Let $K= \{ (x,y,z) \in \mathbb{R}^3 : \sqrt{x^2+y^2} \leq z,\,\, x^2 + y^2 + z^2 = 1 \}$.
I need a parametrisation of $K$ in order to calculate the flux of some function through $K$. I'm not sure how to do it but maybe spherical coordinates will help. (they certainly help in visualising)
Switching to spherical coordinates, let's put $$ \qquad x=r \cos \theta \sin \varphi, \quad y=r \cos \theta \sin \varphi, \quad z = r \cos \varphi$$
where $r \in (0,\infty), \, \theta \in (0,2\pi), \, \varphi \in (0,\pi).$
Then $$(x,y,z)\in K \iff r=1, \quad\theta \in (0,2 \pi), \quad \varphi \in (0,\pi/4).$$
Please help me find a parametrisation of $K$.