Question

Difference between stokes theorem and divergence theorem

Answer 1

Recently learnt the two and I really can't tell the difference. I'm not sure if I'm missing something, but it really seems to me that they evaluate the same thing just using different methods.
With Stokes' Theorem, it seems to me that we evaluate the flux surface integral of a vector field with the double integral of the curl of the vector field dotted with the tangent vector component.
Then with the Divergence Theorem, it seems that we evaluate the same thing, except taking the triple integral of the divergence of the vector field...

I'm not sure if I'm misunderstanding it and I tried searching online but they only tell me methods to use them. Can someone explain in simple terms how to differentiate these?

Here is a question that might help clear my confusion.
Use two methods to calculate the flux integral

∫S(∇×F)⋅dS∫S(∇×F)⋅dS


where F=(y,z,x2y2)F=(y,z,x2y2) and SS is the surface given by z=x2+y2z=x2+y2 and 0≤z≤40≤z≤4.

I don't necessarily need a full method on this, but just (I guess) which theorems relate to this.
It looks like Stokes' Theorem, with the curl there... so this would be Stokes Theorem'. So first method is via the given equation, and then second would be evaluating the line integral with boundary of SS? But what would be the boundary...? Would it be the circle when z=0z=0 or the one at z=4z=4?