In his Ordinay Differential Equations, Arnol'd mentions the following:
Functions of the attached vector that are linear at each given point where thay are attached are called differential 1-forms.
Apparently, he's giving justification of going from the equation ${dx\over dt}=v(x)$ to ${dx\over v(x)} = dt$ and then integrating.
And this is pure confusion to me.
I've never before come across differential forms. So can you please explain in simpler terms?
NOTE: Please make sure that you're explaining it to a Physics undergraduate who is not very familiar yet with his new crush, mathematics.
I'm also attaching the screenshots for context.
Before we even get to differential $1$-forms, you need to be comfortable with the standard differential calculus in higher dimensions. Very roughly speaking, the entire purpose of differential calculus is to study non-linear behaviour of functions by first approximating them by linear functions; this is literally the definition of the derivative. In single variable calculus, the derivative is often introduced as the "slope" of the curve at a point, but while this is geometrically nice, it is not the whole story. I hope you are familiar with the following definition for differentiability:
Rewriting that equation, we get that $\Delta F_p(h) \equiv F(p+h) - F(p) = dF_p(h) + o(h)$, which in words says we can approximate changes in the function $F$ by a linear part, which is $dF_p(h)$, and an additional "remainder" term $o(h)$, which is "small" in the sense that $o(h) / \lVert h\rVert \to 0$ as $h \to 0$.
I'm not sure if this is an appropriate motivation for differential $1$-forms, but the way I like to look at it is as a geometric way of saying what $dF$ does. Suppose we deal with the special case $m=1$. Let $F: \Bbb{R}^n \to \Bbb{R}$ be a differentiable function. Then, for each point $p \in \Bbb{R}^n$, $dF_p$ is a linear transformation from $\Bbb{R}^n$ into $\Bbb{R}$. In symbols, we say that for every $p \in \Bbb{R}^n$, $dF_p \in (\Bbb{R}^n)^*$ (this is common notation for the set of linear maps from $\Bbb{R}^n \to \Bbb{R}$). In other words, from the function $F$, we have constructed a new function $dF: \Bbb{R}^n \to (\Bbb{R}^n)^*$, which to each point $p$ assigns the differential at that point, $dF_p$.
Once again, the way you should think about $dF$ is that if you give me a point $p$, then $dF_p$ is a linear transformation which locally approximates the function near the point $p$. i.e if $h$ is a "displacement" vector with very small length, then if you are displaced slightly from the point $p$ to the point $p+h$, then the function changes by an amount $F(p+h) - F(p) \approx dF_p(h)$. So, $dF_p$ in a sense "measures and approximates" small changes to the function $F$ in a neighbourhood of the point $p$.
Now, a differential $1$-form is pretty much what $dF$ does. The following definition only works in $\Bbb{R}^n$ (I'm going to simplify some things by not going into tangent spaces of manifolds etc, and canonically identify tangent spaces of $\Bbb{R}^n$)
Notice how this is exactly the same thing as what $dF$ does. So a differential $1$-form $\omega$ is a function which for each point $p$, assigns a linear transformation $\omega_p: \Bbb{R}^n \to \Bbb{R}$. Just to emphasise again, $\omega$ can be thought of as a "function of two-variables" $\omega_{(\cdot)}(\cdot)$. In the lower slot, we input the point $p$ of interest. This leaves us with a function of $1$-variable $\omega_p(\cdot)$, which is linear. In this remaining slot, we can feed it vectors $h$, so that finally $\omega_p(h)$ is a real number.
By the way, I'm sure Arnol'd has an entire chapter later on related to differential forms, so perhaps it might be helpful to read up on that as well.