Question

Do Problem 5 for the coordinate systems indicated in Problems 10 to 13. Parabolic.

Short answer

The required values are mentioned below.

$$\begin{gathered} \nabla U=\frac{1}{\sqrt{u^2+v^2}}\frac{\partial U}{\partial u}{\hat{e}}_u+\frac{1}{\sqrt{u^2+v^2}}\frac{\partial U}{\partial v}{\hat{e}}_v+\frac{1}{uv}\frac{\partial U}{\partial \varphi }{\hat{e}}_{\varphi } \\ \nabla .V=\frac{1}{(u^2+v^2)uv}\frac{\partial }{\partial u}\left(\sqrt{u^2+v^2}uv{V}_1\right)+\frac{1}{(u^2+v^2)uv}\frac{\partial }{\partial u}\sqrt{u^2+v^{2}uv{V}_2)+\frac{1}{uv}\frac{\partial V_3}{\partial \varphi }} \\ {\nabla }^{2}U=\frac{1}{(u^2+v^2)uv}\frac{\partial }{\partial u}\left(uv\frac{\partial U}{\partial u}\right)+\frac{1}{\left(u^2+v^2\right)uv}\frac{\partial }{\partial v}\left(uv\frac{\partial U}{\partial u}\right)+\frac{1}{u^2{v}^2}\frac{{\partial }^{2}U}{\partial \varphi } \\ \nabla \times V=\frac{1}{uv\sqrt{u^2+v^2}}\left\{\frac{\partial }{\partial v}\left(uv{V}_3\right)-\frac{\partial }{\partial \varphi }\left(\sqrt{u^2+v^2{V}_2}\right)\right\}{\hat{e}}_u+\frac{1}{uv\sqrt{u^2+v^2}}\left\{\frac{\partial }{\partial \varphi }\sqrt{u^2+v^2}{V}_1)-\frac{\partial }{\partial u}\left(uv{V}_3\right)\right\}{\hat{e}}_v \\ +\frac{1}{u^2+v^2}\left\{\frac{\partial }{\partial u}\left(\sqrt{u^2+v^2}{V}_2\right)-\frac{\partial }{\partial v}\left(\sqrt{u^2+v^2}{V}_1\right)\right\}{\hat{e}}_{\varphi } \end{gathered}$$

Step-by-step solution

Given Information

The Parabolic.

Definition of cylindrical coordinates.

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.

Determining the value of ∇U, ∇.V , ∇2U and ∇×Vand  

The scale factors are given below.

$$\begin{gathered} h_u=\sqrt{u^2+v^2} \\ {h}_v=\sqrt{u^2+v^2} \\ {h}_{\varphi }=uv \end{gathered}$$

Find$\nabla U$in Parabolic coordinates.

$$\begin{gathered} \nabla U=\frac{1}{h_u}\frac{\partial U}{\partial u}{\hat{e}}_u+\frac{1}{h_u}\frac{\partial U}{\partial v}{\hat{e}}_v+\frac{1}{h_u}\frac{\partial U}{\partial \varphi }{\hat{e}}_{\varphi } \\ \nabla U=\frac{1}{\sqrt{u^2+v^2}}\frac{\partial U}{\partial u}{\hat{e}}_u+\frac{1}{\sqrt{u^2+v^2}}\frac{\partial U}{\partial u}{\hat{e}}_v+\frac{1}{uv}\frac{\partial U}{\partial \varphi }{\hat{e}}_{\varphi } \end{gathered}$$

Find the divergence of V.

$$\begin{gathered} \nabla .V=\frac{1}{(u^2+v^2)uv}\left\{\frac{\partial }{\partial u}\left(\sqrt{u^2+v^2}uv{V}_1\right)+\frac{\partial }{\partial v}\left(\sqrt{u^2+v^2}uv{V}_2\right)+\frac{\partial }{\partial \varphi }\left((u^2+v^2)V_3\right)\right\} \\ \nabla .V=\frac{1}{(u^2+v^2)uv}\frac{\partial }{\partial u}\left(\sqrt{u^2+v^2}uv{V}_1\right)+\frac{1}{(u^2+v^2)uv}\frac{\partial }{\partial u}\sqrt{u^2+v^{2}uv{V}_2)+\frac{1}{uv}\frac{\partial V_3}{\partial \varphi }} \end{gathered}$$

