Question

Write out $\mathbf{\nabla }\mathit{U}\mathbf{,}\mathbf{\nabla }\mathbf{.}\mathit{V}\mathbf{,}{\mathbf{\nabla }}^{\mathbf{2}}\mathbf{.}\mathit{U}\mathbf{,}\mathbf{\nabla }\mathbf{\times }\mathit{V}$in spherical coordinates

Short answer

The values are mentioned below.

$\begin{array}{c}\nabla \cdot \mathbf{V}=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2{V}_1\right)+\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta V_2\right)+\frac{1}{r\sin \theta }\frac{\partial V_3}{\partial \varphi }\\ {\nabla }^{2}U=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial U}{\partial r}\right)+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial U}{\partial \theta }\right)+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^{2}U}{\partial {\varphi }^2}\\ \nabla \times \mathbf{V}=\frac{1}{r^2\sin \theta }\{\frac{\partial }{\partial \theta }\left(r\sin \theta V_3\right)-\frac{\partial }{\partial \varphi }\left(r{V}_2\right)\}{\widehat{e}}_r+\frac{1}{r\sin \theta }\{\frac{\partial }{\partial \varphi }\left(V_1\right)-\frac{\partial }{\partial r}\left(r\sin \theta V_3\right)\}{\widehat{e}}_{\theta }+\frac{1}{r}\{\frac{\partial }{\partial r}\left(r{V}_2\right)-\frac{\partial }{\partial \theta }\left(V_1\right)\}{\widehat{e}}_{\varphi }\end{array}$

Step-by-step solution

Given Information

Given are the spherical coordinates.

Definition of spherical coordinates

The coordinate system primarily utilized in three-dimensional systems is the spherical coordinates of the system. The spherical coordinate system is used to find the surface area in three-dimensional space.

Find the value

The scale factors are given below.

$\begin{array}{c}{h}_{\varphi }=r\sin \theta \\ h_r=1\\ h_{\theta }=r\end{array}$

Formula states that $\nabla U=\frac{\partial U}{\partial r}{\widehat{e}}_r+\frac{1}{r}\frac{\partial U}{\partial \theta }{\widehat{e}}_{\theta }+\frac{1}{r\sin \theta }\frac{\partial U}{\partial \varphi }{\widehat{e}}_{\varphi }$.

Find the divergence of V.

$\begin{array}{c}\nabla \cdot \mathbf{V}=\frac{1}{r^2\sin \theta }\{\frac{\partial }{\partial r}\left(r^2\sin \theta V_1\right)+\frac{\partial }{\partial \theta }\left(r\sin \theta V_2\right)+\frac{\partial }{\partial \varphi }\left(r{V}_3\right)\}\\ \nabla \cdot \mathbf{V}=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2{V}_1\right)+\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta V_2\right)+\frac{1}{r\sin \theta }\frac{\partial V_3}{\partial \varphi }\end{array}$

Calculate the Laplacian of U

$\begin{array}{c}{\nabla }^{2}U=\frac{1}{r^2\sin \theta }\{\frac{\partial }{\partial r}\left(r^2\sin \theta \frac{\partial U}{\partial r}\right)+\frac{\partial }{\partial \theta }\left(\frac{r\sin \theta }{r}\frac{\partial U}{\partial \theta }\right)+\frac{\partial }{\partial \varphi }\left(\frac{r}{r\sin \theta }\frac{\partial U}{\partial \varphi }\right)\}\\ {\nabla }^{2}U=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial U}{\partial r}\right)+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial U}{\partial \theta }\right)+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^{2}U}{\partial {\varphi }^2}\end{array}$

The curl of V is mentioned below.

$\nabla \times \mathbf{V}=\frac{1}{r^2\sin \theta }\{\frac{\partial }{\partial \theta }\left(r\sin \theta V_3\right)-\frac{\partial }{\partial \varphi }\left(r{V}_2\right)\}{\widehat{e}}_r+\frac{1}{r\sin \theta }\{\frac{\partial }{\partial \varphi }\left(V_1\right)-\frac{\partial }{\partial r}\left(r\sin \theta V_3\right)\}{\widehat{e}}_{\theta }+\frac{1}{r}\{\frac{\partial }{\partial r}\left(r{V}_2\right)-\frac{\partial }{\partial \theta }\left(V_1\right)\}{\widehat{e}}_{\varphi }$

The values are mentioned below.

role="math" localid="1664343316129" $\begin{array}{c}\nabla \cdot \mathbf{V}=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2{V}_1\right)+\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta V_2\right)+\frac{1}{r\sin \theta }\frac{\partial V_3}{\partial \varphi }\\ {\nabla }^{2}U=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial U}{\partial r}\right)+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial U}{\partial \theta }\right)+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^{2}U}{\partial {\varphi }^2}\\ \nabla \times \mathbf{V}=\frac{1}{r^2\sin \theta }\{\frac{\partial }{\partial \theta }\left(r\sin \theta V_3\right)-\frac{\partial }{\partial \varphi }\left(r{V}_2\right)\}{\widehat{e}}_r+\frac{1}{r\sin \theta }\{\frac{\partial }{\partial \varphi }\left(V_1\right)-\frac{\partial }{\partial r}\left(r\sin \theta V_3\right)\}{\widehat{e}}_{\theta }+\frac{1}{r}\{\frac{\partial }{\partial r}\left(r{V}_2\right)-\frac{\partial }{\partial \theta }\left(V_1\right)\}{\widehat{e}}_{\varphi }\end{array}$