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?
Boundedness of Surfaces in $\mathbb R^3$
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real-analysis
vector-spaces
surfaces
2 Answers
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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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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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0Haha! Nice associations...! – 2012-05-23