On Calculus on Manifolds Spivak claims
[T]he integral of a $1$-form over a $1$-chain is called a line integral.
Which seems to imply that line integrals (as they are usually defined) can be written in terms of $1$-forms and $1$-chains (and vice-versa).
Say I have $1$-chain $c:[0,1]\to\mathbb{R}^n$ and a vector field $F:\mathbb{R}^n\to\mathbb{R}^n$. We may compute the line integral of $F$ over $c$ as $$\int_cF:=\int_0^1F(c(t))\cdot c'(t)\ dt.$$
What $1$-form $\alpha$ allows me to write the integral above as $$\int_c\alpha \ ?$$
Similarly, given a $1$-form $\alpha$ and a $1$-chain $c$, what vector field $F$ allows me to write $\int_c\alpha$ in terms of $F$?