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

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 =

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.

find the upward flux of F={x2,y2,z2} through z= sqrt(x2+y2), 0≤ z ≤2

Use the divergence theorem to calculate the flux of F = (x - 9y)i + (y – 5z)j + (z – 3x)k out of the unit sphere.