MATH 210
Homework due 03/16/2012
1. Show that the vector field:
F(x, y, z) = h− sin x, cos y , 1i
is conservative and compute its line integral along the path:
r(t) = hπ+t2(t−1)6, π/2 + t3(t−1)9,3t2+ sin(πt)i
where 0 ≤t≤1.
2. Compute the vector line integral:
ZC
(3 −4y)dx + (6 −4x)dy
where Cis the broken line with vertices:
(0,0),(1,1),(2,0),(3,1),(4,0),(5,1),(6,0),(7,1),(8,0),(9,1),(10,0)
3. Prove or disprove that the vector field:
F(x, y) = y
x2+y2−y, −x
x2+y2−x
is conservative.
4. Show that the vector field F(x, y, z) = h2xy +z, x2+ 3y2z , y3+x+ 1i
is conservative and compute its line integral along the path:
r(t) = ht5, t7, t9i0≤t≤1
5. Find the minimum and the maximum of the function
f(x, y) = x−3y+z
on the surface of the sphere of radius 1 centered at the origin.
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