TITLE: What is a differential form?
QUESTION [32 upvotes]: can someone please informally (but intuitively) explain what "differential form" mean? I know that there is (of course) some formalism behind it - definition and possible operations with differential forms, but what is the motivation of introducing and using this object (differential form)?
I have heard that they somehow generalize integration, are used for integration of manifolds and can evaluate k-dimensional integrals in n-dimensional space ($k \leq n$), but is it really true and is it the main motivation of introducing this object into mathematics?
Thank you for explanation

REPLY [2 votes]: Where do Differential forms come from?
Answer: One day Élie Cartan was having a lot of Absinthe and suddenly he had a strange idea: "What if I take the change of variable formula and represent it without that annoying Jacobian determinant?" First Mr. Cartan expressed, as usual, the integrand and the region with new coordinate function $X' = ϕ(X)$. Then he formally expressed the differentials with new differentials, $dx'= ∑\frac{\partial (x')}{\partial x*}dx$. Now he could extract the Jacobian determinant if he treated differentials in a very bizarre way,$ dx*dy = -dy*dx$, just like with matrix columns in determinant. So he could get rid of the Jacobian determinant $Det[J]$ and encode that information into the differentials: $$Det[J]*d(x_1)∧...∧d(x_n)=Det[J]Det[x_1,..., x_n]=Det[J*X]=Det[X']=Det[x'_1,..., x'_n]=d(x'_1)∧...∧d(x'_n)$$
$$  \to    ∫f(X')*d(x'_1)∧...∧d(x'_n)= ∫f((ϕ(X))*Det[J_ϕ]*d(x_1)∧...∧d(x_n)$$
As a true Frenchman Cartan quickly axiomatized and generalized his powerful invention, and left future generations wondering what the Hell's going on...
From gem of a YouTube comment in the comments of this lecture by Ted shiffrin