As I was studying the smoothness of surfaces I have come across two techniques for showing that a surface is smooth.
If the surface $S$ is the set of points which satisfy $f(x,y,z) = a $, where a $\in \mathbb{R}$ then, from my present understanding, one could:
- Show that $\nabla f \neq 0 $ for all points on $S$
- Construct multiple injective and continuously differentiable coordinate charts which together map to the whole of $S$
My questions related to that are:
- Is what I've described accurate?
- It seems to me that the first way of doing things is generally more straight forward. Is there limitations to that way of doing things? Any particular scenarios where it would be easier to work with coordinate charts?
For reference, some of the treads that I've been looking at already are:
Prove that this is a smooth surface (using the $\nabla \neq 0 $ idea)
Show that it is a smooth surface (using the coordinate charts idea)
Cheers!
-Lemon
If partial derivatives
$$ \frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}$$
are continuous, the function is smooth.
In Monge form $Z=f(X,Y) $ a vector cross product normal to surface
$$ Z_x X Z_y $$
should be continuous and non-zero.