Question

Use equations (9.2), (9.8), and (9.11) to evaluate the following expressions. In spherical coordinates $\mathbf{\nabla }\mathbf{\times }\mathbf{\left(}\mathit{r}{\mathit{e}}_{\mathbf{\theta }}\mathbf{\right)}\mathbf{,}\mathbf{\nabla }\mathbf{\left(}\mathit{r}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\mathbf{\right)}\mathbf{,}\mathbf{\nabla }\mathbf{.}\mathit{r}$.

Short answer

The required values are mentioned below.

$$\begin{gathered} \nabla \times \left(r{\hat{e}}_{\theta }\right)=2r{\hat{e}}_{\varphi } \\ \nabla \left(r\cos \theta \right)=\cos \theta {\hat{e}}_r-\sin \theta {\hat{e}}_{\varphi } \\ \nabla ·r=3 \end{gathered}$$

Step-by-step solution

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.

Determining the value of ∇×(reθ),∇(rcosθ),∇.r 

Findthe value of $\nabla \times \left(r{\hat{e}}_{\theta }\right)$.

role="math" localid="1659261996986" $$\begin{gathered} \nabla \times \left(r{\hat{e}}_{\theta }\right)=-\frac{1}{r^2\sin \theta \partial \varphi }\left(r^2\right){\hat{e}}_r+\frac{1}{r}\frac{\partial }{\partial r}{r}^2{\hat{e}}_{\varphi } \\ \nabla \times \left(r{\hat{e}}_{\theta }\right)=2r{\hat{e}}_{\varphi } \end{gathered}$$

Findthe value of .

role="math" localid="1659262028625" $$\begin{gathered} \nabla \left(r\cos \theta \right)=\frac{\partial }{\partial r}\left(r\cos \theta \right){\hat{e}}_r+\frac{1}{r}\frac{\partial }{\partial \theta }\left(r\cos \theta \right){\hat{e}}_r \\ \nabla \left(r\cos \theta \right)=\cos \theta {\hat{e}}_r-\sin \theta {\hat{e}}_r \end{gathered}$$

Findthe value of .

$\nabla \cdot r=\frac{1}{r^{2}sin\theta }\frac{\partial }{\partial r}\left(r^{2}sin\theta r\right)=3$

The required values are mentioned below.

role="math" localid="1659262073399" $$\begin{gathered} \nabla \times \left(r{\hat{e}}_{\theta }\right)=2r{\hat{e}}_{\varphi } \\ \nabla \left(r\cos \theta \right)=\cos \theta {\hat{e}}_r-\sin \theta {\hat{e}}_r \\ \nabla \cdot r=\frac{1}{r^{2}sin\theta }\frac{\partial }{\partial r}\left(r^{2}sin\theta r\right)=3 \end{gathered}$$