I'm a little confused about the divergence and the curl for two-dimensional vector fields. In 3d, I understand the curl as $d:\Omega^1(M^3)\to\Omega^2(M^3)$ and the divergence as $d:\Omega^2(M^3) \to \Omega^3(M^3)$.
But what is the analog in 2d? It seems the curl is the operator $d:\Omega^1(M^2)\to \Omega^2(M^2)$, and then what could the divergence be?
I recall using before the divergence theorem for two-dimensional vector fields...that $$\int_D\text{div}\,\sigma=\int_{\partial D}\sigma\cdot \nu, \tag{1}$$
where $\nu$ is the unit normal to the two dimensional region $D$. Where does this fit in with generalized Stokes' theorem, if $d:\Omega^1(M^2)\to \Omega^2(M^2)$ is the curl, not the divergence?
I'm thinking now as I write this that (1) just treats the 2d vector field as a 3d field with third component zero.