Understanding Stokes' theorem and the fundamental theorem of calculus I don't know a lot about differentials and boundaries, so it may be out of my grasp, but is there perhaps a simple way of understanding the Stokes' theorem for the FTC?

$\displaystyle\int_a^b f(x)\,\mathrm{d}x =
\int_{[a,b]} f(x)\,\mathrm{d}x =
\int_{[a,b]} \mathrm{d}F(x) =
\int_{\partial[a,b]} F(x)
\left\{\begin{array}{l}\displaystyle
\neq \int_{\{a,b\}}F(x) = F(a)+F(b)
\\\displaystyle \overset{?}{=} \int_{(*)}F(x) = -F(a)+F(b)
\end{array}\right.\\$

So I don't really understand why $\partial[a,b] \neq \{a,b\}$, and what the blank $(*)$ is supposed to be. So I guess the question is, what are the prerequisites to understand that part?

In the general Stokes' theorem $∫_M dω = ∫_{∂M} ω$, $M$ is an oriented manifold with boundary, and $∂M$ is the boundary with a particular induced orientation.  In the case of $M = [a,b]$ the usual orientation is "going left to right", and $∂M$ has underlying set $\{ a, b \}$ but the orientation comes out to be +1 at $b$ and -1 at $a$.  Taking that orientation into account gives $∫_{∂M} F = -F(a) + F(b)$.
