Math 2210-1
Chapter 17 Review
1. (constrained optimization from a previous chapter) Find the maximum of f (x, y) = 4x2 − 4xy + y 2
subject to the constraint x2 + y 2 = 1.
2. Sketch the vector field F~ (x, y) = 3x~ı + y~
3. Find the divergence of the following vector fields:
(a) F~ (x, y) = (x2 − y 2 )~i + 2xy~j
(b) F~ (x, y, z) = (−x + y)~i + (y + z)~j + (−z + x)~k
4. Compute the curl of the following vector fields:
(a) F~ (x, y, z) = (x2 − y 2 )~i + 2xy~j
(b) F~ (x, y, z) = x2~i + y 3~j + z 4~k
2
(c) F~ (x, y, z) = ex~i + cos y~j + ez ~k
5. Evaluate the following line integrals
Z
(2x + 9z) ds where C is the curve x = t, y = t2 , z = t3 , 0 ≤ t ≤ 1.
(a)
C
(b) F~ (x, y, z) = x~i + 2zy~j + x~k and C is given by ~r(t) = t~i + t2~j + t3~k for 1 ≤ t ≤ 2.
6. Decide whether or not the following vector fields could be gradient fields, and justify your answer.
(a) F~ (x, y) = (12x2 + 3y 2 + 5y)~ı + (6xy − 3y 2 + 5x)~
(b) F~ (x, y, z) = (2xy + z 2 )~ı + x2~ + (2xz + π cos πz)~k
2
7. Using the Fundamental Theorem of Line Integrals, compute the line integral of ∇f = 2xex sin y~i +
2
ex cos y~j along the curve joining the points (0, 0) and (1, π2 )
I
8. Use Green’s Theorem to compute (x2 +4xy dx+(x2 +3y) dy, where C is the ellipse 9x2 +16y 2 = 144.
C
9. Compute the following flux integrals:
p
(a) F~ = −xz~i − yz~j + z 2~k; S is the cone z = x2 + y 2 for 0 ≤ z ≤ 6.
(b) F~ = x2~i + (x + ey )~j − ~k; S is the rectangle y = −4, 0 ≤ x ≤ 7, 0 ≤ z ≤ 3, oriented in the
negative y-direction.
R
~ where F~ = x2~i + (y − 2xy)~j + 10z~k
10. Use the Divergence Theorem to evaluate the flux integral S F~ · dA,
and S is the sphere of radius 5 centered at the origin, oriented outward.
Review your homework and the other two review sheets!