I look for an example of differential form $\omega$ of degree $r$ defined on a surface $ M \subset \mathbb{R}^n $ of dimension $m$ such that its derivative $D\omega$ can not be defined on the entire surface $ M $.
Background. Let $\omega$ be a differential form of degree $r$ and class $C^k$ ($k\geq 1$) defined on a surface $M\subset\mathbb{R}^n$ of class $C^s$ ($s > k $) of dimension $ m $.
To fix the ideas take $\omega$ as an application of $M$ in $\bigcup_{q\in M} \bigwedge^r(T_qM)$ which at each point $ p\in M $ associates a $r$-linear application in $\bigwedge^r(T_pM)$. I am assuming that the tangent plane $ T_{p^{\prime}} M $ varies to the tangent plane $ T_{p^{\prime \prime}} M $ in a differentiable manner (with regularity $ C^k $) as $ p^\prime $ varies to $p^{\prime \prime}$.
Let's fix an arbitrary point $ p\in M $. Let a parametrization $ \varphi: \overline{U} \to \widetilde{U} $ of $M$ and a curve $\overline{\alpha}: (-\epsilon, + \epsilon) \to \overline{U}$ satisfying $$ (\ast) \left\{ \begin{array}{c} \varphi(u)=p, \\ \overline{\alpha}(0)=u. \end{array} \right. $$ The total derivative of $\omega $ should work as follows: each vector $ (\varphi\circ\overline{\alpha})^\prime (0)$ tangent to the curve $ \varphi \circ \overline{\alpha} :(-\epsilon, +\epsilon) \to M $ at the point $p \in M $ is taken by the derivative of $ \omega$ in the vector $ (\omega \circ\varphi\circ\overline{\alpha})^\prime(0) $ tangent to the curve $ \omega\circ\varphi \circ \overline{\alpha}: (- \epsilon, + \epsilon) \to \bigcup_{q \in M} \bigwedge (T_qM)$ at the point $\omega(p)$.
Suppose a second parametrization $ \psi: \overline{V} \to \widetilde{V} $ and a second curve $ \overline{\beta}: (- \epsilon, + \epsilon)\to \overline{V}$ satisfying the following properties: $$ (\ast\ast) \left\{ \begin{array}{c} \widetilde{U}\cap\widetilde{V}\neq \emptyset\\ \psi(v)=\varphi(u) \\ \overline{\beta}(0)=v \end{array} \right. $$ In order, for the total derivative of $ \omega $ to be well defined over the entire surface $ M $, the following property must be satisfied: for every $ p \in M $, for all parametrization $ \varphi $ and every curve $ \overline{\alpha} $ fulfilling the conditions $ (\ast) $ worth the equality $$ (\omega\circ\varphi\circ\overline{\alpha})^{\prime}(0)= (\omega\circ\psi\circ\overline{\beta})^\prime(0). $$ for any parametrization $ \psi $ and any curve $ \overline{\beta}$ that satisfies $ (\ast \ast) $.
So the example to be looked for is an example where there is a $ M $ surface and a differential form $ \omega: M \to \bigcup_{q \in M} \bigwedge^r(T_qM) $ for which at a point $ p \in M $ there is a parameterization $ \varphi $ and a curve $ \overline{\alpha}$ satisfying $ (\ast) $ and a parameterization $ \psi $ and a curve$ \overline{\beta}$ satisfying $ (\ast\ast) $ such that $$ (\omega\circ\varphi\circ\overline{\alpha})^{\prime}(0)\neq (\omega\circ\psi\circ\overline{\beta})^\prime(0). $$ Is there any classic example with the above property? Is there any reference where I can find such an example?
I was unable to find any reference who could be of some help in answer my question. The references I researched were as follows:
Analysis On Manifolds by James R. Munkres
An Introduction to Manifolds by Loring W. W. Tu
Vector Analysis by Klaus Jänich
A Visual Introduction to Differential Forms and Calculus on Manifolds by Jon Pierre Fortney
Mathematical Analysis II by V. A. Zorich
I would really appreciate if someone can give me some directions in the literature and, if possible give me an exemple.