Let $\vec{F}=(f,g):\mathbb{R}^2\to \mathbb{R}^2$ be a smooth vector field and let $\gamma$ be a smooth non-self-intersecting closed curve in $\mathbb{R}^2$.
I would like to know what is the definition of the following expression: $$\oint_{\gamma}\left(f\,{\rm d}x + g\,{\rm d}y\right)$$
found in this question. Similar expressions appear here, or in the statement of Green's theorem. I don't understand how we can integrate $f$ only w.r.t. $x$, for instance.
Thank you.
The typical definition is $$\int_{\gamma}(f\,\mathrm{d}x+ g\,\mathrm{d}y) = \int_{a}^{b}\vec{F}(\gamma(t))\cdot\dot{\gamma}(t)\,\mathrm{d}t,$$ where $\gamma\colon[a,b]\to\mathbb{R}^{2}$ and the dot above the $\gamma$ denotes (component-wise) differentiation.
The idea is that the integrand is $\vec{F}\cdot\mathrm{d}\gamma$ by abuse of notation and the chain rule, and then writing $\mathrm{d}\gamma = (\mathrm{d}x,\mathrm{d}y)$ gives the notation in question.