Find the Laplacian of U.

$$\begin{gathered} {\nabla }^{2}U=\frac{1}{(u^2+v^2)uv}\left\{\frac{\partial }{\partial u}\left(uv\frac{\partial U}{\partial u}\right)+\frac{1}{\left(u^2+v^2\right)uv}\frac{\partial }{\partial v}\left(uv\frac{\partial U}{\partial u}\right)+\frac{1}{u^2{v}^2}\frac{{\partial }^{2}U}{\partial \varphi }\right\} \\ {\nabla }^{2}U=\frac{1}{(u^2+v^2)uv}\frac{\partial }{\partial u}\left(uv\frac{\partial U}{\partial u}\right)+\frac{1}{\left(u^2+v^2\right)uv}\frac{\partial }{\partial v}\left(uv\frac{\partial U}{\partial u}\right)+\frac{1}{u^2{v}^2}\frac{{\partial }^{2}U}{\partial \varphi } \end{gathered}$$

Find curl of V.

$$\begin{gathered} \nabla \times V=\frac{1}{uv\sqrt{u^2+v^2}}\left\{\frac{\partial }{\partial v}\left(uv{V}_3\right)-\frac{\partial }{\partial \varphi }\left(\sqrt{u^2+v^2{V}_2}\right)\right\}{\hat{e}}_u+\frac{1}{uv\sqrt{u^2+v^2}}\left\{\frac{\partial }{\partial \varphi }\sqrt{u^2+v^2}{V}_1)-\frac{\partial }{\partial u}\left(uv{V}_3\right)\right\}{\hat{e}}_v \\ +\frac{1}{u^2+v^2}\left\{\frac{\partial }{\partial u}\left(\sqrt{u^2+v^2}{V}_2\right)-\frac{\partial }{\partial v}\left(\sqrt{u^2+v^2}{V}_1\right)\right\}{\hat{e}}_{\varphi } \end{gathered}$$

Hence, the required values are mentioned below.

role="math" localid="1659339435729" $$\begin{gathered} \nabla U=\frac{1}{\sqrt{u^2+v^2}}\frac{\partial U}{\partial u}{\hat{e}}_u+\frac{1}{\sqrt{u^2+v^2}}\frac{\partial U}{\partial v}{\hat{e}}_v+\frac{1}{uv}\frac{\partial U}{\partial \varphi }{\hat{e}}_{\varphi } \\ \nabla .V=\frac{1}{(u^2+v^2)uv}\frac{\partial }{\partial u}\left(\sqrt{u^2+v^2}uv{V}_1\right)+\frac{1}{(u^2+v^2)uv}\frac{\partial }{\partial u}\sqrt{u^2+v^{2}uv{V}_2)+\frac{1}{uv}\frac{\partial V_3}{\partial \varphi }} \\ {\nabla }^{2}U=\frac{1}{(u^2+v^2)uv}\frac{\partial }{\partial u}\left(uv\frac{\partial U}{\partial u}\right)+\frac{1}{\left(u^2+v^2\right)uv}\frac{\partial }{\partial v}\left(uv\frac{\partial U}{\partial u}\right)+\frac{1}{u^2{v}^2}\frac{{\partial }^{2}U}{\partial \varphi } \\ \nabla \times V=\frac{1}{uv\sqrt{u^2+v^2}}\left\{\frac{\partial }{\partial v}\left(uv{V}_3\right)-\frac{\partial }{\partial \varphi }\left(\sqrt{u^2+v^2{V}_2}\right)\right\}{\hat{e}}_u+\frac{1}{uv\sqrt{u^2+v^2}}\left\{\frac{\partial }{\partial \varphi }\sqrt{u^2+v^2}{V}_1)-\frac{\partial }{\partial u}\left(uv{V}_3\right)\right\}{\hat{e}}_v \\ +\frac{1}{u^2+v^2}\left\{\frac{\partial }{\partial u}\left(\sqrt{u^2+v^2}{V}_2\right)-\frac{\partial }{\partial v}\left(\sqrt{u^2+v^2}{V}_1\right)\right\}{\hat{e}}_{\varphi } \end{gathered}$$