Classfying quadratic surface $z=x^2+2xy+y^2$.

Question

I'm trying to classify the following quadratic surface in $R^3$, but I'm not sure how to do it.

$$z = x^2 + 2xy + y^2$$

I can reflect that to the PDE classifying method but I still couldn't reach a legit solution for the following exercise.

Answer 1

$$\large z=x^2+2xy+y^2$$ This is a parabolic cylinder!

We can apply the transformation $x^\prime=x+y$ and $y^\prime =x-y$ to get it exactly in the form:

$$ z = {x^\prime}^2.$$

If you prefer $$\large x^2+xy+y^2$$ as the title seems to imply,

then with the transformation $$\begin{aligned} x&=x^\prime+y^\prime \\ y&=x^\prime - y^\prime \end{aligned}$$

we have

$$z= 3{x^\prime}^2 + {y^\prime}^2$$

an elliptic paraboloid.

http://mathworld.wolfram.com/QuadraticSurface.html

http://mathworld.wolfram.com/ParabolicCylinder.html

http://mathworld.wolfram.com/EllipticParaboloid.html