Verification of Gauss' Divergence Theorem visualisation

Question

I'm having trouble visualising what the information provides specifically this part: Φ : [0, 1] × [0, 2π] → R^3

Does this mean the hemisphere has height of 1 in the z-axis and a radius of 2pi in the xy plane?

Answer 1

This part $\phi:[0,1]\times[0,2\pi]\to\Bbb R^3$ only means that $\phi$ is a function that takes two arguments, the first from $[0,1]$ and the second from $[0, 2\pi]$ and assigns them a vector in $\Bbb R^3$.

Together with the next line where the assignment is specified, $\phi$ describes the upper half of the unit sphere, i.e. the part where the third coordinate $z\ge0$.
Specifically, $(r\cos\theta, \, r\sin\theta)$ is a point in the unit disk in the plane, of distance $r$ from the origin, and we take the point above it in the hemisphere by calculating $z=+\sqrt{1 - r^2}$ by the Pythagorean theorem.