Verify Stokes\' Theorem by computing both: integrate integrate S ( Times F) . n

Question

Verify Stokes' Theorem by computing both: integrate integrate S ( Times F) . n dS and integrate s F . dr where the surface S and the vector field F are given below. In picking an orientation of S, assume that you are located at the point (0,0,10^3) and the normal to the surface is pointing more toward you than away from you. Choose your orientation for S in a manner consistent with this choice. Let S:={(x,y,z) : x^2 + y^2 + z^2 = 25, z LE 0} and F(x,y,z) = (x,z,-y) Let S := { (x, y, z) : z = 36 - (x^2 + y^2), z GE 0 } and F(x, y, z) = (y^3, -x^3, z^2).

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