Question

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

Short answer

The required values are mentioned below.

$$\begin{gathered} \nabla ·{\hat{e}}_r=\frac{1}{r} \\ \nabla ·{\hat{e}}_{\theta }=0 \\ \nabla ·{\hat{e}}_r=0 \\ \nabla ·{\hat{e}}_{\theta }=\frac{1}{r}{\hat{e}}_{\varphi } \end{gathered}$$

Step-by-step solution

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 ∇.er,∇.eθ,∇×er,∇×eθ 

Find the value of $\nabla .e_r,\nabla .e_{\theta }.$.

role="math" localid="1659260592443" $$\begin{gathered} \nabla ·{\hat{e}}_r=\frac{1}{r}\frac{\partial }{\partial r}\left(r·1\right) \\ \nabla ·{\hat{e}}_r=\frac{1}{r} \\ \nabla ·{\hat{e}}_{\theta }=\frac{1}{r}\frac{\partial }{\partial r}\left(1\right) \\ \nabla ·{\hat{e}}_{\theta }=0 \end{gathered}$$

Findthe value of $\nabla \times e_r,\nabla \times e_{\theta }.$.

$$\begin{gathered} \nabla ·{\hat{e}}_r=\frac{\partial }{\partial r}\left(1\right){\hat{e}}_{\theta }-\frac{1}{r}\frac{\partial }{\partial r}\left(1\right){\hat{e}}_z \\ \nabla ·{\hat{e}}_r=0 \\ \nabla ·{\hat{e}}_{\theta }=\frac{1}{r\theta \sin }\frac{\partial }{\partial r}\left(r·1\right){\hat{e}}_r+\frac{1}{r}\frac{\partial }{\partial r}\left(r·1\right){\hat{e}}_{\varphi } \\ \nabla ·{\hat{e}}_{\theta }=\frac{1}{r}{\hat{e}}_{\varphi } \end{gathered}$$

The required values are mentioned below.

$$\begin{gathered} \nabla ·{\hat{e}}_r=\frac{1}{r} \\ \nabla ·{\hat{e}}_{\theta }=0 \\ \nabla ·{\hat{e}}_r=0 \\ \nabla ·{\hat{e}}_{\theta }=\frac{1}{r}{\hat{e}}_{\varphi } \end{gathered}$$