Use Stokes Theorem to evaluate a Double Integral

Question

Use Stokes’ Theorem to evaluate double integral S curl F.dS. F(x,y,z)=e^xyi+e^xzj+x^zk, S is the half of the ellipsoid 4x^2+y^2+z^2=4 that lies to the right of the xz-plane, oriented in the direction of the positive y-axis



Answer 1

Actually Stoke's theorem can be used to evaluate the Electric or magnetic flux passing through the surface or sometimes the flow from a surface as in the case of fluid flow. The Line integral along the "boundary curve" can be substituted for the difficult double integral of the curl F dot ds. S bounded below by the xy plane and on the right by the xz plane thus forming a boundary curve as the line of intesection of S and the xy plane (z=0). Let r be a vector from the origin to this boundary curve where x=Acost and y=Bsint becoming r=.25costi +2sintj. dr becomes (-.25sinti+2costj)dt. F becomes 0 i - e^.5costsintj+k and F dot dr = -2coste^.5costsintdt. The limits of this integration are from t=0 to t=pi. I leave the rest to you