Take a real differentiable manifold of class $C^k$ and $\omega$ a differential form of degree $q$ on $M$. The definition that I'm using is the following:
$\omega$ is a function on $M$ such that for each $x \in M$, $\omega(x)$ is a multilinear alternating function on $(T_xM)^q$ with values in $\mathbb{R}$.
Now the professor talk about $C^k$ forms but no definition is given explicitly. I have the following guess. I know that given a point $x$ "locally" (in local charts) in a neighborhood of $x$ the form is given by $\omega = \sum\limits_{I} u_I dx_I$ where $I$ is a multi-index and $u_I: M \to \mathbb{R}$. Now I think that $\omega$ is $C^k$ if for each $x$ every $u_I$ is of class $C^k$.
I didn't find a text where this definition is given explicitly. Can anyone help me to understand it?