I'm wondering if it is possible to use Green's theorem to prove that the rotation of a conservative vector field F (x,y,z)=(F_{1}(x,y,z), F_{2}(x,y,z), F_{3}(x,y,z)) is equal to 0
And if so, how?
I'm wondering if it is possible to use Green's theorem to prove that the rotation of a conservative vector field F (x,y,z)=(F_{1}(x,y,z), F_{2}(x,y,z), F_{3}(x,y,z)) is equal to 0
And if so, how?
Well, since your vector field has three components, you would use the three dimensional version, the "Stokes theorem":
$\int_C \vec{F}\cdot d\vec{r}= \int_A\int \nabla\times \vec{F} d\vec{s}$.
Now whether you can use that to prove that "the rotation of a conservative vector field is equal to 0 depends on exactly how you have defined "conservative vector field". I believe that many texts use "rotation is equal to 0" or rather "the integral around a closed path is 0" as the definition of "conservative vector field.
If, instead, you define a "conservative vector field" to be one such that its curl is identically 0, so that the right hand side is 0, then, yes, the Stoke's theorem will give that result.
If you define it by assuming that it has a potential function who's gradient is equal to the vector field. And id you were to take that functions antiderivative and calculate that function, lets call it K, from one point in space to the same point so K (end point) to K (start point) you'd get zero.