Section 10.34 of Baby Rudin is saying that exact forms are closed:
Let $\omega$ be a $k$-form in an open set $E \subset \mathbb{R}^n$. If there is a $(k-1)$-form $\lambda$ in $E$ such that $\omega = d\lambda$, then $\omega$ is said to be exact in $E$.
If $\omega$ is of class $C'$ and $d\omega =0$, then $\omega$ is said to be closed.
Theorem 10.20(b) shows that every exact form of class $C'$ is closed.
I am a bit suspicious of the last sentence. If $\omega$ is exact and of class $C'$, then there exist a $(k-1)$-form $\lambda$ such that $\omega =d\lambda$. But to use Theorem 10.20(b) to deduce that $d\omega=0$, we need that $\lambda$ is of class $C''$.
So I let $\lambda = \sum_I b_I (x) dx_I $, so that $d\lambda = \sum_I \sum_i (D_i b_I )(x) dx_i \wedge dx_I $ is of class $C'$.
To deduce that $b_I$'s are of class $C''$, we need $(D_i b_I )$'s are of class $C'$, but I am not able to get this because we only have the 'sums' which have same $dx_i \wedge dx_I$ are of class $C'$.
Since my explanation is not clear, let me give you an example. ($k=2$)
Let $\lambda = a(x,y,z)dx +b(x,y,z)dy + c(x,y,z) dz$
We want to show that if $\omega = d\lambda$ is of class $C'$, $d^2 \lambda =0$. We have that
\begin{equation} \omega = d\lambda = \left ( \frac{\partial a}{\partial y} - \frac{\partial b}{\partial x} \right ) dx \wedge dy + \left ( \frac{\partial b}{\partial z} - \frac{\partial c}{\partial y} \right ) dy \wedge dz + \left ( \frac{\partial c}{\partial x} - \frac{\partial a}{\partial z} \right ) dz \wedge dx \end{equation} is of class $C'$. If $a, b, c$ are of class $C''$, (which means $\lambda$ is of class $C''$) we easily obtain (by calculation or using theorem 10.20(b)) $d^2 \lambda =0$.
But, I think $ \frac{\partial a}{\partial y} - \frac{\partial b}{\partial x} , \frac{\partial b}{\partial z} - \frac{\partial c}{\partial y} , \frac{\partial c}{\partial x} - \frac{\partial a}{\partial z}$ can be of class $C'$ without $a, b, c$ being of class $C''$, and so $d^2 \lambda$ might not be equal to $0$.
Lastly, here are some definitions and theorems in Rudin.
Theorem 10.20(b) of Baby Rudin:
If ($k$-form) $\omega$ is of class $C''$ in $E$, then $d^2 \omega =0$.
Definition of $k$-forms:
$k$-form in $E$ is a function $\omega = \sum a_{i_1 \cdots i_k}(x ) dx_{i_1} \wedge \cdots \wedge dx_{i_k}$, which assigns to each $k$-surface $\Phi$ in $E$ a number $\int_{\Phi} \omega = \int_{D} \sum a_{i_1 \cdots i_k}(\Phi (u) ) \frac{\partial (x_{i_1} , \cdots , x_{i_k} )}{\partial (u_1 , \cdots , u_k)} du$ where $D$ is parameter domain of $\Phi$.
$\omega$ is said to be of class $C'$ (or $C''$) if $a_{i_1 \cdots i_k}$s are all of class $C'$ (or $C''$).
But to be more precise, I think $\omega$ should said to be of class $C'$ (or $C''$) if $b_I$, which is summation of $a_{i_1 \cdots i_k }$ where $ \{ i_1 , \cdots , i_k \} = I $, are of class $C'$ (or $C''$).
Thanks you for reading my question. I hope to see some answers here.
And one more small question: I think these kind of suspicion are really small thing, and sometimes I think these thoughts make me waste time. Should I just ignore these small suspicions?
Thanks in advance.