CLP-3 Multivariable Calculus

For $Q(x,y,z)$ to be a polynomial of degree two, it must be of the form

for some constants $A\text{,}$ $B\text{,}$ $\cdots \text{,}$ $J\text{.}$ Each constant $z$ cross section of a quadric surface has an equation of the form

If $A=B=D=0$ but $g$ and $h$ are not both zero, this is a straight line. If $A\text{,}$ $B\text{,}$ and $D$ are not all zero, then by rotating and translating our coordinate system the equation of the cross section can be brought into one of the forms

²

This statement can be justified using a linear algebra eigenvalue/eigenvector analysis. It is beyond what we can cover here, but is not too difficult for a standard linear algebra course.

There are similar statements for the constant $x$ cross sections and the constant $y$ cross sections. Hence quadratic surfaces are built by stacking these three types of curves.

We have already seen a number of quadric surfaces in the last couple of sections.