# What exactly is the curly d?

Our lecturer hastily referred to the boundary of a set M as ∂M, then he dropped it. It sounded very interesting, but he said it was outside the scope of the course. We have also been told that the curly d:s do not allow manipulation in the same way as regular differentials. But given something like ∂z/∂x = 2x + y, you're allowed to move the ∂x to the right side and integrate. So what gives? What exactly does the curly d operator do to a function?