91Ó°ÊÓ

The parabolic 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.

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

Find $∇U$in Parabolic coordinates.

$$\begin{gathered} ∇U=\frac{1}{h_u}\frac{∂U}{∂u}{\stackrel{Áåœ}{e}}_u+\frac{1}{h_v}\frac{∂U}{∂v}{\stackrel{Áåœ}{e}}_v+\frac{1}{h_z}\frac{∂U}{∂z}{\stackrel{Áåœ}{e}}_z \\ ∇U=\frac{1}{\sqrt{u^2+v^2}}\frac{∂U}{∂u}{\stackrel{Áåœ}{e}}_u+\frac{1}{\sqrt{u^2+v^2}}\frac{∂U}{∂v}{\stackrel{Áåœ}{e}}_v+\frac{∂U}{∂z}{\stackrel{Áåœ}{e}}_z \end{gathered}$$

Find the divergence of V.

$$\begin{gathered} ∇.V=\frac{1}{u^2+v^2}\left\{\frac{∂}{∂u}\left(\sqrt{u^2+v^2}{V}_1\right)+\frac{∂}{∂v}\left(\sqrt{u^2+v^2}{V}_2\right)+\frac{∂}{∂z}\left(\left(u^2+v^2\right)V_3\right)\right\} \\ ∇.V=\frac{1}{u^2+v^2}\frac{∂}{∂u}\left(\sqrt{u^2+v^2}{V}_1\right)+\frac{1}{u^2+v^2}\frac{∂}{∂v}\left(\sqrt{u^2+v^2}{V}_2\right)+\frac{∂V_3}{∂z} \end{gathered}$$

Find the Laplacian of U.

role="math" localid="1659269426501" $$\begin{gathered} {∇}^{2}U=\frac{1}{u^2+v^2}\left\{\frac{∂}{∂u}\left(\frac{∂U}{∂u}\right)+\frac{∂}{∂v}\left(\frac{∂U}{∂u}\right)+\frac{∂}{∂z}\left(\left(u^2+v^2\right)\frac{∂U}{∂z}\right)\right\} \\ {∇}^{2}U=\frac{1}{u^2+v^2}\frac{{∂}^{2}U}{∂u^2}+\frac{1}{u^2+v^2}\frac{{∂}^{2}U}{∂v^2}+\frac{{∂}^{2}U}{∂z^2} \end{gathered}$$

Find curl of V.

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

Hence, the required values are mentioned below.

$$\begin{gathered} ∇U=\frac{1}{\sqrt{u^2+v^2}}\frac{∂U}{∂u}{\stackrel{Áåœ}{e}}_u+\frac{1}{\sqrt{u^2+v^2}}\frac{∂U}{∂v}{\stackrel{Áåœ}{e}}_v+\frac{∂U}{∂z}{\stackrel{Áåœ}{e}}_z \\ ∇.V=\frac{1}{\sqrt{u^2+v^2}}\frac{∂}{∂u}\left(\sqrt{u^2+v^2}{V}_1\right)+\frac{1}{u^2+v^2}\frac{∂}{∂v}\left(\sqrt{u^2+v^2}{V}_2\right)+\frac{∂V_3}{∂z} \\ {∇}^{2}U=\frac{1}{\sqrt{u^2+v^2}}\frac{{∂}^{2}U}{∂u^2}+\frac{1}{u^2+v^2}\frac{{∂}^{2}U}{∂v^2}+\frac{{∂}^{2}U}{∂z^2} \\ ∇×V=\left[\begin{array}{c}\frac{1}{\sqrt{u^2+v^2}}\left\{\frac{∂V_3}{∂v}-\frac{∂}{∂z}\left(\sqrt{u^2+v^2}{V}_2\right)\right\}{\stackrel{Áåœ}{e}}_u+\frac{1}{\sqrt{u^2+v^2}}\left\{\frac{∂}{∂z}\left(\sqrt{u^2+v^2}{V}_1\right)\right\}{\stackrel{Áåœ}{e}}_v\\ +\frac{1}{\sqrt{u^2+v^2}}\left\{\frac{∂}{∂u}\left(\sqrt{u^2+v^2}{V}_2\right)-\frac{∂}{∂v}\left(\sqrt{u^2+v^2}{V}_1\right)\right\}{\stackrel{Áåœ}{e}}_z\end{array}\right] \end{gathered}$$