I am having some troubles with p-forms, since the definition isn't that easy. For example if $n = (n_{1},n_{2},n_{3})$ is the unit exterior norm to a oriented 2-manifold $N \subset \Bbb{R}^3$, we call $dA = n_{1}dx_{2} \wedge dx_{3} + n_{2}dx_{3} \wedge dx_{1} + n_{3}dx_{1} \wedge dx_{2}$. If $F = (f_{1},f_{2},f_{3})$ is a vector field $C^1$, then with $g= f_{1}dx_{2} \wedge dx_{3} + f_{2}dx_{3} \wedge dx_{1} + f_{3}dx_{1} \wedge dx_{2}$, $$ \left.g\right|_N = (F \cdot n)dA. $$
And I don't know how to proceed, since the definitions is so confuse. Later, this is been used to prove that, at the same conditions, if $N$ is compact and $\partial N$ being $C^1$, $$ \iiint_{N}(\nabla F )dxdydz = \iint_{\partial N}F\cdot n \ dA $$
I know how to prove this with "classic" analysis, but I need to prove with the above result. Thanks in advance to everyone. Any doubts, please get in touch!