For each of the following, find a (piecewise) smooth parametrization of S which agrees with the induced orientation, and compute S F Tds a) S is the truncated paraboloid y 9 x2 z2, y 0, with outward pointing normal, and F(x, y, z) (x2y, y2x, x y z) b) S is the portion of the plane x 2 y z 1 which l...

a The boundary is 9 x 2 z 2 y 0 with counterclockwise orientation when viewed from far out the posit ...

For each of the following, find a (piecewise) smooth parametrization of S which agrees with the induced

Question:

For each of the following, find a (piecewise) smooth parametrization of ϑS which agrees with the induced orientation, and compute ∫ϑS F • Tds.
a) S is the truncated paraboloid y = 9 - x2 - z2, y > 0, with outward-pointing normal, and F(x, y, z) = (x2y, y2x, x + y + z).
b) S is the portion of the plane x + 2 y + z = 1 which lies in the first octant, with normal which points away from the origin, and F(x, y, z) = (x - y, y - x, xz2).
c) S is the truncated paraboloid z = x2 + y2, 1 < z < 4, with outward-pointing normal, and F(x, y, z) = (5y + cos z, 4x - sin z, 3x cos z + 2y sinz).