| Hwk No. |
Date Assigned |
Assignment |
Date Due |
| 1 |
1/15 |
- 11.1 # 44, 47, 54
- 11.2 # 70
|
1/22 |
| 2 |
1/17 |
- 11.2 # 7
- 11.3 # 12, 14, 16
|
1/22 |
| 3 |
1/22 |
|
1/24 |
| 4 |
1/24 |
- Let u(t)=p+at and v(t)=q+bt, where p, q, a and b are vectors in three dimensional
space. Let theta(u,v) be the angle between u and v. Show that in the limit
as t -> + or - infinity, cos(theta(u,v)) -> cos(theta(a,b)).
- 11.5 # 1-4 (Explain how you match the equations with the figures.)
- 11.5 # 3 (show the curve lies on the given cone)
- 11.5 # 11, 31, and 42
|
1/29 |
| 5 |
1/29 |
- 11.6 # 6 (arc length)
- 11.6 # 27 (curvature, 2D)
- 11.6 # 34 (curvature, 3D)
- 11.6 # 64 (Kepler & Newton)
|
1/31 |
| 6 |
1/31 |
- 11.7 # 9 (3D graph sketching)
- 11.7 # 42, 43, 48 (characterizing surfaces via planar intersections)
- 11.7 # 51 (Intersection of two surfaces)
|
2/5 |
| 7 |
2/5 |
- 11.8 # 19 (Convert Cartesian coordinates to cylindrical and polar)
- 11.8 # 35 (Graph of an equation in cylindrical coords)
- 11.8 # 55 (Describing a surface in cylindrical & spherical coords)
- 11.8 # 59 (Flying from Fairbanks to St. Petersburg)
|
2/7 |
| 8 |
2/7 |
- 12.2.5, 12.2.15 (Finding a function's domain.)
- 12.2.30 (Drawing a graph.)
- 12.2.32 (Drawing level curves.)
- 12.2.44 (Drawing a level surface.)
- 12.2.53-58 (Matching. Be sure to explain how you arrive at each match.)
| 2/12 |
|
| 9 |
2/12 |
- 12.3 # 29 (finding limits)
- 12.3 # 31 (finding a region of continuity)
- 12.3 # 42 (finding a limit -- similar to example from class)
- 12.3 # 52 (a more challenging limit)
| 2/19 |
|
| 10 |
2/19 |
- 12.4 # 45-50 (matching practice)
- 12.4 # 56 (heat equation in 2D)
- 12.4 # 58 (Laplace's equation in 2D)
- 12.4 # 65 (finding the point at which a surface is horizontal)
| 2/21 |
|
| 11 |
2/21 |
- 12.5 # 21 (finding the highest or lowest point on a surface -- justify your answer carefully)
- 12.5 # 31 (minimizing the distance to a plane, within a region)
- 12.5 # 46 (constrained maximum volume problem -- begun in class)
- 12.5 # 59 (another volume maximization problem, slightly more involved)
| 2/26 |
|
| 12 |
2/28 |
- 12.6 # 28 (no calculator allowed for problems from section 12.6)
- 12.6 # 34 (estimating the maximum error of a compound measurement)
- 12.6 # 31 (estimating the location of a point on a curve)
- 12.7 # 31 (Use the chain rule to find the plane tangent to a surface)
- 12.7 # 38 (Dependence of resistance on three parallel resistances)
- 12.7 # 53 (Hint: use equation 10; see the implicit function theorem)
- 12.7 # 54 (Apply the implicit function theorem!)
| 3/4 |
|
| 13 |
3/4 |
- 12.8 # 21 (directional derivatives)
- 12.8 # 30 (using the gradient to find the line tangent to a curve)
- 12.9 # 17 (optimizing a function of 3 variables, given 2 constraints)
- 12.9 # 37 (inscribed circle of maximal area, using Lagrange multipliers)
| 3/6 |
|
| 14 |
3/6 |
- 12.8 # 27 (another directional derivative problem, for practice!)
- 12.10 # 1 (classifying critical points of f(x,y))
- 12.10 # 4 (classifying critical points of f(x,y))
- 12.10 # 23 (a case where the discriminant is zero)
| 3/18 |
|
| |
|
- - - - S P R I N G - - - B R E A K - - -
| |
|
| 15 |
3/18 |
- 13.1 # 12 (practice w/ double integrals)
- 13.1 # 19 (practice w/ double integrals)
- 13.1 # 32 (check that the order of integration doesn't matter)
- Show that if z(x,y)=f(x)g(y), then the double integral of z(x,y)dxdy
over the region a<=x<=b and c<=y<=d is equal to the product of the integral of f(x)
over [a,b], times the integral of g(y) over [c,d].
- 13.2 # 3 (double integral where boundary involves one of the variables)
- Bonus problem: Find the area under the surface z=1-(x^2+y^2) bounded by the region z >= 0.
| 3/25 |
|
| 16 |
3/25 |
- 13.2 # 25
- 13.3 # 13
- 13.3 # 27
- 13.4 # 11
- 13.4 # 15
- 13.4 # 35 (but use Pappus' Theorem instead of the hint!)
- 13.5 # 15
- 13.5 # 43
- Suppose z=f(x,y) is a function of (x^2+y^2) only. Assume that f(x,y)>=0
as long as (x^2+y^2) >= a^2. Calculate the volume of the solid formed by the
surface with height z=f(x,y) over the region (x^2+y^2) <= a^2, in two different
ways:
- Integrate a system of cylindrical shells of thickness "dr", height
z and circumference 2*pi*r.
- Use Pappus' theorem.
Then show that these two approaches give the same answer.
| 4/1 |
| 17 |
4/1 |
- 13.6 #1 (triple integral over a rectangular region)
- 13.6 #7 (triple integral over a more complicated region)
- 13.6 #38 (moment of inertia of a sphere)
- 13.6 #40 (another moment of inertia)
- 13.7 # 1 (centroid of a shape with cylindrical symmetry)
- 13.7 # 2 (moment of inertia of same)
- 13.7 # 30 (volume of a solid defined in spherical coordinates)
- 13.7 # 31 (moment of inertia of a solid with some rotational symmetry)
| 4/8 |
|
| 18 |
4/8 |
Surface area integrals:
- 13.8 #1
- 13.8 #3
- 13.8 #9
- 13.8 #13
| 4/10 |
|
| 19 |
4/10 |
- 13.9 # 1 Practice finding the Jacobian
- 13.9 # 3 More practice finding the Jacobian
- 13.9 # 7 Jacobian & area
- 13.9 # 9 Finding an area in the plane
- 13.9 # 15 Finding a volume in 3 space
- 13.9 # 19 Using Change-of-Variables with a nonconstant function
- 13.9 # 20 Another look at the ellipsoid problem
- 13.9 # 21 Another look at the ellipsoid, continued.
| 4/15 |
|
| 20 |
4/17 |
- 14.1 # 7 (sketching a vector field)
- 14.1 # 11-14 (matching gradients with vector fields)
- 14.1 # 15 (finding curl and divergence)
- 14.1 # 32 (prove that div(curl(F)) = 0)
- 14.2 # 3 (2 kinds of line integrals: ds, and dx or dy)
- 14.2 # 11 (line integral of T dot F ds)
- 14.2 # 17 line integral of f(x,y,z) ds along curve C
- 14.2 # 19-20 (centroid & moments of inertia of a piece of wire)
- 14.2 # 37 work moving on a sphere in a central force field
| 4/24 |
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