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.
For the surface integral I got,
integrate [x=0 - 1] , [y=0 , 2-2x] ( (1/14)*(54 - 38x - 27y) dx dy )
I got the y limit setting z=0 in the plane equation , ==> (y/2) + x = 1==> 2x+y=2 ==> y=2x-2
(Am i correct???)
When working it the other way around using curl, i got
Curl F = -i + 2k
N = unt normal vector to the surface =k
N.k = 2/7
(Curl F). N = -2/7
Fnally got the integrate [(-2/7) ÷ (2/7) dx dy] , limits : [x=0 - 1] , [y=0 , 2-2x]
Finally I got -1 as the answer here.
Could someone point out my mistakes please?
Many thanks!