so, ncatlab [writes](http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/Liouville-Poincar%C3%A9+1-form) that the canonical one-form $\theta\in\Omega(T^* X)$ is defined as the unique such form such that $\sigma^*\theta=j(\sigma)$, for any smooth section $\sigma\in\Gamma(T^* X)$, and with $j$ the isomorphism $j:\Gamma(T^* X)\to \Omega^1(X)$.
Anyone understand why this makes sense? For one, I thought differential one-forms were *defined* as smooth sections of the cotangent bundle, so wouldn't $\Gamma(T^* X)=\Omega^1(X)$? And then, $\sigma:X\to T^* X$ and $\theta:T^*X\to T^* T^* X$, so $\s…