I got a question from my textbook which is as follows
A hemispherical shell is placed on the $xy$ plane centered at the origin. For a vector field $$ \vec{E}=\frac{-y\widehat{e}_{x}+x\widehat{e}_{y}}{x^{2}+y^{2}} ,$$ the value of integral $$\int _{S}\left( \vec{\nabla }\times \vec{E}\right) \cdot d\vec{a}$$ over the hemispherical surface is _________. (where $\widehat{e}_{x}$ and $\widehat{e}_{y}$ are unit vectors in cartesian coordinate system).
I found that while calculating curl,it blows up at (x,y)=(0,0) so I tried approaching with simple line integral.I used polar system to find the line integral which comes out to be 2$\pi$.So does this means stokes theorem is invalid in this case or we have to modify it a bit to get to the result?If so please suggest the method by using stokes theorem.
The strict formulation does not apply, because curl does not have a finite value at $(0,0,z)$.
But in physics we save the equations and the Stokes theorem rule in this kind of situation, by using distributions. In this case, curl of the field is proportional to Dirac delta distribution, so we have
$$ \nabla \times \vec{E} = 2\pi \delta(x)\delta(y)\mathbf e_z. $$ This is correct in distribution sense, because when we integrate both sides over the surface and apply the Stokes theorem transformation to the left-hand side, we obtain the same result $2\pi$.