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

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

The given values are mentioned below.

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

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.

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

The formula states the equation mentioned below.

$\left[\begin{array}{c}dx\\ dy\end{array}\right]=\left[\begin{array}{cc}a\frac{1+\cos hu\cos v}{{\left(\cos hu+\cos v\right)}^2}& -a\frac{\sin hu\sin v}{{\left(\cos hu+\cos v\right)}^2}\\ a\frac{\sin hu\sin v}{{\left(\cos hu+\cos v\right)}^2}& a\frac{1+\cos hu\cos v}{{\left(\cos hu+\cos v\right)}^2}\end{array}\right]\left[\begin{array}{c}du\\ dv\end{array}\right]$

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

$$\begin{gathered} a_u=a\frac{1+\cos hu\cos v}{{\left(\cos hu+\cos v\right)}^2}{e}_x-a\frac{\sin hu\sin v}{{\left(\cos hu+\cos v\right)}^2} \\ {a}_v=a\frac{\sin hu\sin v}{{\left(\cos hu+\cos v\right)}^2}{e}_x+a\frac{1+\cos hu\cos v}{{\left(\cos hu+\cos v\right)}^2} \end{gathered}$$

More values are given below.

$$\begin{gathered} {\mathit{e}}_{\mathbf{u}}\mathbf{}\mathbf{=}\mathbf{}\mathit{a}\frac{\mathbf{1}\mathbf{+}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{h}\mathbf{u}\mathbf{}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{v}}{\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{h}\mathbf{}\mathbf{u}\mathbf{+}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{v}}{\mathit{e}}_{\mathbf{x}}\mathbf{-}\mathit{a}\frac{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{h}\mathbf{u}\mathbf{}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{v}}{\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{h}\mathbf{u}\mathbf{+}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{v}} \\ {\mathit{e}}_{\mathbf{v}}\mathbf{=}\mathit{a}\frac{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{h}\mathbf{u}\mathbf{}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{v}}{\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{h}\mathbf{u}\mathbf{+}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{v}}{\mathit{e}}_{\mathbf{x}}\mathbf{+}\mathit{a}\frac{\mathbf{1}\mathbf{+}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{h}\mathbf{}\mathbf{u}\mathbf{}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{v}}{\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{h}\mathbf{u}\mathbf{+}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{v}} \end{gathered}$$

More values are given below.

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

$d{s}^2=\frac{a^2}{{\left(\cos hu+\cos v\right)}^2}d{u}^2+\frac{a^2}{{\left(\cos hu+\cos v\right)}^2}d{v}^2$

The value of $dV$is given below.

$$\begin{gathered} dV=h_u{h}_{v}dudv \\ dV=\frac{a^2}{\cos hu+\cos v}dudv \end{gathered}$$

The value of $dS$is given below.

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