I am reading Fulton's book on algebraic topology, and I find myself puzzled by the style of proof in his book, especially in the first two parts. Most proofs seems to miss a lot of details, or maybe lack rigor. To illustrate I will take the proof of Proposition 2.16 as an example.
I have no problem with the subdivision part, but I find myself unsatisfied from there on. At first the author considers 'the integral of $\omega\circ H$ around the boundary of $R_{i,j}$', but in the rest of the proof he switches over to 'the integral of $\omega$ along the bottom and right side of the original rectangle' or '$\int_{\partial R}\omega$', which is confusing because $\omega$ is defined in the image $H(R) \subset U$, not in the preimage.
According to my understanding, there are two ways to fix this. One is to consider the integrals of $\omega \circ H$ around the original rectangles. But the author did not previously give an explicit definition of $\omega \circ H$. It does not seem obvious to me that how one should define $\omega \circ H$ such that its integral around the original rectangle is equal to $\int_{H(\partial R_{i,j})}\omega$.
Another way is to consider the integrals of $\omega$ around the images $H(R_{i,j})$, and take the sum of these integrals. But to do this, it seems to me that the author leaves lots of details to fix, mostly about the behaviour of $R_{i,j}$ under the homotopy $H$, e.g. showing the orientation of the original boundaries are preserved by $H$ in order that the integrals along the inside edges canceled.
I wonder whether these 'gaps' are my illusions due to misunderstanding, or that the author did leave some details in the proof? Maybe the first two parts are only an introduction to provide some motivations, so the rigor of proofs are sacrificed here? Are there any supplementary materials that might help me to understand these proofs?
Remember that you define the integral of a $1$-form $\omega$ on a parametrized curve $\gamma\colon [a,b]\to M$ by pulling back and integrating $\gamma^*\omega$ over $[a,b]$. When you write "integrals of $\omega\circ H$" that is not correct; you need the integral of $H^*\omega$. And all the action is, as Fulton wrote, taking place on the rectangle $R$. If $H_s$ is the restriction of $H$ to $[a,b]\times \{s\}$, he's arguing why $\int_{[a,b]} H_1^*\omega = \int_{[a,b]}H_0^*\omega$, and this is precisely $\int_{[a,b]} \delta^*\omega = \int_{[a,b]}\gamma^*\omega$.
If you're not used to the basics of differential forms, pullbacks, and integrals, you might find some of my YouTube lectures (linked in my profile) helpful. Eventually, there are a few lectures involving homotopy and topological applications, as well.