Question

For the vector fied than the vecto and outgoing arrows. Her can use the formula for F to confirm t n rigtppors that the veciors that end near P, are shorter rs that start near p, İhus the net aow is outward near Pi, so div F(P) > 0 Pi is a source. Near Pa, on the other hand, the incoming arrows are longer than the e the net flow is inward, so div F(P2) 0 and P2 is a sink. We FIGURE 4 The vector field F-x'i+yj his impression. Since F--x2і + y2j, we have F2 2y, which is positive when y x. So the points above the line y-x are sources and those below are sinks. 16.9 EXERCISES 1-4 Verify that the Divergence Theorem is true for the vector field 7. F(x, y. z) - 3xyi + xe F on the region E S is the surface of the solid bounded by the cylinder y 2 + z2 1 and the planes x--1 and x-2 S is the sphere with center the origin and radius2 S is the ellipsoid x 2y 322-4 S is the surface of the tetrahedron enclosed by the coordinate E is the cube bounded by the planes x-0,x 1,y 0 E is the solid enclosed by the paraboloid z -y and the plane z-9 E is the solid ball x2 + y2 + z2 E is the solid cylinder y2+z2 9,0sx s2 10. F(x, y, z) zi+ yj+ zxk 3. F(x, y, z) - (z, y, x), 16 planes and the plane 4. F(x, y, z)-(x2 -y, z), where a, b, and c are positive numbers 5-15 Use the Divergence Theorem to calculate the surface integral F dS; that is, calculate the flux of F across S. 5. F(x, y, z)-xye'i + xyYj-ye' k, 11. F(x, y, z)-(2x? +y?)1+(Уз + z*)jt3y": k, S is the surface of the solid bounded by the paraboloid z .. 1-x2-y2 and the xy-plane S is the surface of the box bounded by the coordinate planes 12. F(x, y, 2)- (xy +2xz)i + ( and the planes x-3, y = 2, and z + y)j (xy- 22)k, S is the surface of the solid bounded by the cylinder x2 + y 2-4 and the planes z-y-2 and z-0 6. F(x, y, z) = x2yzi + xy":j + xyz" k. lr/r, where r S is the surface of the box enclosed by the planes- 0, -a,y- O,y b,z 0, andz c, where a, b, and c are positive numbers x1+yj+rk, S consists of the hemisphere z-i-x2-y2 and the disk 13. F x2 + y2 < 1 in the xy-plane
NO.25 in 16.7 and NO.12 in 16.9 please.

Answer 1

2TT 3

in 2TT 2