> MAT 267 Test #3a Spring 2008 N. Brewer Name:_________________________
Instructions:The exam is worth a total of 100 points; please make sure your exam has all pages before you begin.Show all work in detail or your answer will not receive any credit. Include appropriate units on all questions that apply. Write neatly and box all answers.Please use the back of your test if you need scratch paper.
Part I - Conceptual Questions (25 points)
Instructions: Give short complete answers to the following.
1.[5 points] How do you check whether or not a vector field F = < P, Q > is conservative?
2. [5 points] What is the triple integral formula for the volume of a 3D object E ?
3. [5 points] What are the conversion formulas to convert (x, y, z) into spherical coordinates (ρ, θ, φ)?
4. [5 points] What does it mean if F is conservative?
5. [5 points] How do you determine if a critical point is a local max., local min. or saddle point?
Part II - Show all your work. (75 points)
1.[10 points] Find the potential function f for the gradient vector field F = < >
2. [15 points] Calculate the work done by 
along the curve C given by r(t) = < 
W = 
C
3. [10 points] Find and classify all critical points for the following function:
4. [15 points] Evaluate
where D is the region bounded by 
and 
and 
. D
5. [10 points] Use Green's Theorem 
to evaluate
C D
where D is the region bounded by the circle 
C
6. [15 points] Let E be bounded by the spheres 
and 
. Evaluate
E
+2 Bonus: If you turned in all your homeworks, you get this bonus.