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Stokes and Gauss' Theorem

1. A vector field v(x, y, z) is given by the formula

v(x, y, z) = xyˆx − yˆ2y.

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 first 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.)

2. 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

(see attachment for formula)

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?)

3. Show that (see attachment for formula)

where V is the volume enclosed by the surface S, and r = xˆx + yˆy + zˆz. (Hint: This is similar to the
previous problem, but using Gauss' theorem instead of Stokes' theorem.)

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