Question

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

Short answer

The required values are mentioned below.

$$\begin{gathered} \nabla U=\frac{\cos hu+\cos v}{a}\frac{\partial U}{\partial u}{\hat{e}}_u+\frac{\cos hu+\cos v}{a}\frac{\partial U}{\partial v}{\hat{e}}_v \\ \nabla .V=\frac{{(\cos hu+\cos v)}^2}{a^2}\left\{\frac{\partial }{\partial u}\left(\frac{a}{\cos hu+\cos v}{V}_1\right)+\frac{\partial }{\partial v}\left(\frac{a}{\cos hu+\cos v}{V}_2\right)\right\} \\ {\nabla }^{2}U=\frac{{(\cos hu+\cos v)}^2}{a^2}\left\{\frac{{\partial }^{2}U}{\partial u^2}+\frac{{\partial }^{2}U}{\partial v^2}\right\} \\ \nabla \times V=\frac{{(\cos hu+\cos v)}^2}{a^2}\left\{\frac{\partial }{\partial u}\left(\frac{a}{\cos hu+\cos v}{V}_2\right)-\frac{\partial }{\partial v}\left(\frac{a}{\cos hu+\cos v}{V}_1\right)\right\}{\hat{e}}_z \end{gathered}$$

Step-by-step solution

Given Information

The Bipolar.

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 ∇×V

The scale factors are given below.

$$\begin{gathered} {\mathrm{h}}_{\mathrm{u}}=\frac{\mathrm{a}}{\cosh \mathrm{u}+\cos \mathrm{v}} \\ {\mathrm{h}}_{\mathrm{v}}=\frac{\mathrm{a}}{\cosh \mathrm{u}+\cos \mathrm{v}} \end{gathered}$$

Find $\nabla U$in Parabolic coordinates.

$$\begin{gathered} \nabla \mathrm{U}=\frac{1}{{\mathrm{h}}_{\mathrm{u}}}\frac{\partial \mathrm{U}}{\partial \mathrm{u}}{\hat{\mathrm{e}}}_{\mathrm{u}}+\frac{1}{{\mathrm{h}}_{\mathrm{v}}}\frac{\partial \mathrm{U}}{\partial \mathrm{v}}{\hat{\mathrm{e}}}_{\mathrm{v}} \\ \nabla \mathrm{U}=\frac{\cosh +\cos \mathrm{v}}{\mathrm{a}}\frac{\partial \mathrm{U}}{\partial \mathrm{u}}{\hat{\mathrm{e}}}_{\mathrm{u}}+\frac{\cosh +\cos \mathrm{v}}{\mathrm{a}}\frac{\partial \mathrm{U}}{\partial \mathrm{v}}{\hat{\mathrm{e}}}_{\mathrm{v}} \end{gathered}$$

Find the divergence of V.

role="math" localid="1659336684243" $\nabla .V=\frac{{(\cos hu+\cos v)}^2}{a^2}\left\{\frac{\partial }{\partial u}\left(\frac{a}{\cos hu+\cos v}{V}_1\right)+\frac{\partial }{\partial v}\left(\frac{a}{\cos hu+\cos v}{V}_2\right)\right\}$

Find the Laplacian of U.

${\nabla }^{2}U=\frac{{(\cos h+\cos v)}^2}{a^2}\left\{\frac{{\partial }^{2}U}{\partial u^2}+\frac{{\partial }^{2}U}{\partial v^2}\right\}$

Find curl of V.

$\nabla \times V=\frac{{(\cos hu+\cos v)}^2}{a^2}\left\{\frac{\partial }{\partial u}\left(\frac{a}{\cos hu+\cos v}{V}_2\right)+\frac{\partial }{\partial v}\left(\frac{a}{\cos hu+\cos v}{V}_1\right)\right\}{\hat{e}}_z$

Hence, the required values are mentioned below.

$$\begin{gathered} \nabla U=\frac{\cos hu+\cos v}{a}\frac{\partial U}{\partial u}{\hat{e}}_u+\frac{\cos hu+\cos v}{a}\frac{\partial U}{\partial v}{\hat{e}}_v \\ \nabla .V=\frac{{(\cos hu+\cos v)}^2}{a^2}\left\{\frac{\partial }{\partial u}\left(\frac{a}{\cos hu+\cos v}{V}_1\right)+\frac{\partial }{\partial v}\left(\frac{a}{\cos hu+\cos v}{V}_2\right)\right\} \\ {\nabla }^{2}U=\frac{{(\cos hu+\cos v)}^2}{a^2}\left\{\frac{{\partial }^{2}U}{\partial u^2}+\frac{{\partial }^{2}U}{\partial v^2}\right\} \\ \nabla \times V=\frac{{(\cos hu+\cos v)}^2}{a^2}\left\{\frac{\partial }{\partial u}\left(\frac{a}{\cos hu+\cos v}{V}_2\right)-\frac{\partial }{\partial v}\left(\frac{a}{\cos hu+\cos v}{V}_1\right)\right\}{\hat{e}}_z \end{gathered}$$