"Guessing" the Shape of a Function based on its Gradient-Vector Field

Question

I was looking at this link here: https://en.wikipedia.org/wiki/Gradient

This page contains visualizations of two different functions along with the Gradient-Vector Fields:

• The second picture (picture on the right hand side, orange color) seems to be straightforward: We can clearly see that this function has a global minimum that is located somewhere in the middle of the plot. If I understand correctly, the "blue arrows" are showing the Gradient-Vector field : For instance, take any "blue arrow" and project it orthogonally upwards on to the function - if you were to "reverse the direction" of a given "blue arrow" (i.e. "negative direction"), this "reversed direction" would now point in the direction towards the minimum of the function.

• I am now looking at the first picture (picture on the left hand side, red/blue/green/yellow colors). Just based on this picture, I think the corresponding function two regions where the derivative is 0 - and the (negative direction of these) arrows are pointing towards these regions (I think these are Saddle Points?).

However, I am not sure about this. I tried to make a 3D plot of this function [ x * 2.718^(-x^2 + y^2) ]:

Here is a 3D plot of the same function but from a different perspective:

I think my assertion is correct? Is what I have described the actual relationship between the surface of a function and its Vector-Gradient field?

Thank you!

Answer 1

This is a "soft-answer".

Looking at the gradient field, you should be able to have a reasonable guess on the shape of the "level curves", that is, the subsets $f^{-1}(\{c\})$ (where $f$ is your function, and $c$ is arbitrary).

Indeed:

• the level subsets are everywhere orthogonal to the gradient (in order to see this, compute $(f \circ \gamma)'(0)$ where $\gamma : ]-\epsilon,\epsilon[ \rightarrow f^{-1}(\{c\})$ is any parametrized curve drawn on the level subset);
• therefore, around a point where the gradient doesn't vanish, the level subsets look like hyperplanes;
• at a point $x$ where the gradient vanishes, things get a little more complicated: if all the arrows point towards $x$, or if all the arrows seem to point away from $x$, the level subsets are locally topological spheres; if some arrows point towards $x$ and some point away from $x$, it may be a saddle point, etc.

Once you have a fairly good idea of what the level subsets look like, you can easily guess the shape of the graph (hikers do this all the time when looking at their maps)!

Answer 2

You made a fatal mistake in your 3d picture, omitting the parenthesis around $-(x^2+y^2)$ in reality your left picture has a "mountain" at the right side and the corresponding valley at the left side. You can see this in the picture itself also. on the right side all the arrows go up towards the midpoint, on the left side they go down