# Motivation and intuition about differential forms [duplicate]

So I have tried to get motivation behind the formal definition of differential forms and that what I understood and I want to make sure that I’m on the right track:

So we want to integrate over a manifold which generalizes integration on $$\mathbb{R}^n$$, so we want to talk about something that looks similar enough to $$\mathbb{R}^k$$ locally, so we define a manifold and then we want to use our knowledge of $$\mathbb{R}^k$$ and look at the manifold locally and define the tangent space to each point(and now we look at a space that is isomorphic to $$\mathbb{R}^k$$ and way easier to use). But now we have a new problem, we have a different coordinate system to each tangent space and we want to get rid of it so we define $$dx_1,dx_2,\ldots,dx_k$$ to be the basis (which will be the partial derivatives to each point $$p$$) for each tangent space and then we want to integrate over the manifold we want a way to define volume independent of coordinates so we will use the determinant which is an alternating multilinear function and it will approximate the $$k$$ dim volume at the point $$p$$ we chose for the tangent space.

So we basically got that a differential form basically gives to each point $$p$$ on a set defined on a $$k$$ dim manifold in $$\mathbb{R}^{n}$$ a volume function which is defined on the tangent space. And when we talk about $$1$$ form for example so each differential form will have the form of a linear combination of $$a_1(p)dx_1+a_2(p)dx_2+\ldots+a_n(p)dx_n$$ because each of the coefficients is dependent of $$p$$ we will get that each of them is a function of $$p$$, and because we the $$1$$ dim volume is length we got that $$1$$ form is measuring a length over a set in $$1$$ dim manifold on $$\mathbb{R}^n$$ so when we will integrate we will get the line integral. And in the same analogy we can have the surface integral but now we will measure an area over a set on a manifold.

Is my intuition correct? Or am I really far behind?

• I wrote notes about the intuition behind Calculus on manifolds and differential forms here: github.com/danielvoconnor/calc_on_manifolds_intuition Commented Sep 4, 2023 at 5:45
• I read a little bit of it and I’m trying to understand if I I’m right about it, the actual reason you want the function to be multilinear and alternate it’s because you want to measure the k dim Parallelepiped, which is basically just a determinant which acts on the k vectors which we are going to call them dx1,dx2,…dxk and because the partial derivatives span the tangent space so each dxi acts as a projection to a partial derivative( so we can think of the partial derivatives as a system coordinates to each tangent space at each point on the set at the manifold) Commented Sep 4, 2023 at 6:08
• And further more from my understanding the intuition behind the wedge product is just evaluating the k dim volume of the Parallelepiped volume which spans between two dx? So for example dxdy is going to be just an area, so for example the determinant of k vectors that are going to be presented in the basis of the partial derivatives of the basis of tangent space is going to be just the determinant and is going to be written as dx1dx2dx3…dxk is it all right? Commented Sep 4, 2023 at 6:20
• All right. $\phantom{.}$ I hope you don't stop here. Learn as much as you can about differential forms. Commented Sep 4, 2023 at 6:42