Give the spherical coordinates representation of the following solids:
(i) Inside the sphere $ \ x^2+y^2+z^2=4 \ $ and outside the cylinder $ \ x^2+y^2=1 \ $
(ii) Inside the sphere $ \ x^2+y^2+z^2=4 \ $ and above the plane $ \ z=3 \ $
Answer;
(i)
$ 0 \leq \rho \leq 2 \\ 0 \leq \phi \leq \frac{\pi}{4} \\ 0 \leq \theta \leq 2 \pi \ $
Am I right?
Help me out
Write the equations for the coordinates
\begin{eqnarray} x &=& \rho \sin \theta \cos\phi \\ y &=& \rho \sin \theta \sin\phi \\ z &=& \rho \cos \theta \end{eqnarray}
So that the cylindrical surface can be written as
\begin{eqnarray} x^2+y^2 &=& \rho^2\sin^2\theta\cos^2\phi + \rho^2\sin^2\theta\sin^2\phi \\ &=& \rho^2\sin^2\theta(\cos^2\phi + \sin^2\phi) \\ &=&\rho^2\sin^2\theta \geq 1 \end{eqnarray}
Or equivalentely
$$ \rho |\sin\theta| > 1 $$