# What is the flaw in my thinking for the graph of this function?

Consider the map

$$f: \mathbb R^2 \to \mathbb R, (x,y) \mapsto x^3 + y^3 + xy$$

This defines a surface in $\mathbb R^3$. Let's consider some level set $f(x,y) = c$:

(see here page 67)

I think of these pictures as viewed from above looking down on the x-y-plane. Using these 3 pictures to imagine what the surface should look like lead me to believe that the surface should look like this:

(apologies, this is the best I could do with the online drawing tool)

To verify this I then used an online plotter which yielded this:

Now my question is:

How is it possible that the level lines of the last graph yield the level lines in the first three pictures? I do not see how this is possible.

(the range of the plot I used was $-1 \le x,y \le 1$ and $-1.3 \le z \le 3$, varying the range does not seem to change the graph)

• I curves in the last picture are intersections of the surface with planes $\{x = x_0\}$ and $\{y = y_0\}$, or if you like, the coordinate curves of the graph parameterization of the surface. In particular, they are not level curves. – Travis Jun 11 '15 at 4:46

In general it's helpful to remember how changes in the topology of the level curves correspond to local features of the function: a loop appearing/disappearing corresponds to a local extremum, while a transition across a self-intersection like you see in this example at $c=0$ corresponds to a saddle point.