Math 21D
Practice Midterm #2
1. Consider the curve parametrized by r(t) = et
costi + et
sin tj, with 0 ≤ t ≤ 4π.
a. Compute the total arc length of the curve.
b. Find the curve’s unit tangent vector T.
c. Find the curve’s principal normal vector N.
d. Find the curve’s curvature κ.
2. Let C be the curve in the plane parametrized by r(t) = (t, t2, t3), −1 ≤ t ≤ 1. Let F be the vector field
F(x, y) = 2x
2
i − zj + xyk.
Compute the circulation of F along C.
3. Let F be the vector field xi + (y − x)j. Compute the flux of F across the unit circle centered at the origin.
4. Let F be the vector field
F(x, y, z) = (3y2 + 6z)i + (6xy + 2z)j + (6x + 2y).
a. Use the curl test to verify that F is a conservative vector field.
b. Find a potential function for F.
c. Now suppose F is a force field. Compute the work F does on a particle moving along a straight line joining (0, 0, 0) and (1, 2, 3).