Thinking on $1-$forms as functions.

Question

Let $M$ be a differentiable manifold and $\mathrm{T}M$ be its tangent bundle. I would like to think about $1-$forms as functions from $\mathrm{T}M\to \mathbb{R}$ and evaluate their differentiation. So, Let $\gamma:(-\epsilon,\epsilon) \to \mathrm{T}M $ be a curve then what is $$\frac{\mathrm{d}}{\mathrm{d} t}|_{t=0}\beta(\gamma(t)),$$ for a $1-$form $\beta$ on $M$? Is there any global form for this?

Answer 1

I would like to think about $1-$forms as functions from $\mathrm{T}M\to \mathbb{R}$

That is what a $1$-form is.

If you have a coordinate system, through which the $1$-form is locally written as $$f_1\operatorname{d}\!x_1+\ldots+f_n\operatorname{d}\!x_n$$ where the $f_i$ are functions of $x_1,\ldots, x_n$, then this means that $\beta$ is represented by the vector $(f_1,\ldots, f_n)$ at the given point (here $\gamma(0)$). Hence $$\frac{\mathrm{d}}{\mathrm{d} t}|_{t=0}\beta(\gamma(t))$$

is the inner product of the vectors $(f_1,\ldots, f_n)$ and $\gamma'(0)$.