- #1

Salmone

- 101

- 13

The vector field ##\vec{u}## is defined as follows: ##\vec{u}(\vec{r})=(\frac{a}{r})^3(\vec{\omega} \times \vec{r})## with ##a## being a constant, ##\eta## being a constant, ##\vec{r}## a position vector and ##\omega## the angular speed constant in modulus.

Doing calculations I've obtained ##\vec{\Pi}=-\frac{6\eta a^3}{r^4}\hat{r}(\vec{\omega} \times \vec{r})## with ##\hat{r}## being a unit vector.

**First**is this result right?

Then I want to compute a surface integral: I wanna compute ##\int_S\vec{\Pi} \cdot d\vec{S}## over a sphere of radius ##R>0## with ##d\vec{S}## being ##r^2sin(\theta)d\theta d\phi \hat{r}##.

##\hat{r}## point in the same direction of the radius of the sphere over which I'm integrating.

To do this I thought to compute the dot product as ##-\frac{6\eta a^3}{r^4}(\vec{\omega} \times \vec{r})\hat{r} \cdot \hat{r}r^2sin(\theta)d\theta d\phi=-\frac{6\eta a^3}{r^4}(\vec{\omega} \times \vec{r}) r^2sin(\theta)d\theta d\phi## then ##\vec{\omega} \times \vec{r}=\omega rsin(\theta)## and the integral becomes ##-\frac{6\eta a^3}{r}\omega\int_0^{2\pi}d\phi\int_0^{\pi}d\theta sin^2(\theta)## but this integral should be equal to zero.

**Where am I wrong?**