MATH 241 FALL 2006- COURSE LOG

W   8/23        Syllabus, course policies
Limits in two variables (at the origin)- definition, 2 criteria:
checking along lines, bounding above by a function of distance
HW- sect 11.2:  11, 13, 15, 25, 34

F  8/25         Limits- more examples
Partial derivatives: definition, geometric interp., examples
HW-  sect 11.3: 17, 21, 23, 47, 49, 63, 76

M  8/28         Geometry in three dimensions
Projections, frames and analytic geometry; scalar product
HW- section 9.2: 31, 37, 38; section 9.3:  21, 37, 29, 45

T  8/29           Problem session
11.2  34, 36 11.3 63, 76   9.2 38       9.3 37

W 8/30           properties of scalar product; spherical coords. on the unit sphere
the reflection problem; answer expressed in vector notation
vector equations of lines and planes
HW-sect 9.5 17, 23, 24, 26, 29, 37, 38

F 9/1               problems on lines and planes, including versions of:
9.5: 5, 10, 18, 25(on 8/30), 28, 38
(these should all be considered HW problems)
eqn for line segment joining two points; midpoint and median plane
handout: reflection and refraction on a general plane (PDF)
(an example in vector geometry- includes an exercise. Can you solve it?)

M 9/4              Labor Day (no classes)
T  9/5             Quiz 1

W  9/6              Directional derivative/ First-order Taylor approximation in 1 and 2 variables/
Computation of directional derivative in terms of partial derivatives/ directions
of max increase/decrease and of no first-order change.
HW:  (11.6) 11, 13, 18, 19, 21

F 9/8                 Eqn of tangent plane to a graph/ differentiability, linearization and differentials/
relationship w/ 1st order Taylor approx. and directional derivative
HW:  (11.4) 3, 11, 15, 19, 22, 26, 31, 32

The quiz on 9/12 will be based on the HW from sections 11.4 and 11.6

M 9/11                Gradient vector; level sets, tangent plane (line) to a level set
HW: (11.6)  29, 37, 41, 46, 49

W 9/13                chain rule: function along curve, function composed with map (change of coordinates.)
example: the gradient in polar coordinates
HW: (11.5) 3,5,17,19

F 9/15                   implicit function theorem for functions and maps
problems from handout: implicitly defined functions, chain rule for maps
HW: (11.5) , 23, 27, handout

The test on Tuesday will include material up to 9/15- both theory questions (based on the lectures)
and problems (at the level of the HW, or slightly harder) may be on the test. Waiting until Monday
evening to study for the test would be a really bad idea.

M 9/18                 Discussion of HW problems/ parametrization of surfaces (example)/
gradient in spherical coordinates (start)

T  9/19                 EXAM 1 (with solutions)

W 9/20               gradient in spherical and cylindrical coordinates; vector product
CHANGE IN EXAM SCHEDULE AND GRADE COMPUTATION
The course average will be based on the best three of 4 one-hour exams;
there will be no final exam. Percentages are (highest to lowest grade): 30%,30%,20%
The  remaining 20% correspond to the quiz average (highest 5 quizzes)
EXAM DATES:  9/19, 10/10, 11/7, 11/28
Quiz dates: 9/26, 10/17, 10/24, 11/14, 12/5
Problem sessions: 10/3, 10/31, 11/21

F 9/22                  critical points in two variables: local max/min, saddle points
Hessian quadratic form; quadratic form associated to a symmetric 2X2 matrix:
notions of pos/neg definite, indefinite, degenerate vs. non-degenerate.
Exam 1 returned
HW (11.7) 2,3,4,5,6,7,9,10
(these problems will be tested in the quiz of 9/26)
Correction to the original syllabus: the deadline to drop with a W is 10/3

M 9/25               test for degenerate and pos/def quadratic froms; classification of critical points

T 9/26                 Quiz 3
absolute max/over a bounded region, and over an unbounded region with boundary
HW(11.7) 26, 35, 37, 39, 41, 47

W 9/27                 Second-order Taylor formula (inc. remainder estimate) in two variables/
classification of critical points in three variables

F  9/29                  Lagrange multipliers/ two-constraint problems
Homework:  handout; see also (11.8) 5, 7, 9, 15, 19, 29

M 10/2                  finding the index of a quadratic form in three variables/
level sets in a neighborhood of a CP (for index 0,1,2,3)

T  10/3                  problem session (based on handout)

W 10/4                  parametrization of surfaces- basis for the tangent plane
HW (10.5) 19, 21, 23, 27 (11.4) 33, 35

F  10/6                vector fields-  gradient vector fields: necessary condtions/
the associated system of DEs and integral curves/ gradient vf.'s have
no closed integral curves/ radial v.f's are always conservative  (potential=antiderivative)
HW (13.1)  5,6,35,36  (13.3) 3,5,6 (13.5) 11, 12

Material included in Exam 2 (scheduled for Tuesday 10/10)- lectures from 9/18 to 10/6
Problems: review HW problems from text and handout/ Theory: review class notes

M  10/9                discussion of homework problems

T   10/10             EXAM 2

W  10/11             vector fields as force fields: 2nd order eqn of motion, conservation of energy
(for conservative v.f.). Work done by a v.f. along a curve and definition of line integral; examples
Independence of path for conservative v.f., and converse
HW (13.2) 5, 12, 15, 17, 19 (13.3)  1,11,13,15,21
Change: on 10/17 we'll have a problem session (not a quiz)

F 10/13                Fall break (no classes)

