I just had a general question about differential forms.
Background. Let $f = f_k$ and define $f^{k-1}$ on $I^{k-1}$ by $f_{k-1}(x_1, \dots, x_{k-1})= \int_{a_k}^{b_k} f_{k}(x_1, \dots, x_{k-1}, x_k) \ dx_{k}$
Note that $f$ is real continuous on $I^{k}$.
Question. What is the intuitive meaning of this? Rudin says that after $k$ steps we arrive at a number $f_0$ which is the integral of $f$ over $I_k$. Is this similar to the notion of the exterior derivative? Basically if I want to integrate something in $\mathbb{R}^{100}$ I just have to work backwards?
Source. Principles of Mathematical Analysis by Rudin