I want to calculate:
$$\iiint_V div (\overrightarrow F \cdot \space dV) $$
with $\overrightarrow F=x^3\hat i+ y^3 \hat j+z^3\hat k$ and Surface of sphere given as $x^2+y^2+z^2=r^2$
So, first I calculate $div.\overrightarrow F=3(x^2+y^2+z^2)$
Then using spherical co-ordinates, we know:
$x= rsin\theta \cdot cos\phi$
$y= rsin\theta \cdot sin\phi$
Now what I don't understand is the following given according the textbook:
$dxdydz=r^2sin\theta \cdot dr \cdot d\theta \cdot d\phi$
Where does $dxdydz$ come from?
and How are these limits are calculated?
- For $r$, the limit is $0$ to a
- For $\theta$ the limit is $0$ to $\pi$
- For $\phi$ the limit is $0$ to $2\pi$
Using these limits, the result can be calculated using integration.