right) \\ {e}_v=\frac{1}{\sqrt{\sin h^{2}u+{\sin }^{2}v}}\left(-i\sin hu\cos v+j\cos hu\sin v\right) \\ {e}_z=k \end{gathered}$$

More values are given below.

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

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

$d{s}^2=a^2\left(\sin h^{2}u+{\sin }^{2}v\right)\left(d{u}^2+d{v}^2\right)+d{z}^2$

The value of $dv$is given below.

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

The value of $dS$is given below.

$ds=\left(ia\sin hu\cos v+ja\cos hu\sin v\right)du+\left(-ia\cos hu\sin v+ja\sin hu\cos v\right)dv+kdz$