This may be a dumb question since I'm only new to the form language.
I'm currently reading a book on symplectic geometry where in the first chapter it's proved that the canonical form of cotangent bundle is intrinsic (i.e. independent of the coordinate chosen). This theorem confused me. Since form is defined independent of coordinates chosen, isn't it natural for any form to be intrinsic?
Sometimes forms are defined only in coordinates, with the requirement that the definition does not depend of the choice of coordinates, e.g. the differential of a function $df|_U:=\frac{\partial f}{\partial x^i}dx^i$ with coordinates $(x^1,\dots,x^n)$ on $U$. You know that it results in a $1$-form $df$ defined on the whole manifold $M$, but you had to choose coordinates and to patch them in order to prove it.
On a cotangent bundle $T^*M$, choosing coordinates $(x^1,\dots,x^n)$ on $M$ and associated coordinates $(x^1,\dots,x^n,\xi^1,\dots,\xi^n)$ on $T^*M$, the $2$-form $\omega|_U:=\sum_{i=1}^ndx^i\wedge d\xi^i$ is a priori such a coordinate-dependent form. But it appears that we are able to give a coordinate-free definition of $\omega$ (see for example 2.3 in Ana Cannas da Silva's Lectures on Symplectic Geometry), as the (opposite of the) exterior derivative of the $1$-form $\alpha:p\mapsto(d\pi_p)^*\xi$ where $p=(x,\xi)\in T^*M$ and $\pi:T^*M\to M$ is the usual projection.