MATH 241 SPRING 2006- COURSE LOG

1/11W  Course policies
Problem list 1:  parametrized curves, vector-valued functions of one variable
list 1 (Freire's copy)

1/13F    list 1 (cont.)
acceleration and curvature
(link to a PDF file written by Jeffery Cooper, author of the `Matlab companion'- obtained from
J.Cooper's web page )

1/16 M     Martin Luther King, Jr. holiday (no classes)

1/18 W      list 1: unit tangent, unit normal and curvature in the plane

1/20   F    list 1: unit normal, curvature, components of acceleration (in space);  plane curves given implicitly
hyperbolic sinh and cosh (summary)
LIST 1 DUE:  Wednesday 1/25 (at the start of class)

1/23  M    conceptual point: curvature and the 2nd order Taylor approximation
The mean-value inequality (list 1, problem 12)
Functions of 2 variables: graphs and level sets (MATLAB: use of `surf' and `contour')
List 2 given out
List 2 (Freire's copy)

1/24  T        Q&A sesssion
RULE: I will answer questions about the HW problems or the theory, but I won't SOLVE
homework problems. Students with HW questions will be asked to show how much they have
done on the problem and where they got stuck (possibly at the `blackboard'). `Can't even get  started'
is not acceptable.
Grading scale for homework sets: 0, 1, or 2 pts per problem

