By the book's reasoning the two forms of Green's theorem are equivalent because if let F= G1 for the tangential form, we'd obtain the equation of the normal form of green's theorem and if assumed F=G2 in the Normal Form, we'd obtain the equation of the Tangential Form.
How does being able to assume different vector fields F and plugging that vector field F in to obtain the other counterpart/side of Green's theorem/ the other side of the coin of Green's theorem imply that the tangential forma and normal form are equivalent?
Why/how are there two versions of Green's theorem that are equivalent? The two forms don't look the same to me.
We're substituting different vector field Fs in to make the circulation convertible to normal form and vise versa to show that theorem 4 equivalent to theorem 5 and that doesn't make sense or sound correct as a proof.
Below: they used this switching technique of vector fields that I'm uncomfortable with.
I don't think what they're doing makes sense:
Best Answer
The equivalence between these two forms of the theorem isn't a statement that the values of those integrals are equal. They are not. However, the theorem says that equations (3) and (4) hold for all vector fields $F$, including those which produce the alternate form. So they are equivalent in that proving that one of the equations holds for all $F$ implies that the other holds for all $F$, and vice versa.