>>13739196>>13739124>"not a math student">wants to learn cohomologyUhhhhhh.... good luck...?
I'll leave you with these, though (even though I don't really understand the subject):
Say you have some vector field F, and curl(F) = 0. If you know that this is true on a region "with no holes" (partial derivatives exist and continuous, etc.), then you can conclude that F = grad(f) for some scalar function f (meaning F is conservative). So if you integrate over a closed loop, the integral will be zero. (because the two ends are the same)
However, problems arise when the region DOES have holes. Cohomology is a way of detecting holes. And when you have holes in your region, then it's possible that F is not conservative, and it's possible that integrals on closed loops are not zero. Paraphrasnig Terry Tao: de Rham Cohomology measures the failure of the fundamental theorem of calculus in higher dimensions.
For example, if you integrate on a closed loop containing the origin (the origin being a hole), then you don't get zero (I think you get 2pi). What are the other vector fields that have a curl of zero on this punctured plane? Because if we know that, we might know all the fields that have this problem. If you take any scalar function's gradient, then it obviously has a curl of zero, so we don't care about those. And if you have any functions with a curl of zero, you can form linear combinations of them that also have a curl of zero; so they form a vector space.
Turns out, on the punctured plane this space of fields is 1-dimensional, meaning the only vector fields with a curl of zero are scalar multiples of F that I showed you above (up to adding conservative fields, which we don't really care about).
There's a lot more to it, but I only gave you the baby parts (which are the only thing I know).
