According to my professor, this formula is called the integral formula of Gauss and Green. I've tried searching for it online but I could find nothing similar to it.
B here is a region that's bounded by the simple closed curve γ(t).
I don't really understand what is it that we get after applying this formula. I mean, when we calculate the definite integral of f over the Intervall [a,b], we get the area under the curve of f from x=a to x=b. And double integrals are used either to calculate the area of a region or the volume above it that's bounded by the surface of a multi-variable function. But what is it that we calculate using this formula?
This formula is supposed to be something slimier to the fundamental theorem of calculus but in multi-variable calculus. I cannot see how.
Thanks for all your help. And forgive me for the grammatical mistakes, if there is any. English isn't my first language.
Rewrite the right-hand side as a line integral $$ \int_a^b\det\pmatrix{f_1&\gamma_1'\\f_2&\gamma_2'}\mathrm{dt}= \int_A^b f_1\gamma_2'-f_2\gamma_1'\mathrm{dt}=\int_{\partial B}f_1\mathrm{dy}-f_2\mathrm{dx} $$ where $\partial B$ is the positively-oriented boundary of $B,$ parameterized as $(\gamma_1(t),\gamma_2(t)),a \leq t\leq b$
Now it's somewhat reminiscent of the fundamental theorem of calculus isn't it? On the right hand side, we're integrating some function over the boundary. On the left-hand side we're integrating some kind of derivative of that function over the whole area.
This idea can be made precise and generalized to any number of dimensions and in other ways, but that takes a lot of work.
Integrals have many more interpretations and uses than simply areas or volumes. If they didn't, they wouldn't be nearly as important as they are.