(10 points) Let S be the upper hemisphere of the unit sphere x² + y²+2² = 1. Let F(x, y, z) = (x, y, z). Calculate the surface integral J F F-dS. S

Describe the loci zz−2z−2z+8 = 0 expressed in terms of conjugate coordinates z, z.

Write the sphere equationil ada lo enoitaspo onte g bas rotoov bai (d) x2 + y? + 2? + 2x – 4y – 6z = 11 in standard form, and hence find the center and radius of this sphere.

Find both parametric and rectangular representations for the plane tangent to r(u,v)=u2i+ucos(v)j+usin(v)kr(u,v)=u2i+ucos⁡(v)j+usin⁡(v)k at the point P(4,−2,0)P(4,−2,0).One possible parametric representation has the form⟨4−4u⟨4−4u ,  , 4v⟩4v⟩(Note that parametric representations are not unique. If your first and third components look different than the ones presented here, you will need to adjust your parameters so that they do match, and then the other components should match the ones expected here as well.)The equation for this plane in rectangular coordinates has the form x+x+  y+y+  z+z+  =0

Find the area of the part of the surface x = 2z + y² – 3, that lies above the triangle in the XV-plane with vertices (0,0), (2,1), and (0,1). 3.