- #1

- 2

- 0

I'm trying to determine ##\vec{\bigtriangledown }\times \vec{a}## , where ##\vec{a}=\vec{\omega }\times \vec{r}##, being ##\vec{\omega }## a constant vector, and ##\vec{r}## the position vector, using this definition:

##\vec{curl}(\vec{a})=\lim_{V\rightarrow 0}\frac{1}{V}\oint _{S}\vec{n}\times \vec{F}da##

where ##\vec{n}## is the unit vector perpendicular to the surface.

First of all, I know the answer is ##2\vec{\omega }##. That's what I'm trying to get but I can't.

I've tried to do it with a sphere. So, ##V=\frac{4}{3}\pi r^3## , ##\vec{n}=\vec{u_{r}}## and ##\vec{F}=\vec{a}##.

##\vec{\omega }## is the angular velocity, so it's always perpendicular to ##\vec{r}##, then ##\vec{a}=\vec{\omega }\times \vec{r}=\omega r \vec{u_{\varphi}}##, because I have chosen ##\vec{u_{\omega}}=−\vec{u_{\theta}}##. [##\theta## is the zenith angle, and ##\varphi## is the azimuth angle.]

##da## would be ##r^2 sin\theta d\varphi d\theta## .

So now I have:

##\vec{curl}(\vec{a})=\lim_{r\rightarrow 0}\frac{1}{\frac{4}{3}\pi r^3}\int_{0}^{\pi}\int_{0}^{2\pi} \omega r^3 sin\theta (\vec{u_{r}}\times \vec{u_\varphi)} d\varphi d\theta##

where ##\vec{u_{r}} \times \vec{u_\varphi}=\vec{u_\omega}##.

The final result is ##3\vec{u_\omega}##. What am I doing wrong?

Thanks in advance and Merry Christmas