33 points my notes question part points submissions used use stokes' theorem to evaluate s curl f · ds. f(x, y, z) = zeyi + x cos(y)j + xz sin(y)k, s is the hemisphere x2 + y2 + z2 = 16, y ≥ 0, oriented in the direction of the positive y-axis.

f x y z 4y cos (< x >) y x sin z z xy k s is the hemisphere x2 y2 z2 25 z ≥ 0 oriented upward

use stokes theorem to evaluate f dr

s is the hemisphere x 2 y 2 z 2 16

consists of the top and the four sides but not the bottom of the cube with vertices oriented outward

stokes theorem triangle with vertices

f xyz x y 2 i y z 2 j z x 2 k where k is the triangle with vertices

∫ f ∙ dr c when f x y z 2xi 3zj xk and c is the triangle with vertices 0 0 0 1 1 1 and 0 0 2

s is the part of the paraboloid z 1 − x2 − y2 that lies above the xy plane oriented upward