Find all level sets of the function:
f(x,y,z) = x^2 + y^2 - z^2
Help please!
If $\displaystyle \lambda\in\mathbb{R}$ is a fixed number, the level sets of $\displaystyle f:\mathbb{R}^3 \to \mathbb{R},\;\;f(x,y,z)=x^2+y^2-z^2$ at level $\displaystyle \lambda$ consists of all points in $\displaystyle \mathbb{R}^3$ which are solution of the equation $\displaystyle f(x,y,z)=\lambda$. If $\displaystyle \lambda=0$ the level surface is a cone centered at the origin opening along the z axis. If $\displaystyle \lambda>0$, then the level surface is a hyperboloid of one sheet. Finally, if $\displaystyle \lambda < 0$, the level surface is a hyperboloid of two sheets.