Calculus for Science and Engineering III

MATH 223 -- Thomas Section

Syllabus

Latitude & Longitude data (Section 11.8, cylindrical and spherical coordinates)

Practice Exercises

"*" denotes additional recommended optional exercises, not to be handed in.
Hwk No. Date Assigned Assignment Date Due
1 8/25
  • 11.1 # 44, 47, 54
8/29
2 8/27
  • 11.2 # 7, 13, 70
  • 11.3 # 1, 12, 14, 16, 25
9/3
3 9/3
  • 11.4 # 35,37,40,56
9/8
4 9/8
  • 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 -> (+ infinity) 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
9/12
5 9/12
  • 11.6 # 6 (arc length)
  • 11.6 # 27 (curvature, 2D)
  • 11.6 # 34 (curvature, 3D)
  • 11.6 # 64 One step in deriving the relations of Newton's and Kepler's Laws.
9/15
6 9/15
  • 11.7 # 9 (3D graph sketching)
  • 11.7 # 42, 43, 48 (characterizing surfaces via planar intersections)
  • 11.7 # 51 (Intersection of two surfaces)
9/19
7 9/17
  • 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)
9/22
8 9/25 Having recovered from the exam on 9/24 ... Please note the change in due date!
  • 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.)
  • 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)
WED. 10/1
9 9/29
  • 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)
10/3
10 10/1
  • 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 -- continued from handout)
  • 12.5 # 59 (another volume maximization problem, slightly more involved)
  • * 12.5 # 47, 48, 49, 54
10/6
11 10/6
  • 12.6 # 28 (no calculator allowed for problems from section 12.6)
  • 12.6 # 31 (estimating the location of a point on a curve)
  • 12.6 # 34 (estimating the maximum error of a compound measurement)
  • * 12.6 # 29, 35, 40, 42
10/10
12 10/8
  • 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!)
10/13
13 10/13
  • 12.8 # 21 (directional derivatives)
  • 12.8 # 30 (using the gradient to find the line tangent to a curve)
  • 12.8 # 27 (another directional derivative problem, for practice!)
10/17
14 10/15
  • 12.9 # 17 (optimizing a function of 3 variables, given 2 constraints)
  • 12.9 # 37 (inscribed circle of maximal area, using Lagrange multipliers)
10/22
  • - - - F A L L - - - B R E A K - - - 10/20-21
15 10/22
  • 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)
10/24
16 10/27
  • 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].
10/29
17 10/29
  • 13.2 # 3 (double integral where boundary involves one of the variables)
  • 13.2 # 25
  • 13.3 # 13
  • 13.3 # 27
  • 13.4 # 11
  • 13.4 # 15
  • 13.4 # 35 (try using Pappus' Theorem instead of the hint!)
  • * (from class): Find the area under the surface z=1-(x^2+y^2) bounded by the region z >= 0.
11/3
18 11/3
  • 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:
    1. Integrate a system of cylindrical shells of thickness "dr", height z and circumference 2*pi*r.
    2. Use Pappus' theorem.
    Then show that these two approaches give the same answer.
  • 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)
11/7
19 11/5
  • 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)
11/10
20 11/10
    Surface area integrals:
  • 13.8 #1
  • 13.8 #3
  • 13.8 #9
  • 13.8 #13
11/14
21 11/12
  • 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.
11/17
22 11/23
  • 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
11/26
optional ---
  • 14.3 # 1,3,5,17,25,30,31,36
  • 14.4 # 1,17,21,34
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Updated: September 29, 2008