$\mathrm{Grad}$, $\mathrm{rot}$ ($\mathrm{curl}$) and $\mathrm{div}$ are the notions of the 3-dimensional vector analysis. They replace the notion of the exterior derivative of $k$-forms in the case of $k=0,1$ and $2$, respectivelly.
But, what is the situation in the 2-dimensional vector analysis?
On a 2-dimensional differentiable manifold, the exterior derivative of a $0$-form is also an $1$-form, so it can be identified with a vector field, just as in the 3-dimensional vector analysis. So, $\mathrm{Grad}$ is a good name for it in this case too. But in 2-dimensions, the exterior derivative of a $1$-form is a $2$-form, and the latter can be identified with a $0$-form, so this operation assigns a scalar field to a vector field, so it corresponds to the 3-dimensional $\mathrm{div}$. But the name of the operation that corresponds to the exterior derivative of a $1$-form in the 3-dimensional vector analysis is $\mathrm{rot}$ ($\mathrm{curl}$). Then what is the name of the integrand on the right side of Green's theorem? $\mathrm{rot}$ ($\mathrm{curl}$), or $\mathrm{div}$, or something else?