7. Use the divergence theorem to calculate the outward flux of F through the closed surface S. (a) ♬ = r²i + y² + 22k and S is the region cut from the solid cylinder z² + y²

16. Use the Divergence Theorem to find the total outward flux of F = xi + 2yj + 3zk across the surface of the sphere defined by 4 = x² + y² + z².

Find the centroid

Suppose F(x, y) = (x² + 6y, 2x − 2y²). Use Green's Theorem to calculate the circulation of around the perimeter of the triangle C oriented counter-clockwise with vertices (12, 0), (0, 6), and (-12,0). F·dr =

2) Find the outward flux of F across the region D where F(x, y, z) = (In(x² + y²), –tan-(), z Vx² + y² ) and the region D is the thick-walled cylinder 1 < x + y < 2 and –1 < z< 2.

Suppose F(x, y) = (x² + 3y, 7x − 7y²). Use Green's Theorem to calculate the circulation of F around the perimeter of the triangle C oriented counter-clockwise with vertices (6, 0), (0, 3), and (-6, 0). Jo F. dr =

Show that the circulation of F = (x², y², z(x² + y²)) around any closed curve on the surface of the cone z² = x² + y² is equal to 0.

az. Suppose F = (2xz + 3y²) a, + (4yz²) a;. (a) Calculate S[F·dS, where S is the shaded surface in Figure 1. (c) Based on your results for parts (a) and (b), what named theorem do you think is being satisfied here, if any? (b) Calculate SF· dl, where C is the A → B → C → D → A closed path in Figure 1. az C C (0,1,1) D (0,0,0) (A ay В ax Figure 1: Figure for Problem 1.

Use the Divergency Theorem to evaluate fF. Nds and find the outward flux of F through the surface of the solid bounded by the graphs of the equations: F(x, y, z) = (xy² + cosz)i + (x²y + sinz)j + e³k; S: z =√x² + y², z = 2.

Calculate F - dS where F = (e" sin y, e" cos y, yz²) S is the surface of the box bounded by the planes x = 3, y = 2, z = 1, and the coordinate planes.