homework and exercises - Is curl of a vector a scalar quantity in 2 spatial dimensions? If it is so, then somebody help me understanding Maxwell's equations in 2+1 D

I have seen on wikipedia that in 2 spatial dimensions, Green's theorem, Gauss's divergence and Stokes theorems are equivalent and it makes sense. When I tried to write Maxwell's equations in 2+1 spacetime dimensions, I got many questions and confusions. Here I give a brief summary.

In 3+1 D we have

$$\oint_{\bf{A}} {\bf E}\cdot d{\bf A}=\int_V \nabla\cdot\bf {E} dV= \int_V \rho dV.$$ In 2+1D it might be written as $$\oint_\ell {\bf E}\cdot d\bf{\ell}=\int_A \nabla\cdot\bf {E} dA=\int_A \rho dA.$$ $A$ is scalar area in 2+1 D. In 3+1 D $E\cdot dA$ gives the flux. What does ${\bf E}\cdot d\bf{\ell}$ give us?

Secondly in 3+1 D we have

$$\oint {\bf E}\cdot d\vec{\ell}=\int_A (\nabla\times\bf {E})\cdot dA=-\frac{\partial}{\partial t}\int_A \bf B\cdot dA.$$ I think it can be written in 2+1 D as $$\oint {\bf E}\cdot d\vec{\ell}=\int_A (\nabla\times\bf {E}) dA=-\frac{\partial}{\partial t}\int_A B dA,$$ where $B$ is a scalar in 2+1 D and I think the cross product term is also a scalar.

Either I am wrong at writing these expressions or some thing goes messy in 2+1 D. From above expressions we can see that

$$\nabla \cdot \bf E = \nabla \times \bf E$$ which is the main prob for me to understand.