Differential forms, integration and the generalized Stokes theorem (part III)
The last ingredient we need for the generalized Stokes theorem is the exterior derivative of a differential form. Consider a $k$-form $\omega$ in $\mathbb{R}^n$. Its exterior derivative $d\omega$ will be a $(k+1)$-form, i.e. a function of a point $p$ and $k+1$ vectors $v_1, \ldots, v_{k+1}$, which we can think of as forming the parallelogram \[ P_p(v_1, \ldots, v_{k+1}) = \{p + t_1 v_1 + \cdots + t_{k+1} v_{k+1}\ |\ 0 \leq t_i \leq 1, \ i = 1, \ldots, k+1\} \] By this point it should not be particularly surprising to see the definition: \[ d\omega_p(v_1, \ldots, v_{k+1}) = \lim_{h \to 0} \frac{1}{h^{k+1}} \int_{\partial P_p(hv_1, \ldots, hv_{k+1})} \omega \] Observe that the boundary of the $(k+1)$-parallelogram $P_p(v_1, \ldots, v_{k+1})$ is a union of $k$-dimensional parallelograms, and so it makes sense to integrate the $k$-form $\omega$ over it. Furthermore note that, if $v_1, \ldots, v_{k+1}$ is an orthonormal set, then \[ h^{k+1} = \text{vol}(P_p(hv_1, \ldots, hv_{k+1})) \] ...