Given a scalar field $f({\bf r})= \dfrac{\sqrt{4-x^2-y^2-z^2}}{(x^2+z^2)}$ calculate it's domain.
I have no idea where to start. I have tried to calculate $\nabla f$.
I believe the domain refers to the numerator. Thus $x^2+y^2+z^2 $ has to be less than or equal to $4$.
Is this then a ball of radius 2?
The gradient has nothing to do with this... You just need to see what are the points for which the expression is well defined:
$$ D = \{(x,y,z)\in\mathbb{R}^3: 4-x^2-y^2-z^2 \ge 0 \wedge x^2+z^2 \ne 0\} $$