I'm trying to solve a problem which is giving me some troubles, it's about applying Stokes' Theorem on the surface defined by:
$ x = 0, y = 0, z = 0 , (x + y + z) = 2 $
of the field
$ F = (x, -y, z) $
Whith the orientation n pointing towards the axis origin O.
The surface Σ should look like a triangular pyramid, with one of the vertices in the origin O. So I thought to just consider the pyramid face opposed to the vertex in O, so the one with the equation
$ (x + y + z) = 2 $
So i calculated
$ \int_{\delta (+) \sum }^{ } F * dr $
With Stokes' Theorem, considering
$ z = 2 - x - y $
and it returns
$ (\int_{0}^{2 - x}\int_{0}^{2} 2) = 4 $
So 4 or -4 depending on the orientation...
Then i'm requested to compute
$ \int_{\delta (+) \sum }^{ } F * dr $
with a direct method, using line integrals. Here comes my problem: how do I set the line integrals? Should I just consider
$ (x + y + z) = 2 $
or the other faces too?
I'm sorry for the bad English... thanks in advance!