19 please Find the net outward flux of F = a Times r across any smooth closed su

Question

19 please Find the net outward flux of F = a Times r across any smooth closed surface in R^3, where a is a constant nonzero vector r = (x, y, z). Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. F = (x, -2y, 3z); S is the sphere {(x, y, z): x^2 + y^2 + z^2 = 6}. F = (x^2, 2xy, y^2); S is the surface of the cube cut from the first octant by the planes x = 1, y = 1, z = 1. F = (x, 2y, z); S is the boundary of the tetrahedron in the first octant formed by the plane x + y + z = 1. F = (x^2, y^2, z^2); S is the sphere {(x, y, z); x^2 + y^2 + z^2 = 25}. F = (y - 2x, x^3 - y, y^2 - z); S is the sphere {(x, y, z): x^2 + y^2 + z^2 = 4}.

Explanation / Answer

19)F=<x,2y,z>

divF=1+2+1 =4

x+y+z=1

0<=z<=1-x-y

0<=y<=1-x

0<=x<=1

flux = EdivF dv

= [0 to 1][0 to 1-x][0 to 1-x-y] 4 dz dy dx

= [0 to 1][0 to 1-x][0 to 1-x-y] 4 z dy dx

= [0 to 1][0 to 1-x]4(1-x-y-0) dy dx

= [0 to 1][0 to 1-x]4(1-x-y) dy dx

= [0 to 1][0 to 1-x]4(y-yx-(1/2)y2) dx

= [0 to 1]4((1-x)-(1-x)x-(1/2)(1-x)2)-0 dx

= [0 to 1]4(1/2)(x2-2x+1) dx

= [0 to 1]2(x2-2x+1) dx

=[0 to 1]2((1/3)x3-x2+x)

=2((1/3)13-12+1) -0

=2/3

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