How does Green's theorem and Stokes' theorem generalize the fundamental theorem of Calculus

Question

I've read in few places that Green's theorem $$ \oint_C L dx + M dy = \iint_{D} \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right) dx dy $$ is a generalization of fundamental theorem of calculus. And same with Stokes' theorem. I assume all these can be described in some differential geometry language which I am not so familiar with. Could someone explain how Green and Stokes theorem are generlisation of the FTC? It would be appreciated! thank you!

Answer 1

Let $D$ be the rectangle $[a,b] \times [0,1]$, say, and suppose that $L(x,y)=0$ identically in $D$, and $M(x,y)=f(x)$ depends only on $x$.

Then the formula $$ \oint_C L dx + M dy = \iint_{D} \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right) dx dy $$ reduces to $$f(b)-f(a)=\int_a^b f'(x) \, dx \,.$$