Questions concerning the Magnetic Field

Is the flux through a surface and a "close casting" the same thing? Are both these quantities equal to 0?

Also, how can one tell if an arbitrary vector field is a candidate for a magnetic field? For example, could the given a field represent a magnetic field?

$$F = A x,, mathbf{hat{x}} + B y ,,mathbf{hat{y}}.$$

My answer was yes, since $nabla cdot mathbf{F}= 0$.