Why the level surface is constant?

Question

Why level surfaces are of the form (,)= or (,,)=?

Are there any good visual representations why it is that?

And why the velocity vector stays constant as well?

Answer 1

It is just the definition of the level curve or level surface.

For example if $$ f(x,y)=x^2+y^2$$ The level curves are the sets defined by $$x^2+y^2=C$$ where $C$ is a constant.

As you notice for positive values of $C$ we get circles and for negative values of $C$ the set is empty while for $C=0$ we get a single point.

These are the curves that you get by cutting the paraboloid $$z=x^2+y^2$$ with the horizontal planes $$z=C$$

Answer 2

If you have a function $f:\mathbb{R}^n \to \mathbb{R}$, and you have some value $c \in \mathbb{R}$, then the set of points

$$ \{ x \in \mathbb{R}^n \mid f(x) = c\}$$

is the level surface for the value c.

Thus, if you say "the level surface is constant," there must be some value $c$ assumed. But, given such a value $c$ it means that the value of the function $f$ is always $c$ for points on the level surface.