Tutorial Exercise Let S be hemisphere x2 + y2 + z2 = 16 with z > 0, oriented upward. Let F(x, y, z) = xPeyzi + y2exZj +

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Tutorial Exercise Let S Be Hemisphere X2 Y2 Z2 16 With Z 0 Oriented Upward Let F X Y Z Xpeyzi Y2exzj 1 (42.1 KiB) Viewed 87 times

Tutorial Exercise Let S be hemisphere x2 + y2 + z2 = 16 with z > 0, oriented upward. Let F(x, y, z) = xPeyzi + y2exZj + z2exYk be a vector field. Use Stokes' theorem to evaluate slo curl F. ds. Step 1 of 3 Recall that by Stokes' theorem, Slee curl F. ds = fe F. dr, where S is an oriented surface with boundary C that is a simple closed curve with positive orientation. We also recall that orientation of the surface induces the positive orientation of C, which is the orientation observed if one thinks of walking along the curve C with one's head in the same direction as the normal vectors with the surface always to the left. First, we find parametrization r(t) of the boundary curve C for the surface x2 + y2 + 22 = 16, z 20, which is a circle centered at the origin with radius 4 If one imagines walking around the circle with one's head pointing upward in the same direction as the unit normal vectors of S, one must walk counterclockwise counterclockwise to have the hemisphere to the left. A suitable parameterization in R3 of the boundary circle C is given by the following. r(t) = (4 cos(t), 4 sin(t), o), o St 527 Step 2 of 3 We now recall that Ses F. dr = = fºrce F(r(t)). r'(t) dt, where r(t), defined on [a, b], is the parameterization of C. We calculate r(t). r(t) = (4 cos(t), 4 sin(t), o) r(t) = (-4 sin(t), 4 cos(t), o) Also, we have F(x, y, z) = (x2eyz, y exz, z2e\Y), so we find F(r(t)). Fr(t)) = F(4 cos(t), 4 sin(t), 0) Submit Skip (you cannot come back)