Definition of $C^k$ boundary

Question

Can someone give me a resonable definition of $C^k$ boundary, e.g., to define and after give a brief explain about the definition.

I need this 'cause I'm not understanding what the Evan's book said.

Thanks!

Answer 1

In $\mathbb{R^n}$, the boundary of a subset is $C^k$ if it's locally the graph of a $C^k$ function in some direction. So a circle has $C^{\infty}$ boundary because at all points in the positive upper half plane, it's the graph of the function $y=\sqrt{1-x^2}$, which has infinitely many derivatives at every point but the two end points. But those end points are in the graph of $x=\sqrt{1-y^2}$ or $x=-\sqrt{1-y^2}$, which also has infinitely many derivatives.

Answer 2

One possible definition is this: Locally, i.e., in a neighbourhood of any point on the boundary, the boundary is the graph of a $C^k$ function – possibly after the coordinate system has been rotated.

Equivalently, the boundary is locally the image of a $C^k$ embedding of $\mathbb{R}^{n-1}$ into $\mathbb{R}^n$.