# How do you sketch f(x,y)=arcsin(x^2+y^2-2)?

Feb 16, 2015

Hello !

Let S be the surface of equation $z = \setminus \textrm{\arcsin} \left({x}^{2} + {y}^{2} - 2\right)$.

S is a surface of revolution because $z = F \left(r\right)$ where $r = \sqrt{{x}^{2} + {y}^{2}}$. Here, $F \left(r\right) = \setminus \textrm{\arcsin} \left({r}^{2} - 2\right)$.

First, you study the curve of equation $z = \setminus \textrm{\arcsin} \left({x}^{2} - 2\right)$ : you get

Second, you rotate this curve around (0z) axis and you get the surface S :

Remark that $f$ exists only on the domain defined by $1 \setminus \le q {x}^{2} + {y}^{2} \setminus \le q 3$ : it's a disk of radius $\sqrt{3}$ with an circular hole of radius 1.