:Do Problem 5 for the coordinate systems indicated in Problems 10 to 13.Eliptical cylinder.

The Elliptical cylinder.

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 scale factors are given below.

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

Find$\nabla \cup$in Parabolic coordinates.

role="math" localid="1659338050122" $$\begin{gathered} \nabla \cup =\frac{1}{h_u}\frac{\partial \cup }{\partial u}{\hat{e}}_u+\frac{1}{h_v}\frac{\partial \cup }{\partial v}{\hat{e}}_v+\frac{1}{h_z}\frac{\partial \cup }{\partial z}{\hat{e}}_z \\ \nabla \cup \frac{1}{a\sqrt{\sin h^{2}u+\sin h^{2}v}}\frac{\partial \cup }{\partial u}{\hat{e}}_u+\frac{1}{a\sqrt{\sin h^{2}u+\sin h^{2}v}}\frac{\partial \cup }{\partial v}{\hat{e}}_v+\frac{\partial \cup }{\partial z}{\hat{e}}_z \end{gathered}$$

Find the divergence of V.

role="math" localid="1659338986068" $$\begin{gathered} \nabla \times V=\frac{1}{a\sqrt{\sin h^2+{\sin }^{2}v}}\left\{\frac{\partial }{\partial u}\left(a\sqrt{\sin h^2+{\sin }^{2}v{V}_1}\right)+\frac{\partial }{\partial v}\left(a\sqrt{\sin h^2+{\sin }^{2}v{V}_2}\right)\frac{\partial }{\partial z}\left(a^2\left(\sin h^{2}u+{\sin }^{2}v\right)V_1\right)\right\} \\ \nabla ·V=\frac{1}{a^2\left(\sin h^{2}u+{\sin }^{2}v\right)}\frac{\partial }{\partial u}+\left(a\sqrt{\sin h^2+{\sin }^{2}v{V}_1}\right)+\frac{1}{a^2\left(\sin h^{2}u+{\sin }^{2}v\right)}\frac{\partial }{\partial v}\left(a\sqrt{\sin h^{2}u+{\sin }^{2}v{V}_2}\right)+\frac{\partial V}{\partial z} \end{gathered}$$

Find the Laplacian of U.

role="math" localid="1659339151711" $$\begin{gathered} \nabla \cup =\frac{1}{a\sqrt{\sin h^2+{\sin }^{2}v}}\left\{\frac{\partial }{\partial u}\frac{\partial \cup }{\partial u}+\frac{\partial }{\partial v}\left(\frac{\partial \cup }{\partial v}\right)+\frac{\partial }{\partial z}\left(a^2\left(\sin h^2+{\sin }^{2}v\right)\frac{\partial \cup }{\partial z}\right)\right\} \\ {\nabla }^2\cup =\frac{1}{a^2\left(\sin h^{2}u+{\sin }^{2}v\right)}\frac{{\partial }^2\cup }{\partial u^2}+\frac{1}{a^2\left(\sin h^{2}u+{\sin }^{2}v\right)}\frac{{\partial }^2\cup }{\partial v^2}+\frac{{\partial }^2\cup }{\partial z^2} \end{gathered}$$

Find curl of V.

$$\begin{gathered} \nabla \times V=\frac{1}{a\sqrt{\sin h^2+{\sin }^{2}v}}\left\{\frac{\partial V}{\partial V}-\frac{\partial }{\partial Z}a\sqrt{\sin h^2+{\sin }^{2}v{V}_2}\right\}{\hat{e}}_u+\frac{1}{a\sqrt{\sin h^2+{\sin }^{2}v}}\left\{-\frac{\partial }{\partial Z}\left(a\sqrt{\sin h^2+{\sin }^{2}v{V}_1}\right)-\frac{\partial V_3}{\partial u}\right\}{\hat{e}}_v \\ +\frac{1}{u^2+v^2}\left\{\left(\frac{\partial }{\partial u}\sqrt{\sin h^2+{\sin }^{2}v{V}_2}\right)-\frac{\partial }{\partial V}\left(a\sqrt{\sin h^2+{\sin }^{2}v{V}_1}\right)\right\}{\hat{e}}_z \end{gathered}$$

Hence, the required values are mentioned below.