Apologies in advance for my rambling language, but I am having a hard time understanding differential forms as a whole. I’ll try to formulate my thoughts as clear as I could possible do.
What I have learned
- A $p$-form $\omega$ on linear space $\mathbf{V}$ is basically a muti-linear function $$\omega :\mathbf{V}^p\rightarrow \mathbb{R}$$
- A differential form, is a type of anticommutative $p$-form defined on a family of linear space $\left\{ \mathbf{V}_p \right\}_{p\in \mathfrak{M}} $, where $\mathfrak{M}$ is some manifold in $\mathbb{R}^n$. Usually, it’s the tangent space $\left\{ \mathrm{T}\mathfrak{M} _p \right\} $
- A manifold, as far as I understand, in short, in $\mathbb{R}^n$ is something that is “locally flat”, as an object that is stitched by a few images of maps. A manifold’s smoothness is dertermined by the maps.
- The exterior derivative on a domain $D$ in $\mathbb{R}^n$ $$ \mathrm{d}:\bigwedge\nolimits_r^p{\left( D \right)}\rightarrow \bigwedge\nolimits_{r-1}^{p+1}{\left( D \right)}$$$$ \sum_{1\leqslant i_1<\cdots <i_p\leqslant n}{a_{i_1\cdots i_p}\left( x \right) \mathrm{d}x_{i_1}\land \cdots \land \mathrm{d}x_{i_p}}\mapsto \sum_{1\leqslant i_1<\cdots <i_p\leqslant n}{\sum_{1\leqslant j\leqslant n}{\partial _{x_i}a_{i_1\cdots i_p}\left( x \right) \mathrm{d}x_j\land \mathrm{d}x_{i_1}\land \cdots \land \mathrm{d}x_{i_p}}} $$
- The notion of pull-back
I get the gist of the concepts so far, however I don’t understand the next few things quite as good.
An orientable manifold that is a manifold that is smooth and the “stitches” in it is compatible.
Integration of a differential form on a regular smooth manifold $(I,\phi,\mathfrak{M})$ is defined as something akin to $$ \int_{\mathfrak{M}}{\omega}:= \int_I{\phi ^*\omega} $$ I’m fine with this so far, the definition for a more complex manifold gets a bit more technical and frankly I’m not understanding it very well…
Stokes Theorem. Not the crux of my problem, I should understand it pretty well once I have my question resolved.
Questions
I have 3 questions at this point.
(1) The wedge product of the form $\mathrm{d}x_{i_1}\land \cdots \land \mathrm{d}x_{i_p}$ in differential $p$-form $\omega$ is a muti-linear function
$$
\mathrm{d}x_{i_1}\land \cdots \land \mathrm{d}x_{i_p}\left( \xi _1,\cdots ,\xi _p \right) =\begin{vmatrix}
\xi _{1i_1}& \cdots& \xi _{pi_1}\\
\vdots& \ddots& \vdots\\
\xi _{1i_p}& \cdots& \xi _{pi_p}\\
\end{vmatrix}
$$
However when integrating $\omega$, $\mathrm{d}x_{i_1}\land \cdots \land \mathrm{d}x_{i_p}$ cannot be a mutilinear function since
$$
\int_I{\phi ^*\omega}=\int_I{\sum_{1\leqslant i_1<\cdots <i_p\leqslant n}{a_{i_1\cdots i_p}\left( \phi \left( t \right) \right) \begin{vmatrix}
\partial _{t_1}\phi _{i_1}& \cdots& \partial _{t_p}\phi _{i_1}\\
\vdots& \ddots& \vdots\\
\partial _{t_1}\phi _{i_p}& \cdots& \partial _{t_p}\phi _{i_p}\\
\end{vmatrix} \left( t \right)}\,\mathrm{d}t_1\cdots \mathrm{d}t_p}
$$
That’s still not that confusing, but what really throws a wrench in this system is that we also have
$$
\int_I{\phi ^*\omega}=\int_I{\sum_{1\leqslant i_1<\cdots <i_p\leqslant n}{a_{i_1\cdots i_p}\left( \phi \left( t \right) \right)}\,\mathrm{d}\phi _{i_1}\left( t \right) \land \cdots \land \mathrm{d}\phi _{i_p}\left( t \right)}
$$
If I work this expression out, I have
$$
\int_I{\sum_{1\leqslant i_1<\cdots <i_p\leqslant n}{a_{i_1\cdots i_p}\left( \phi \left( t \right) \right) \begin{vmatrix}
\partial _{t_1}\phi _{i_1}& \cdots& \partial _{t_p}\phi _{i_1}\\
\vdots& \ddots& \vdots\\
\partial _{t_1}\phi _{i_p}& \cdots& \partial _{t_p}\phi _{i_p}\\
\end{vmatrix} \left( t \right)}\,\mathrm{d}t_1\land \cdots \land \mathrm{d}t_p}
$$
But it’s not the same since $\mathrm{d}t_1\cdots\mathrm{d}t_p$ is a measure (or, a measure that was integrated over), by Fubini theorem we can change the order of them, whereas $\mathrm{d}t_1\land \cdots \land \mathrm{d}t_p$, is a wedge product, changing the order of the $t$’s will result in a possible negative sign to appear. How should I interpret this? when can I make the change from wedge product to a measure?
(2) What advantage does this type of integration provides? I know there’s Stokes theorem and from that we have the residue theorem, but can’t we achieve that by just using Lebesgue integral?
(3) Compare how we calculate integrals of differential forms with integrals of Hausdorff measure, the similarity is striking. Is there a connection between this two types of integral?
Any forms of help is appreciated, thanks!