91Ó°ÊÓ

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$∇∪$in Parabolic coordinates.

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

Find the divergence of V.

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

Find the Laplacian of U.

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

Find curl of V.

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

Hence, the required values are mentioned below.