Complete the following:
Stokes and Gauss' Theorem
Q1. A vector ?eld v(x, y, z) is given by the formula
v(x, y, z) = xyx^ - y2y^.
Consider a square path in the xy plane which starts at (0,0,0) and moves along the corners (1,0,0), (1,1,0) and (0,1,0). Calculate the path integral of v, i.e. v · dr, and calculate the area integral of the divergence, R ∇ × v · da, and verify that Stokes' theorem holds. (Note: For the ?rst leg of the path, dr = ˆxdx, and for the second leg of the path, dr = ˆydy. The area element here is da = dxdyˆz, integrated over the square.)
Q2. Given a vector t = -ˆxy + ˆyx, use Stokes' theorem to show that the integral around a closed curve of arbitrary shape in the xy plane
1/2 ?t.dr=A
where A is the area enclosed by the curve. (Hint: Use Stokes' theorem to write the integral in terms of the curl of the vector. What does the integral now represent?)
Q3. Show that where V is the volume enclosed by the surface S, and r = xˆx + yˆy + zˆz.
1/3 ∫∫s r.da=V