Use stoke's theorem to evaluate∬m(∇×f)⋅ds where m is the hemisphere x^2+y^2+z^2=9, x≥0, with the normal in the direction of the positive x direction, and f=⟨x^5,0,y^1⟩. begin by writing down the "standard" parametrization of ∂m as a function of the angle θ (denoted by "t" in your answer)

By Stokes' theorem,



where is the circular boundary of the hemisphere in the - plane. We can parameterize the boundary via the "standard" choice of polar coordinates, setting



where . Then the line integral is




We can check this result by evaluating the equivalent surface integral. We have



and we can parameterize by



so that



where and . Then,



as expected.