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 $f(x,y) = c$:

enter image description here

(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)

enter image description here

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

enter image description here

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)

  • 1
    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
up vote 51 down vote accepted

Your sketch doesn't even look like the graph of a smooth function - remember that it should just be a deformed plane with nowhere vertical tangents. Here's an animation I whipped up that might help your intuition for this example:

level curves animation

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.

  • 1
    Thank you so much! This really helped me! I didn't see the bump on the graph that I plotted. – a student Jun 11 '15 at 8:45
  • 6
    This animation is excellent, how did you make it? – m_t_ Jun 11 '15 at 9:24
  • 9
    @mt_: Mathematica. Here's the code for an interactive version: – Anthony Carapetis Jun 11 '15 at 10:21
  • 1
    @astudent High values of x and y hide the bump in the scale. If you plot it to xmax = ymax = 0.5, it is clearly visible. – Davidmh Jun 11 '15 at 13:30
  • 1
    @Mehrdad – Anthony Carapetis Jun 12 '15 at 3:19

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