I am reading a paper on differential forms. They define it as follows:
A differential $k$-form on a domain $U$ is a continuous, infinitely differentiable map $\omega \colon U \to \Lambda^k (\Bbb{R}^{n \ast})$ where $\Bbb{R}^{n\ast}$ is the dual of $\Bbb{R}^n$ as a vector space.
In particular we may let $x_1,...x_n$ be a basis for $\Bbb{R}^n$ and let $dx_i\in \Bbb{R}^{n \ast}$ be the unique linear map $\Bbb{R}^n\to \Bbb{R}$ that satisfies $dx_i(x_j)=\delta_{ij}$
It then goes on to make a comment:
Intuitively, we may consider $dx_1 \wedge ... \wedge dx_k$ to be an oriented, $k$ dimensional volume element
I don't understand this remark. I know $dx$ from calculus is supposed to be an infinitesimal change and I also understand that wedge product is intuitively a way of turning vectors into higher dimensional objects.
But forget about wedge products. I don't understand how $dx$ defined as $\delta_{ij}$ has anything to do with an infinitesimal change.
What is going on here? Why is this abstract definition related to an infinitesimal change?
If you feed a list of $k$ vectors $v_1,\dots,v_k\in\Bbb R^n$ into the $k$-form $dx_1\wedge\dots\wedge dx_k$ you'll get the signed volume of the parallelepiped formed by $v_1',\dots,v_k'$, where $v_j'$ is the projection of $v_j$ into $x_1\dots x_k$-space.