M 10/16              conservative vector fields in connected regions: 3 equivalent conditions/ example: Ampere's
law for the magnetic field, field produced by a straight wire --> the `angle' v.f. in the plane
HW( 13.3)  23,24,29 to 32, 33

T 10/17                discussion of HW problems/ finding the potential for a conservative v.f. in the whole space/
solution of Exam 2

W 10/18               conservative v.f.: simply-connected regions in the plane and in space
(HW: see 10/16)  Exercise on potentials
(contains 4 problems- optional HW set, due 10/24- grade counts as 1 quiz if turned in)

Here is an interesting calculus web site- click on `third semester calculus' for
some examples related to the first part of the course (with nice graphics)
Dr. Vogel's gallery of calculus pathologies

F 10/20               double integrals over rectangles (lecture by Dr. L. Finotti)

HW (12.2) 9, 11, 13, 17, 23, 27, 31
Quiz 4 on 10/24:  based on HW problems listed above for 10/11, 10/16, 10/20

M 10/23             discussion of HW/ double integrals over general regions
HW (12.3) 13, 15, 19, 25, 41, 43, 51

T  10/24               Quiz 4/ double integrals in polar coords. (start)
HW(12.4) 9,13,21,23,27,29,33

W 10/25              double integrals-examples/ Green's theorem for rectangles
HW (12.5): 5, 7, 11 (assume constant density =1)

F  10/27                Green's theorem for general regions/ definition of rot (operator
from vector fields to functions) as the limit of average circulation
HW (13.4)- 7, 13, 15, 21, 27

M 10/30                 divergence thm in the plane- divergence of a v.f./ rot and div under rotations
potentials in simply-connected regions. Quiz 4 returned.
HW(13.5) 23, 31, 33, 34
(assume the vector fields are defined in the plane, and `curl''=`rot'.)

T  10/31              Problem session- HW problems, examples:  behavior of the differential operators
div, rot under multiplication by a function or under composition; Laplace operator

W  11/1               Exercise: rot(v) and Laplacean(f)  do not change under rotation of coordinates  (proposed)
Why surface area and arc length have different definitions/how the area of a
parallelogram changes under projection onto a plane/ formulas for surface area
of graphs and parametrized surfaces.  HW( 12.6) 5, 11, 22, 23, 27
A note on the definition of surface area
(optional reading- adapted from R. Courant, Differential and integral calculus vol. 2)

F  11/3                Surface area, examples: helicoid, surfaces of revol'n, spherical coordinates.
Surface integrals: graphs, parametrized surfaces, spherical coords.
Examples: centroid of a hemisphere, gravitational potential of a sphere.
HW (13.6)  6, 9, 11, 13, 33, 36(a)  (33: done in class)

EXAM 3 on Tuesday, 11/7- lectures from 10/6 to 11/3. Main topics:
vector fields: line integrals, conservative vector fields, potentials, simply-connected regions
integration:  area/centroid of planar regions, double integrals, area/centroid of surfaces, surface integrals
Green's theorem and divergence theorem in the plane/ def. and properties of div and rot.

M 11/6               Review session

T  11/7                EXAM 3

W  11/8              Stokes' theorem- proof for a curve on a general plane, definition of curl, statement in the general case
(oriented surfaces with oriented boundary), the Moebius strip
HW(13.7)  3, 5, 7, 9,16

F 11/10              Orientable and oriented surfaces- examples (level sets, closed embedded surfaces, Moebius
strip). Induced orientation on the boundary and orientattion induced by the boundary-examples
HW (13.5) 17, 18, 19, 20, 24,27

M 11/13             Solution of Exam 3 /discussion of HW problems (inc. conditions for a v.f. to be a curl)

T 11/14              QUIZ 5: HW from 13.5 and 13.7
solution of quiz 5/ example: finding the vector potential of a v.f. with zero divergence

W 11/15              Stokes' thm and conservative v.f. in domains/ Existence of vector potentials when div F=0:
electric field of a point charge (counterexample)/ Flux integrals- Gauss's law, net flow rate.
HW (13.6)  21, 23, 25, 27, 39

F  11/17              The divergence theorem (triple integrals)
HW(13.8) 7, 11, 17, 25 to 32

M  11/20             Electrostatics: charge density, Gauss's law and existence of potentials (div E, curl E); Poisson's
equation for the potential.  magnetic field produced by a current loop; Ampere's law for the circulation
of B. Handout given out.

Revised quiz/exam dates: W 11/22: Quiz 6 (HW from 13.6, 13.8)  W 11/29: Exam 4   M 12/4: Quiz 7 (handout)
(Only the highest 5 quizzes will count twd the course ave)

T     11/21          Discussion of HW problems

W     11/22         Quiz 6/ Current density,  differential expression for Ampere's law

F       11/24         Thanksgiving break (no classes)

M       11/27        Discussion of quiz 6 (returned)- handout: conservation of mass/ charge

T       11/28         Review session

W      11/29         EXAM 4:  lectures from 11/8 through 11/27 ( sections 13.5 through 13.8 in Stewart)
Surface integrals, flux, properties of div and curl, Stokes' theorem, divergence theorem

Review: HW problems listed above. class notes, handout (except problems 10, 12, 13)
EXAM 4

F       12/1           Discussion of remaining problems in handout (Maxwell's eqns for time dependent
fields, wave equations for E and B without source terms)/ solution of exam 4 (returned)

M       12/4           Quiz 7 (optional)/course evaluations

COURSE GRADE DISTRIBUTION:  A=1, B=3, C+=4, C=3 (11 students took all four exams)