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

Determine the flux of F(x, y, z) = < −x2 + x, y, 8x3 − z + 9 > across the surface with an upward orientation. Let the surface be the portion of the paraboloid z = 9 − 4x2 −4y2 on the first octant above the plane z = 1.

Find the flux of F=z2kF=z2k upward through the part of the sphere x2+y2+z2=a2x2+y2+z2=a2 in the first octant of 3-space.

II. Let S be the portion of the cylinder x² - 6x + z = 4, oriented by upward-slanting normal vectors, that is cut by the planes y = 1, y = 4, and z = 0. Find the flux of F(x, y, z) = (xy, 3xz², yz) across S.

Calculate the line integral f (−x²y − 3x + 6y + 7) dx + (7xy + 1) dy, where C' is the rectangle with vertices (−2, −3), (−2, 2), (-4, 2), and (-4,-3) oriented counterclockwise. Enter an exact answer. Provide your answer below: fc(-x²y-3x+6y + 7) dx + (7xy+1) dy =

Use the Divergence Theorem to find the flux of F(x, y, z)=z³ i-x³j+y³ k across the sphere x² + y² + z² = a² with outward orientation. $ = i

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.

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.

Calculate the line integral f (2x²y + 10x + 6y − 6) dx + (−6xy + 7) dy, - where C' is the rectangle with vertices (−1, 1), (−1, 2), (-4,2), and (-4, 1) oriented clockwise. Enter an exact answer. Provide your answer below: fc(2x²y + 10x + 6y - 6) dx + (−6xy + 7) dy =