Each problem is worth 3 points. Be sure to follow the guidelines.
1. Find a parameterization of the part of the plane plane 2x + 3y + z = 6 that lies in the first octant.
2. The equation x2
A + y2
B + z2
C = 1
defines a shape called an ellipsoid. It is the three-dimensional analogue of an ellipse. Find a parameterization of the ellipsoid x2 4 + y2 9 + z2 = 1.
3. Compute the curl and divergence of the vector field 〈xy2z2, x sin(yz), yexy〉.
4. Is 〈x, y, z〉 the curl of a vector field on R3? If it is, find a vector field ˆF whose curl is〈x, y, z〉. If not, explain why.
5. Review problem. Find and classify the critical points of f(x, y) = 10x + x2y − xy −6y.
6. Review problem. Find the curvature of ˆr(t) = 〈t3 − t, t3 − 4, t2 − 1〉 at the point (0, −3, 0).
