# Calc 3, finding level sets of a function

• February 17th 2009, 09:22 AM
Drumiester
Calc 3, finding level sets of a function
Find all level sets of the function:

f(x,y,z) = x^2 + y^2 - z^2

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