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This question can look like a duplicate of this one, but it's kind of different. I'm trying to relate some geometrical meanings I've seem in some books to the definition of differential forms in $\mathbb{R}^n$ as mappings $p \mapsto \omega(p)\in \Lambda^k(\mathbb{R}^n_{\phantom{n}p})$.

Differential forms seems to be object with high geometrical importance, however, I'm failing to grasp what they really represent. Many books, mainly on Physics, try to give one geometrical interpretation for differential forms as "families of surfaces" such that the value on a vector is the number of surfaces the vector crosses.

This confuses me a little. Why do this interpretation makes any sense? I mean, if I want to construct an object with this geometrical property, why it should be a function associating skew-symmetric tensors to each point in space?

Also, vector fields are easy to understand. We know what each vector is at each point, we picture as a small arrow, and we know that they can describe things with directions, they can describe rates of changes, being derivations, and so on. Now this geometrical interpretation they give, do not allows us to picture differential forms at points, just the association at each point.

My understanding was the following: as I see, differential forms replace the classical $dx$, $dA$, $dV$ and so on, that were considered infinitesimal objects. My idea is that in that case, $\omega(p)$ would represent just a small patch of the surfaces $\omega$ represents and because of that, we could think of $\omega$ really relating to those infinitesimal objects. I'm unsure of this intuition, and I can't see how this would leads us towards the rigorous definition of differential forms.

So, what's the true geometrical meaning of differential forms and how this meaning implies that the algebraic definition we give is a good one?

Thanks very much in advance!

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Caveat: This answer is (judiciously!) incomplete, and makes no pretense of giving the One True Geometric Meaning of differential forms. Also, there are so many conventions (regarding spaces and their duals, index placement, and vectors/covectors versus vector fields/$1$-forms) that it's impossible to be notationally and terminologically consistent with other sources you may have read.

A differential $k$-form may be viewed as defining (at each point) a "measuring device for $k$-dimensional oriented volume elements". Loosely, for example, if you view a vector field as a velocity field, then (to coin a phrase) a $1$-form may be viewed as a "(vector-valued) speedometer field".

To give this heuristic principle a precise interpretation in $\mathbf{R}^3$, let $\mathbf{e}_i$ denote the Cartesian frame fields (i.e., the vector fields whose values at each point are the standard basis of $\mathbf{R}^3$); $dx^i$ the (dual) coordinate $1$-forms; $\omega_i$ smooth functions; and $a_i = \omega_i(p)$ the value of $\omega_i$ at a point $p$. A $1$-form $$ \omega = \omega_1\, dx^1 + \omega_2\, dx^2 + \omega_3\, dx^3 $$ defines the linear functional $\omega(p) = a_1\, dx^1 + a_2\, dx^2 + a_3\, dx^3$ on the vector space $T_p\mathbf{R}^3 \simeq \mathbf{R}^3$. If $X = X^1 \mathbf{e}_1 + X^2 \mathbf{e}_2 + X^3 \mathbf{e}_3$ is a vector field, then $$ \omega(X)(p) = \sum_{i,j=1}^3 a_i X^j dx^i(\mathbf{e}_j) = a_1 X^1 + a_2 X^2 + a_3 X^3 $$ may be viewed as the "measurement": $a_1$ times the first component of $X$ plus $a_2$ times the second component plus $a_3$ times the third component.

Analogously, if $\omega_{ij}$ are smooth functions and $a_{ij} = \omega_{ij}(p)$, the $2$-form $$ \omega = \omega_{23}\, dx^2 \wedge dx^3 + \omega_{31}\, dx^3 \wedge dx^1 + \omega_{12}\, dx^1 \wedge dx^2 $$ defines a linear functional $\omega(p) = a_{23}\, dx^2 \wedge dx^3 + a_{31}\, dx^3 \wedge dx^1 + a_{12}\, dx^1 \wedge dx^2$ on the space $\bigwedge^2(\mathbf{R}^3)$ of "oriented $2$-plane elements". If $X = X^{23} \mathbf{e}_2 \wedge \mathbf{e}_3 + X^{31} \mathbf{e}_3 \wedge \mathbf{e}_1 + X^{12} \mathbf{e}_1 \wedge \mathbf{e}_2$, then $\omega(X)(p)$ may be viewed as the "measurement": $a_{23}$ times the projection of $X$ on the $(x_2, x_3)$-plane plus $a_{31}$ times the projection of $X$ on the $(x_3, x_1)$-plane plus $a_{12}$ times the projection of $X$ on the $(x_1, x_2)$-plane.

Note that the components of a vector field $X$ are the projections of $X$ onto the coordinate axes, so the two preceding interpretations are more closely analogous than they may first seem.

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