3- (a) Let $$ \mathbf{F}=f(r) \mathbf{e}_{ heta} $$ be a vector field written in cylindrical coordinates. Draw $$ \mathbf{F} $$ on the $$ z=0 $$ plane for $$ f(r)=r^{2} $$. Find the curl of $$ \mathbf{F} $$ for generic $$ f(r) $$. When does the curl vanish? Does it match your intuition about curl? (b) Repeat part (a) for $$ \mathbf{F}=f(r) \mathbf{e}_{r} $$. 4- The vector field $$ \mathbf{F}=\hat{\mathbf{r}}=\mathbf{e}_{1} $$ (in polar coordinates) has constant magnitude, but not constant direction. So its derivative is non-zero. What are $$ \boldsymbol{
abla} \cdot \mathbf{F} $$ and $$ \boldsymbol{
abla} imes \mathbf{F} $$ ? Repeat for $$ \mathbf{F}=\hat{\boldsymbol{ heta}}=\mathbf{e}_{2} $$.

Question

3- (a) Let $$ \mathbf{F}=f(r) \mathbf{e}_{	heta} $$ be a vector field written in cylindrical coordinates. Draw $$ \mathbf{F} $$ on the $$ z=0 $$ plane for $$ f(r)=r^{2} $$. Find the curl of $$ \mathbf{F} $$ for generic $$ f(r) $$. When does the curl vanish? Does it match your intuition about curl? (b) Repeat part (a) for $$ \mathbf{F}=f(r) \mathbf{e}_{r} $$. 4- The vector field $$ \mathbf{F}=\hat{\mathbf{r}}=\mathbf{e}_{1} $$ (in polar coordinates) has constant magnitude, but not constant direction. So its derivative is non-zero. What are $$ \boldsymbol{
abla} \cdot \mathbf{F} $$ and $$ \boldsymbol{
abla} 	imes \mathbf{F} $$ ? Repeat for $$ \mathbf{F}=\hat{\boldsymbol{	heta}}=\mathbf{e}_{2} $$.

Maxwell Robbins · Accepted Answer

For part (a), the curl of $$ \mathbf{F}=f(r) \mathbf{e}_{	heta} $$ is $$ (f'(r) + \frac{f(r)}{r}) \mathbf{e}_{z} $$, which vanishes when $$ f'(r) + \frac{f(r)}{r} = 0 $$. For $$ \mathbf{F}=f(r) \mathbf{e}_{r} $$, the curl is always zero. For part (b), the divergence of $$ \mathbf{F}=\hat{\mathbf{r}}=\mathbf{e}_{1} $$ is zero, and the curl is also zero.

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