In the book "Differential Forms and applications" by Manfredo do Carmo, he says that a differential $k$-form on a $n$-dimensional smooth manifold $M$ is determined by a choice, for each parametrization $(f_{\alpha},U_{\alpha})$ of $M$, of a differential $k$-form in $U_{\alpha}$ such that whenever we have another parametrization $(f_{\beta},U_{\beta})$ with $f_\beta(U_\beta)\cap f_\alpha(U_\alpha)\neq\emptyset$, we get $\omega_{\alpha}=(f_\alpha^{-1}\circ f_\beta)^\ast\omega_{\beta}$.
Is this true?
Is the usual way of specifying a differential form on a manifold the description of these "local" forms? Thanks.
EDIT
The definition I have of a differential $k$-form in a smooth $n$-manifold $M$ is the following:
A $k$-form on $M$ is a map that for each $p\in M$ associates a $k$-linear alternating map $\omega(p)\in A^{k}(T_pM)$.
It is called a differential $k$-form if any of its local representations $\omega_{\alpha}$ in $U_\alpha$ is a differential $k$-form.