Verification of Stokes Theorem

Question

Could someone explain how do we verify stokes theorem for the vector field F=zi + (2x+z)j + xk taken over the triangular surface S in the plane (x/1)+(y/2)+(z/3)=1 bounded by the planes x=0 y=0 and z=0. Take boundary of the above triangular surface as the path of the line integral.

Answer 1

Sketch the plane in each of the normal planes formed by the basis vectors, xy, xz, yz. Then use this to sketch the 3d plane and get a better visualization of it, parametrize each of the three lines that form the triangle. and use the formula  ∫ F*dr =  ∫ F(r(t))*r'(t) and because they're 3 lines that aren't smoothly connected or easy to parametrize the 3 at once, split it up into 3 integrals.

For Stoke's Theorem (3D Green's Theorem), this states ∫ F*dr =  ∫∫ curlF*dA so solve that like any surface integral, you can use the general formula  ∫ F*dA =  ∫∫ F*gradient(surface which is just the plane x+.5y+(1/3)*z=6)dxdy and just set ur bounds correctly and you should get the same answer

Note: this is not supposed to be easy, it's supposed to be hard/annoying to calculate the line integral the traditional way, so you can see that it is easier with Stoke's theorem.