Mathematics 241 – Multivariable Calculus Exam 3 November 30, 2007

Information and Directions

Do all work in the blue exam booklet. Do not place anything you want considered for credit on this sheet. There are 100 possible points, plus the Extra Credit problem. The Extra Credit problem will be counted only if it raises your average.

I. A) (15) Let F be the vector field

F(x, y, z) = (xz − y^{2}, xy + z, 2x + sin(y)).

Compute the divergence div(F) and the curl curl(F). B) (5) Show that if F = (F_{1}, F_{2}, F_{3}) is any vector field on a domain in R^{3} whose compo-

nent functions have continuous second-order partial derivatives, then div(curl(F)) = 0.

II. All parts of this problem refer to the function

f(x, y) = x^{3} + y^{2} − 6xy + 6x + 3y.

A) (10) Determine all critical points of f. B) (15) Use the Hessian Criterion (second derivative test) to classify each of the critical

points as a local maximum, a local minimum, a saddle point, or a degenerate critical point.

III. (20) A mountain has the shape of the graph z = f(x, y) = 4 − x^{2} − \frac{y^{2}}{4} , where x, y are east-west and north-south map coordinates respectively, and z is the altitude (in thousands of feet). If you hike along a trail on the mountain described by the circle g(x, y) = x^{2}+y^{2} = 1 on the map, what are the highest and lowest altitudes that you reach?

IV. All parts of this problem refer to the region D in the right half-plane (x ≥ 0), bounded by y = x, y = −x and x^{2} + y^{2} = 9. A) (5) Sketch the region D. B) (10) What would the limits of integration be to compute ∫ ∫_{D} f(x, y) dx dy integrating

with respect to y first, then x? C) (10) What would the limits of integration be to compute ∫ ∫_{D} f(x, y) dx dy integrating

with respect to x first, then y? D) (10) Using any applicable method, compute ∫ ∫_{D} x dx dy.

Extra Credit (10) Compute the triple integral of the function f(x, y, z) = e^{(x^{2}+y^{2}+z^{2})^{3/2}} over the unit ball x^{2} + y^{2} + z^{2} ≤ 1 in R^{3}. (A good choice of coordinates is essential here!)