Math 2210 Homework #10
Due at the beginning of recitation on November 22cnd
1. Evaluate the following line integrals using Green’s Theorem. Orient all
curves counterclockwise unless indicated otherwise.
Z
(a)
y 2 dx +x2 dy where C is the boundary of the unit square 0 ≤ x ≤ 1,
C
0 ≤ y ≤ 1.
Z
(b)
xydx + (x2 + x)dy where C is the triangle with vertices (1, 0),
C
(0, 1) and (−1, 0).
Z
(c)
F~ · d~r where F~ (x, y) = hx + y, x2 − yi and C is the boundary of
C
√
the region enclosed by y = x2 and y = x for 0 ≤ x ≤ 1.
Z
(d)
F~ · d~r where F~ (x, y) = ln x + y, −x2 and C is the rectangle
C
with vertices (1, 1), (3, 1), (1, 4) and (3, 4), oriented clockwise.
Z
1
2. Use the formula A =
xdy − ydx to find the area of the following
2 C
regions.
(a) the region between the graph of y = x2 and the x-axis for 0 ≤ x ≤ 2.
(b) the region between x-axis and the cycloid parameterized by ~r(t) =
ht − sin t, 1 − cos ti for 0 ≤ t ≤ 2π
(c) (Optional, skip this one if you want) the region in the first quadrant
3t
inside the folium of Descartes, which is parameterized by x = 1+t
3
2
3t
and y = 1+t
3 for t 6= −1. (Part of the problem is to figure out what
range of t values correspond to the boundary of the region. A hint
for reducing the amount of computation is that d( xy ) = xdy−ydx
.)
x2
3. Find the divergence and curl of F(x, y, z) = xy 2 z 2 i + x2 yz 2 j + x2 y 2 zk
4. Let f be a scalar field and F a vector field. State whether each expression
is meaningful. If not, explain why. If so, state whether it is a scalar field
or a vector field.
(a)
(b)
(c)
(d)
(e)
curl f
grad f
grad(div F)
div (div F)
(grad f )×(div F)
5. Find a parametric representation for the surface of the part of the
√ sphere
x2 + y 2 + z 2 = 36 that lies between the planes z = 0 and z = 3 3
6. Find the area of the surface of the part of the plane x + 2y + 3z = 1 that
lies inside the cylinder x2 + y 2 = 3.