What prompts me to ask this is i am trying to prove the gradient is perpendicular to the level set. I am using w = x^2 + y^2 . When I graph this I get a cup shaped figure in three dimensions and when I graph the level sets they are circles since w is a constant. Is this true so far ??
So now if we follow a path through the level sets is the path on ONE circle or a set of circles? That is where my confusion lies. Then I believe the rest of the proof shows the dot product of the gradient with the velocity vector is o!
But is the gradient perpendicular to each level curve were w = some constant? If so then there would be a gradient for each point of the path through the level sets IF the level set contains more than one level curve? A little nomenclature confusion on my part. Perhaps it is a set of a bunch of circles and the gradient is perpendicular to each constant value of w? If you could use my example it would be helpful.
A figure worth a thousand words.
Considering the change of variables
$$ x = r \cos\theta\\ y = r \sin\theta\\ z = r^2 $$
The level lines $\phi(r_0,\theta) = \{r_0\cos\theta,r_0\sin\theta,r_0^2\}$ and the lines $\psi(r,\theta_0)=\{r\cos\theta_0,r\sin\theta_0, r^2\}$ at point $\left(r_0,\theta_0\right)$ are orthogonal because
$$ < \frac{d\phi}{d\theta}, \frac{d\psi}{dr} > = 0 $$