## Use Green's Theorem to evaluate Z C y 2 + x 3 dx + x 4 dy, where C is the counter-clockwise parametrization of the perimeter of the unit square with vertices at (0, 0),(1, 0),(1, 1) and (0, 1).

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This assignment is to be submitted on MyCourses by 11:59 Friday April 17th.

1. Use Green's Theorem to evaluate Z C y 2 + x 3 dx + x 4 dy, where C is the counter-clockwise parametrization of the perimeter of the unit square with vertices at (0, 0),(1, 0),(1, 1) and (0, 1).

2. Let C be a simple closed curve that bounds a region D to which Green's Theorem applies. Show that Area(D)=Z C xdy = − Z C ydx.

3. Let φ(u, v) = (u−v, u+v, uv) and let D be a unit disc in the uv-plane. Compute the surface area of φ(D).

4. Evaluate the surface integral Z S F · dS, where F(x, y, z) = (x, y, z2 ) and S is parametrized by φ(u, v) = (2 sin(u), 3 cos(u), v) with 0 ≤ u ≤ 2π and 0 ≤ v ≤ 1.

5. Calculate Z S CurlF · dS, where S is the hemisphere x 2 + y 2 + z 2 = 1 with x ≥ 0 and F = (x 3 , −y 3 , 0).