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).
mathematics
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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).
