5. Consider the surface S given by r(u, v) = 3 cos(u) sin(v)i + 3 sin(u) sin(v)j + 3 cos(v)k for 02 and 0≤v. (Hint: What

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5 Consider The Surface S Given By R U V 3 Cos U Sin V I 3 Sin U Sin V J 3 Cos V K For 02 And 0 V Hint What 1 (41.43 KiB) Viewed 14 times

5. Consider the surface S given by r(u, v) = 3 cos(u) sin(v)i + 3 sin(u) sin(v)j + 3 cos(v)k for 02 and 0≤v. (Hint: What kind of surface is this?) (a) (20 pts) Find the unit normal vector for this surface (Hint: when scaling, consider first factoring out sin(v)). Use it to find the tangent plane for the surface at the point corresponding to (u, v) = (0, π/2). (b) (15 pts) Let F(x, y, z) (x+y)i + (x-y+z)j + (z-y)k. Find the both the divergence and the curl for F. Is the vector field conservative? (c) (20 pts) Find the flux of the vector field F from part (b) outward through the surface S given by r(u, v). (Hint: Use one of our integral theorems.) 2 (d) (15 pts) Let q(x, y, z) = x² + y² + z be the charge density for the surface S. Find the total charge Q of the surface (not a volume).