Stoke's theorem and the Divergence theorem.
When I take multivariable calculus in high school, my friends and I came across a question regarding those two theorems. We asked our teacher, but she couldn't help.
so, for a line integral, if the path is closed and the filed F is conservative, the line integral of the field alone the closed path should be zero.
if stokes theorem relates the line integral to the surface integral, and divergence theorem relates surface integral to triple integral over a volume
And, given a non-conservative field F and a closed path, the line integral won't be zero. But if you combine those two theorem and establish an equation, you will get the line integral equal to div(curl(F)) integrated over a volume dxdydz. BUT, div(curl(F)) is always zero. that is, combining those two theorem to evaluate a non-conservative field over a closed path will produce zero as an answer, but in fact only conservative field on closed path gives you zero. would this be a contradiction?
Sorry about my inaccurate description I don't know how to use math symbols on this.
Re: Stoke's theorem and the Divergence theorem.
just found that same question has been asked in math challenge. maybe the symbols and signs may help explaining the question.