Consider the vector field F(x, y, z)=(3xy, 4x2, -3yz+6) Consider also the 3-dimensional region D bounded by the surface S S₁US₂ where 1 = • S₁ = {(x, y, 0) | x² + y² ≤ 1}, the unit disc in the plane z = 0, with boundary circle C = {(x, y, 0) | x² + y² = 1}. S₂ = {(x, y, 1x2 − y²) | x² + y² ≤ 1}, an upside down paraboloid with the same boundary circle C. To visualize all this, think of a bullet standing on its flat end. Let n denote the outward pointing unit normal vector on S. (Note that ñ is only piecewise continuous: it is discontinuous along the common boundary circle C of S₁ and S2; but piecewise continuity is just fine, as it is in Green's Theorem). (a) Verify that divF = 0. (b) Use the divergence theorem to calculate JJ FdS where n is the outward pointing unit normal vector on the surface S. (c) Calculate the surface integral ffs, F.ñ dS using a double integral (Hint: What are the values of F(x, y, z) and of n on the plane z = 0?) (d) Use your previous results to write down the value of the surface integral f√s, F. ñ ds. (Hint: In this problem, there are almost no actual integrals that you have to calculate. If you find yourself buried deep in the calculation of some multiple integral, you are doing it wrong).

For the vector field (x, y, 2) evaluate the line integral along the curve made of three adjacent segments. from (1,2,8) to (3,0,3) from (3, 0,3) to (-1,0, 2) from (-1, 0, 2) to (0,0,7)

f(x,y) = 2x3+4y3-3x2-120x-192y+5 Is f(x,y) a curve, vector field, or surface?

Sketch the vector field represented by F (x, y) = (-x,y)

Let F(x, y) = ⟨xy^2+2x+1, x^2y − 3y⟩ be a vector field. Let C be the curve thatconsists of two line segments C1 and C2, where C1 has vertices (−1, 0) and (0, 1)and C2 has vertices (0, 1) and (1, 0). C is oriented clockwise.(a) Sketch C accurately.(b) Find the work done by F by moving a particle along the path C in the givendirection.

Consider the vector field F(x, y, z) = (3xy, 4xz, -3yz+6) Consider also the 3-dimensional region D bounded by the surface S S₁ US2 where

= Calculate the flux of the vector field F(x, y, z) = (5x + 9)ỉ through a disk of radius 3 centered at the origin in the yz-plane, oriented in the negative x-direction. Flux =

Find the Duf of f(x,y,z) = z In(x+ y) + e3y on the vector u 3i-j-4k at Po(1,0,1)

Show that (x, y) = 5x1y1 – 9x2Y2 for vectors x = (x1, x2) and y = (y1 Y2) in R? defines an inner product.

Find a vector valued function which gives the curve of intersection of  y2 + z2 =16  with the plane 2x+z=-11. (The goal is to parametrize x=x(t), y = y(t), z=z(t) as a function of t, and form a function like r_>(t) = x(t), y(t), z(t)>) The Proffessor then says "The equation y2 + z2 =16 tells us to take  y= y(t)= 4 cost(t) z= z(t) = 4 sin(t)" I don't understand how he got  y= y(t)= 4 cost(t) z= z(t) = 4 sin(t). Please explain in detail.