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GIven an equation such as $ax^2+by^2+cz^2+dxy+exz+fyz=g$ where $a,b,c,d,e,f,g\in \mathbb R$, How can we tell if the surface described is a bounded one without explicitly plotting a graph?

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The surface is bounded iff the eigenvalues of the matrix $\begin{pmatrix} a & d/2 & e/2 \\ d/2 & b & f/2 \\ e/2 & f/2 & c \\ \end{pmatrix}$ have the same sign.

One way to show this is to compute the extrema of function $$(x,y,z)\mapsto x^2+y^2+z^2$$ restricted to the surface. You can do this using Lagrange multipliers very easily.

Alternatively, it is not difficult to see that there is a linear change of variables which takes your function to one of the form $$\alpha X^2+\beta Y^2+\gamma Z^2$$ for which the question is very easy.

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+1 Clever method, I think I'll have to use it sometime. –  Alex Becker May 22 '12 at 23:24

With $$ g > 0: $$ It is bounded if and only if $$ a > 0 $$ and $$ 4 ab - d^2 > 0 $$ and $$ 4 abc + def - a f^2 - b e^2 - c d^2 > 0. $$

This is called Sylvester's Criterion. See also TWEETY. And SYLVESTER.

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Haha! Nice associations...! –  Miserable May 23 '12 at 8:10

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