91影视

The Bipolar.

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.

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

Find $鈭�U$in Parabolic coordinates.

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

Find the divergence of V.

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

Find the Laplacian of U.

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

Find curl of V.

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

Hence, the required values are mentioned below.

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