1/25 W List 1 collected
`               List 2: partial derivatives, gradient vector, directional derivative (problems 1,2,3)1/27  F    List 2: second derivatives along a curve: Hessian quadratic form, second directional                derivative (problems 4,5,6)1/30 M    List 2: differentials; the first order Taylor approximation (problems 7,8,9,10)2/1  W      List 2 (end) gradient of dist(point, plane); List 3 (start): comparing directional derivatives             LIST 2: DUE FRIDAY 2/3             Exam 1: Tuesday 2/14 (Lists 1, 2, 3)  2/3F  List 3:  tangent plane approximation- tangent planes (or lines) to graphs and level sets (problems 2-5)             List 3 (Freire's version)2/ 6 M      List 3: max/min along lines, second order Taylor approx, critical points (Problems 6,7,8)2/8 W      List 3: classification of critical points (2 variables): problems  9, 13, 142/10 F      List 3: absolute minima: criterion for existence of an absolute min/max (problems 11,12)                 MATLAB example: graph, level sets, gradient vector field ( with `quiver'): local picture near a critical point2/13 M     Answer to questions on lists 1,2,3                 vector product, curvature in space (review problem 15)2/14 T     Exam 1; list 3 collected                 Exam 1 (version A)       Exam 1 (version B)2/15 W   List 4: local max/min in 3 variables, ellipsoids (problems 1,2)                List 4 (Freire's version)2/17 F    List 4: saddle points in 3 variables; hyperboloids of 1 and 2 sheets; Hessian criterion (problem 3)                  absolute max/min values in bounded regions (problem 4)2/20 M    Max-min uder constraints, Lagrange multipliers (problems 5,6)2/22 W  Max-min under constraints; two-constraint problems (problems 7,8,9)2/24  F   max/min in a 3D bounded domain (list 4, problem 10); parametirization of surfaces (graphs, ellipsoids, hyperboloids)2/27 M   parametrization of surfaces- basis for the tangent plane (problems 11,.12)2/28 T    Double integrals: definition by Riemann sums-slicing and Fubini's theorem-volume under graphs              (problems 1,2,4 on list 5)- list 4 collected 3/1 W   class cancellled (colloquium talk at Notre Dame)3/3 F       List 5 given out- problems 3,5,6: double integrals-change in order of int, polar coords               Conceptual remark: double integrals of odd functions over symmetric domains vanish                List 5 (Freire's version)EXAM 2- Tuesday 3/14- Lists 4 and 5 (List 5 also due 3/14)3/6 M        List 5-centroid, area of surfaces (problems 7,8,9)                  Centroid of a region decomposed into subregions, including "negative area"                  Brief discussion of contrast between definitions of arc length and area. Details in link below:                  A note on the definition of surface area                  (adapted from Courant/John-optional reading: if you are interested, but have trouble                    understanding  the example, ask me for more details)3/8 W       Area of the projection of a parallelogram onto a plane (cosine factor)                  Area of a piece of a surface given as a level set, as a double integral over a planar projection (Problem 10)3/10 F          Integration of functions on surfaces, with respect to surface area (example: centroids)                   (Review: integration of functions on curves, with respect to arc length).                     List 5, problems 9 (using parametrization), 11 (using projection on a fixed plane)3/13 M          Q&A session for exam 2;  graded list 4 returned3/14 T        Exam2                 List 5 collected3/15 W       Pappus' theorems for areas and volumes of surfaces (resp. solids) of revolution                   Example- finding centroids, area of a torus, volume of a solid torus                    Graded exam 2 returned- solution of exam 23/17 F        Triple integration examples: vol. of intersection of wo cylinders,  region between                    paaboloids, change in order of integration3/20-3/24     SPRING BREAKNOTE: in response to student requests,  from now on I'll include a related section of Stewart for each topic (when there is one). I am not following Stewart,- what is found in that book may be onlydistantly  related to what was done in lecture, and, of course,  the lectures take precedence (i.e, HW andtests are based on the lectures, and whether or not the material is similar (or even related) to what is found in Stewart is irrelevant.)This reference is meant mainly as a source of additional problems, for those who want some.3/27 M       Change of variable in multiple integrals: Jacobian of a mapping                    (Stewart 3rd ed: 12.9)                   List 6 handed out- problems 1, 2 discussed                    List 6(Freire's version)3/29  W       List 6, problems 3,4:  change of vble in triple integrals, integration in spherical coordinates.                    Review of improper integrals in one variable                     (not in Stewart)3/31  F        List 6, problems  5,6,7,8: improper double and triple integrals; implicit differentiation in                    several variables (discussion of the implicit function theorem in one variable)                    (Stewart 3rd ed.: p. 761, ex. 4)4/3    M        List 6, problems 9, 10, 11: chain rule for mappings                    Ex:  gradient vector field in polar and spherical coordinates;  improper Gaussian integral                     (Stewart 3rd. ed: :11.5 (sort of))  LIST  6 IS DUE WEDNESDAY 4/5THIRD TEST: WEDNESDAY, 4/194/5     W      List 6 collected, list 7 handed out     List 7 (PDF file)                    vector fields (notation, examples);  work along a path; definition of line integral                     properties: orientations, concatenation, independence of parametrization                      (Stewart 3rd ed.: 13.1, 13.2)4/7   F      Ex: work and change in kinetic energy                 Line integrals of gradients; conservative vector fields and potentials                 Necessary consition based on mixed partial derivatives.                  Example: radial vector fields are conservative                 (Stewart: 13.3)4/10  M       Three equivalent conditions for conservative vector fields; construction of                 potentials by `partial integration'. Example: a vector field undefined at the                  origin which satisfies the necessary condition but is not conservative.                 Simply-connected regions. Reference: Stewart  13.3 . List 7- problems 5,64./12 W       Green's theorem- use to compute line integrals, areas, centroids.                     Divergence form of Green's theorem- defn of divergence and flux                     Reference: Stewart, 13.4, formula (13) on p.946 for problem 9                      List 7- Problems 7,8,9,10 should be doable with this theory                      List 6 returned4/14 F   "Spring Recess" (no classes)4/17 M          Review problems for exam 3 (handout)                      Review problems (scanned PDF)4/18   T          Stokes' theorem;  questions on lists 6 and 7                       (reference: Stewart sect. 13.7)                       List 7 due4/19              Exam 3 (PDF file)4/21              List 8 handed out- problems 1,2,4,5 discussed                      (reference: Stewart sect 13.5)                      Properties of curl and div                      List 8 (PDF)                      4/24             Divergence theorem in R^3: problems, 6,7,8                    List 8- homework version                      reference: Stewart sect. 13.8                    LIST 8 is due Friday 4/28 (automatic extension to Monday 5/1, BY 5:00 PM)                    (drop it off in the box outside my office)4/25              Problem session: solution of Exam 3,  questions on                       list 8 (1 through 8), course evaluations (sect. 7)                        4/26             discussion of physics applications (problems 9-13)                     conservation of mass and of charge; current density; Maxwell's equations in the static case                     (using vector calculus to pass from integral to differential forms)                     examples : E for a sph. symm charge density (sect 5.),  B for straight-line current (sect. 7)4/28            Maxwell's equations for time-dependent fields  (passing from integral to differential form)                   Derivation of the wave equation for E and B from Maxwell's eqns without charges or currents                   Example: verifying  travelling-wave solutions to the 1D wave equation                    Course evaluations (Sect. 5)                               FINAL EXAM  Section 5 (12:20): Tuesday, May 9, 12:30-2:30                          Section 7(3:35):  Monday, May 8, 5:00-7:00 Material included:: lectures from 4/5 to 4/28 (vector calculus), including lists 7 and lists 8 (and the review handout from 4/17)                          The final may include problems related to the discussion of physics applications (list 8)Exam 4 (final)-version AExam 4 (final)-version B                                                                                        `