Calc 3, finding level sets of a function

**Drumiester** · February 17th 2009, 09:22 AM

Find all level sets of the function:

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

Help please!

**Pedro²** · February 18th 2009, 09:29 AM

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.

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