In the generalized form of stokes theorem it states that the integral of the $k+1$ differential form of an operator over a compact manifold $M$ is equivalent to the integral of the $k$ differential form of that same operator over something referred to as $\partial M$
What is $\partial M$ supposed to be and how do you derive it from $M$?
Let $M$ be an n-manifold with boundary. Then $\partial M$ is the boundary of the manifold. The interior of $M$ is the set of points in $M$ such that they have a neighborhood which is homeomorphic to an open subset in $\mathbb{R}^n$. $M - IntM = \partial M$
If $M$ is a manifold without boundary, then $\partial M = \emptyset$
For example, consider the unit ball $B$ in $\mathbb{R}^3$. For any point (strictly) inside the ball, we can find an open subset of $B$ which contains that point and is an open subset of $\mathbb{R}^3$. For any point on the surface, we cannot. Hence, $\partial B=S^2$, which is a two manifold.