I know that a $C^2$ surface means the 2nd partial derivatives exist and are smooth, but I'm a bit unfamiliar with math notation/colloquia. I want to make sure I understand correctly.
Say we have an equation $F(x_i, x_j, ...) = 0$ for coordinates $x_i, x_j, ...$
I think a good example would be a spherical polar characteristic equation for a topological sphere, so some equation $r(\theta, \phi) = 0$. Since r here describes a sphere, we satisfy being closed, since there's no boundary or "holes" anywhere (which makes it compact).
To ensure r is $C^2$, it must be true that $\frac{\partial^2 r}{\partial \theta^2},\frac{\partial^2 r}{\partial \phi^2}, \text{and} \ \frac{\partial^2 r}{\partial \theta \partial \phi}$ exist and are continuous right? If so, then this means that if higher order derivatives exist and are smooth given that r describes a compact, closed surface, this would mean that r describes a Ck surface, where k is the highest order derivative that exists that's smooth. Now, my question is if a surface is $C^k$, does this automatically mean that it's also $C^{k-1}, C^{k-2}, C^{k-3}, ..., C^1$ ? Or do mathematicians make surfaces confined to only being of a particular smoothness class $C^k$?
What of the following example: $r(\theta, \phi) = A + B sin(\theta) + C cos(\theta)$, for constants A, B, and C. Since there's no $\phi$-dependence, doesn't this mean it wouldn't be $C^2$?