In the book *Mathematical Methods of Classical Mechanics* by V.I. Arnold, the author introduces (p.189) the concept of exterior derivative as *"the principal linear part of the increment"* of the function $$F(\varepsilon)=\int_{\partial V(\varepsilon)} \omega$$

(where $V(\varepsilon)$ is a "curvilinear parallelepiped" with vertexes $x_0, x_0+\varepsilon \xi_1, ..., x_0+\varepsilon \xi_{n+1}$), $\varepsilon \to 0$, which could be shortly written as $$F(\varepsilon)=(d\omega)(x_0)(\xi_1, ...,\xi_{n+1})\varepsilon^{n+1}+o(\varepsilon^{n+1})$$

Then, in order to show the independence of the exterior derivative from the coordinate system, he states that after a change of coordinates, the difference $$\int_{\partial V(\varepsilon)} \omega - \int_{\partial V'(\varepsilon)} \omega$$ (where $V'$ is the curvilinear parallelepiped expressed in new coordinates) *is smaller than $o(\varepsilon^{n+1})$*, and asks to prove it.
Unfortunately I have no clue how to prove it.