I will try to restrain myself! Mathematical concepts do NOT HAVE a specific "physical meaning" because mathematics is not physics. It is true, that the names "divergence" and "curl" come from physics- or more correctly, fluid motion. If \vec{f}(x,y,z) is the velocity vector of a fluid at point (x,y,z), the div f measures the tendency of the flud to "diverge" (move away from) some central point. Similarly, curl f measures the tendency to circulate around some central point.
However, you say "they ask us to calculate area and etc. using Green's theorem".
Well, Green's theorem says
\int\int \left[\frac{\partial Q(x,y)}{\partial x}-\frac{\partial P}{\partial y}\right]dxdy= \oint P(x,y)dx+ Q(x,y)dy
which doesn't use either "div" or "grad"!
You know, surely, \int\int dx dy, taken over a given region in the xy-plane, is the area of that region. To "use Green's theorem" to find the area of a region, you just have to use functions P(x,y) and Q(x,y) such that
\frac{\partial Q(x,y)}{\partial x}- \frac{\partial P(x,y)}{\partial y}= 1.
One obvious choice is P(x,y)= 0, Q(x,y)= x. Integrating \int x dy around the boundary of a region will, according to Green's theorem, be the same as integrating 1 over the region itself- and so will give you the area of the region.
