If $M$ is a smooth manifold, $w\in T^*_pM$, then is it possible to find a smooth function $y:M\to \mathbb{R}$ such that $dy|_p=w$. If it is, is there an easy way to give $y$ explicitly? I have just learned about covectors and I am not really understanding them overly well.
2025-01-13 07:42:44.1736754164
Smooth function $y$ that satisfies $dy|_p=w$ for covector $w$.
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HINT: Start by solving this in $\Bbb R^n$ with $p=0$. Can you write down a function $f$ with $df|_0 = \sum a_i\,dx^i$ (here the $a_i$ are constants)?