Yesterday I posted something about this question but many rightfully pointed out that I gave too little details so here I am with a more complete post.
I was looking for a rigorous proof of this
Theorem Let $\omega$ be a closed $C^1$ differential linear form defined on an open set $\Omega\subseteq\mathbb R^n$. Given two homotopic curves $\gamma_0$ and $\gamma_1$ in $\Omega$, then $\int_{\gamma_0} \omega=\int_{\gamma_1}\omega$.
Some days ago I eventually found a brilliant pdf where a proof is given: two technical lemmas are previously demonstrated and I had no issues understanding them but when the proof of the above theorem came I cannot help myself in its last lines. I still have no idea where the answer may be hidden, so I copy down here those propedeutic lemmas and a summary of the proof until the incriminated point.
Lemma 1 Let $\Omega$ be an open set in $\mathbb R^n$ , $\gamma_0$, $\gamma_1$ two closed curves in $\Omega$ and $H : [a, b]×[0, 1] \to\Omega$ an homotopy from $\gamma_0$ to $\gamma_1$ . Then, for all $\varepsilon > 0$ exists a $C^\infty$ homotopy $H^∗ : [a, b] × [0, 1] \to \Omega$ such that $$|H^*(t,\lambda)-H(t,\lambda)\|\leq\varepsilon,\ \ \forall\,(t,\lambda)\in[a,b]\times[0,1]\\ \int_a^b\|\partial_t H^*(t,\lambda)-\partial_t H(t,\lambda)\|dt\leq\varepsilon,\ \ \text{for }\lambda\in\{0,1\}.$$
Lemma 2 Let $\omega$ be a closed $C^1$ linear differential form defined on an open set $\Omega\subseteq\mathbb R^n$ and $H^*:[a,b]\times[0,1]\to\Omega$ a $C^2$ homotopy. Then, the function $$I^*:\lambda\in[0,1]\mapsto \int_a^b \langle\omega(H^*(t,\lambda)),\partial_t H^*(t,\lambda)\rangle dt$$ results constant ($\langle\cdot,\cdot\rangle$ is the duality product of $\mathbb R^n$).
The theorem's proof goes like this.
Let $\gamma_0$ and $\gamma_1$ be homotopic curves in $\Omega$ and $H : [a, b] × [0, 1] → Ω$ an homotopy from $\gamma_0$ to $\gamma_1$. Furthermore, let $$M = \max\{\|H(t, λ)\| : (t, λ) ∈ [a, b] × [0, 1]\} \text{ e } L = \max\{\text{length } \gamma_0, \text{length } \gamma_1\}.$$ Fixed an arbitrary $ε ∈ (0, 1)$, consider a $C^\infty$ homotopy $H^∗ : [a, b]×[0, 1] → Ω$ that satisfies Lemma 1 with respect to $H$. Then, from the Lemma 2 we also have $I^*(0)=I^*(1)$ and so $$\left|\int_{\gamma_0} \omega-\int_{\gamma_1} \omega\right|\leq\left|\int_{\gamma_0}\omega-I^*(0)\right|+\left|\int_{\gamma_1}\omega-I^*(1)\right|.$$ Now, let's call $u_λ(t) = H(t, λ)$ and $v_λ(t) = H^∗(t, λ)$ for all $t ∈ [a, b]$ and $λ \in \{0, 1\}$: for such $λ$, we obtain $$\int_{\gamma_\lambda} \omega-I^*(\lambda)=\int_a^b\langle\omega(u_\lambda(t))-\omega(v_\lambda(t)),u'_\lambda(t)\rangle dt+\int_a^b\langle\omega(v(t)),u'_\lambda(t)-v'_\lambda(t)\rangle dt.$$
Till here, everything is clear: in the following lines will be used a norm over $(\mathbb R^n)^*$ defined over the generic functional $L=\sum_{i=1}^n a_i dx_i\in(\mathbb R^n)^*$ as $$|L|=\left(\sum_{i=1}^n a_i^2\right)^{\frac 12}.$$
From Lemmas 1-2, observe that $\|v_\lambda(t)\|\leq\|u_\lambda(t)\|+\|u_\lambda(t)-v_\lambda(t)\|<M+\varepsilon<M+1$ and so $$\left|\int_{C_\lambda}\omega-I^*(\lambda)\right|\leq\int_a^b|\omega(u_\lambda(t))-\omega(v_\lambda(t))|\|u'_\lambda(t)\| dt+\int_a^b|\omega(v(t))|\|u'_\lambda(t)-v'_\lambda(t)\| dt\leq \\ \leq\sup_{t\in[a,b]}|\omega(u_\lambda(t))-\omega(v_\lambda(t))|\int_a^b \|u'_\lambda(t)\| dt+\sup_{t\in[a,b]}|\omega(v(t))|\int_a^b\|u'_\lambda(t)-v'_\lambda(t)\| dt\leq\\\leq\varepsilon L+(M+1)\varepsilon.$$
My problem is right there, in the last inequality you see: I don't understand why $\sup_{t\in[a,b]}|\omega(u_\lambda(t))-\omega(v_\lambda(t))|\leq\varepsilon$ and $\sup_{t\in[a,b]}|\omega(v(t))|\leq M+1$. I'm really concerned on the first one, since the latter can be undone for $|\omega(v(t))|$ being continuous over the compact $[a,b]$ (thanks to Weierstrass theorem). So, I tried to make some work on compactness of $[a,b]$ and continuity or even Lipschitz continuity of $\omega$ coefficients but nothing conclusive: do you have any idea how to explain that inequality or may you have some other way to conclude that $\sup_{t\in[a,b]}|\omega(u_\lambda(t))-\omega(v_\lambda(t))|<C\varepsilon$ for any $C\geq0$ (this is also optimal for proof's aim)?