Exterior derivative independence from coordinate systems 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.
 A: A remark, too long for a comment. To check that the exterior derivative is a geometric operation, coordinate-free, it seems better to define first the Lie derivative of a form $\omega$ with respect to a vector field $X$: you define easily
$$
\mathcal L_X(\omega)=\frac{d}{dt}(\Phi_X^t)^*(\omega)_{\vert t=0},
$$
where $\Phi_X^t$ is the flow of the vector field $X$. Then you can define the exterior derivative inductively by taking as a definition the Elie Cartan formula,
$$
\mathcal L_X(\omega)=d\omega \lrcorner X+d(\omega \lrcorner X),
$$
where $\lrcorner$ stands for the interior product. You know what is $df$ when $f$ is a function (0-form); using the above formula, you get for $\omega_{p+1}$ a $(p+1)$-form,
$$
d_{p+1}\omega_{p+1} \lrcorner X=\mathcal L_X(\omega_{p+1}) -d_p(\omega_{p+1} \lrcorner X),
\tag{$\ast$}$$
where $d_q\omega_q$ is the exterior differentiation of a $q$ form. Indeed $(\ast)$ gives you directly a geometric definition of $d_{p+1}\omega_{p+1} $ from the knowledge of $d_p$. Let us just recall that for a $q+1$ form $\omega$
$$
\langle\omega\lrcorner X,Y_1\wedge\dots\wedge Y_q\rangle=
\langle\omega,X\wedge Y_1\wedge\dots\wedge Y_q\rangle.
$$
