Submit your Mathematica file  (and a pdf version)  onto Canvas.   You  can  convert  a  cell to from math input to  text  and vary the font when  explaining your work.   Use  complete sentences to describe your results. Please see the files Chapter16 .4 .nb and Chapter16 .5 .nb for examples. Be sure to provide work to justify how you set up your integrals.

1. Green’s Theorem: Let F(x,y) = ⟨2x3 − y3 , x3 + y3 ⟩ and C be the unit circle.

(a) Graph the vector field along with C in Mathematica.

(b) Verify Green’s theorem for \C F · dr. This means computing a vector line integral

and an area integral, then showing that they give the same value. You can do this in Mathematica or on paper.

(c) Verify the flux form of Green’s theorem for \C F · nds (see p. 1032).  This means computing a vector line integral for the flux across C  and an area integral, then showing that they give the same value. An example of a flux integral is on p. 978. You can do this in Mathematica or on paper.

2. Parameterized  surfaces:  Let G(u,v)  =  (u cos v, u sinv, v) for 0  ≤  u  ≤   1 and 0 ≤ v ≤ 2π .

(a) Graph the parameterized surface in Mathematica using ParametricPlot3D.     (b) Compute the area of the surface. You can do this in Mathematica or on paper.

3. Surface Integrals of Vector Fields : Let S be the upper hemisphere

{(x,y,z)|x2 + y2 + z2  = 1, z ≥ 0}

oriented by the normal pointing out of the sphere.

For each vector field

F1 (x,y,z) = ⟨x,y,0⟩ ,     F2 (x,y,z) = ⟨y,x,0⟩ :