Class Diary for M247, Spring 2009, Jochen Denzler
Wed Jan 07:
Intro to MV facts; graphs; representation by level curves. Informal hwk
given to try and sketch graphs of some examples
Fri Jan 09:
Discussion of graphing examples. Poar, cylindrical, and spherical coordinates.
Points (x,y), vectors [delta x delta y]T, and how they connect when
we choose an origin. Adding vectors.
Mon Jan 12:
Dot product; norm; Cauchy-Schwarz inequality.
Tue Jan 13: (75min)
Proof of Cauchy-Schwarz inequality; discussion of hwk assignments; key
abstract properties of vector addition, dot product, norm. Hwk 1-3 due Fri,
4-5 Tue (see main page for hwk) Limits and
continuity started.
Wed Jan 14:
Precise definition for limit and continuity; basic properties and examples: As
(x,y)-> (0,0), lim (x^2y)/(x^2+y^2)=0, but lim (xy)/(x^2+y^2) DNE.
Fri Jan 16:
Multi-variable continuity cannot be reduced to single-variable componentwise
continuity; dsicussions towards pertinent hwk. Vector valued functions (of one
and several variables) and their continuity. Balls, open sets.
Hwk: 6-8 by Wed, 9-11 by Fri
Mon Jan 19:
MLK DAY
Tue Jan 20: (75 min)
Q&A re Hwk. Open sets; boundary of a set; closed sets. Partial deriatives
of a mv function. Differentiable in MV calculus should mean existence of a
tangent plane. Why partial derivatives fall short of capturing this notion.
Wed Jan 21:
A few more examples for insufficiency of partial derivative. Linear
inhomogeneous (affine) functions. Matrices begun.
Fri Jan 23:
Matrix arithmetic. Working towards total derivative, and how it is a natural
generalization of the single variable derivative.
Mon Jan 26:
The total derivative of a vector valued mv fct at a point defined as a
matrix. Its entries identified as partial
derivatives. Hwk 12-14 due Wed
Tue Jan 27: (75 min)
Examples; review; proof that continuous partial derivatives imply total
differentiability.
Wed Jan 28:
Questions regarding pending hwk problems; idea of directional derivative.
Hwk: 12-14 extended till Fri. Also do 15,16 till Fri.
Fri Jan 30:
Directional derivative. Gradient, and its geometric meaning.
Hwk 17-20 due Wed.
Mon Feb 02:
Question on how to find level sets addressed. Gradient orthogonal to level
sets. Vanishing of gradient necessary for local minima and maxima in an open
set.
Tue Feb 03: (75 min)
The chain rule via multiplication of total derivatives (Jacobi matrices),
and interpretation in analogy to single variable case. What it means for the
partial derivatives.
Wed Feb 04:
Questions and Answers
Fri Feb 06:
EXAM 1
Mon Feb 09:
no class (overtime from 2 past Tuesdays)
Tue Feb 10:
no class (overtime from 2 more past Tuesdays)
Wed Feb 11:
no class (to be made up by future overtime Tuesdays)
Fri Feb 13:
no class (to be made up by future overtime Tuesdays)
Mon Feb 16:
Review of derivatives and chain rule. Hwk 21-28 due Fri
Tue Feb 17: (75min)
Examples on MV chain rule like deriving
integral0xg(x,t)dt. Proof of the chain rule
(calculational part).
Wed Feb 18:
Proof of chain rule finished (proof logic, epsilon, delta). Comments on
pending hwk. More on notation (curly-d1 etc)
Fri Feb 20:
Lots of hwk questions. (Deadline extended). Ck defined.
Mon Feb 23:
For Ck, partial derivatives of order up to k commute (proof
omitted). Counterexample when the C2 hypothesis fails. Hessian
matrix. Minimax problems begun.
Tue Feb 24: (75min) necessary vs sufficient conditions for interior
relative minima or maxima in terms of the Hessian. Examples.
Wed Feb 25:
positive definiteness of a symmetric matrix defined. Hurwitz condition given
without proof (proved for 2x2; 2x2 and 3x3 determinants defined ad
hoc). Gershgorin sufficient condition.
Fri Feb 27:
Global extrema exist for cont fcts on a bounded and closed set; an example;
Mon Mar 02:
Some Ramifications: Hurwitz test for positive definiteness may proceed with any
order of variables. Global minimum of a cont function on a closed but unbounded
set (by showing that candidates outside a certain bounded set never had a
chance). New hwk handed out and discussed. Hwk 32-38 due Monday. (37,38
needs tomorrow's material. Concerning # 34: May turn in shared solution in
groups and may use symbolic algebra software.
Tue Mar 03: (75min)
Constrained minima and maxima: Lagrange multiplier method. -- Implicit
function theorem barely started.
Wed Mar 04:
Implicit function theorem illustrated and explained for function R^2 -> R.
Fri Mar 06:
class canceled for sickness; will use 2 more 75 min Tuesdays to make up.
Mon Mar 09:
Questions on pending hwk.
Tue Mar 10: (75 min)
Another example on the implicit function theorem; this time 1 or 2 equations in
3 variables. Inverse matrix introduced.
Wed Mar 11:
Riemann integral of continuos functions over rectangle; Fubini for this case.
Fri Mar 13:
Still more questions about pending hwk.
Mon Mar 16:
SPRING BREAK
Tue Mar 17:
SPRING BREAK
Wed Mar 18:
SPRING BREAK
Fri Mar 20:
SPRING BREAK
Mon Mar 23:
Review and questions.
Tue Mar 24: (75min)
Riemann integral over non-box domains via extension by 0. Examples.
Wed Mar 25:
EXAM 2
Fri Mar 27:
Integration and area element in polar coordinates. Hwk 42,43 due Mon; hwk
39-41 later next week (preferrably Wed)
Mon Mar 30:
area of parallelograms, 2x2 determinants, cross product in 3-space. Hwk
39-46 due Fri
Tue Mar 31: (75min)
Scalar triple product and volume of parallelepiped. Proof of substitution
formula in 2 dim.
Wed Apr 01:
Evaluation of Gaussian integral by MVC methods. An example for a general
coordinate transformation: area of the region covered by drawing a parabola
with a thick pen.
Fri Apr 03:
Area of thick parabola finished up. Surfaces in 3-space and how to find their
area. Surface integrals of scalar functions.
Mon Apr 06:
Moment of inertia (for bodies and surfaces).
Tue Apr 07: (75 min)
flux integrals: vector valued functions integrated over surfaces, and some
physical interpretations thereof. Hwk: 47-51 due Mon
Wed Apr 08:
Oriented surfaces; definition of flux integrals given recisely. Coordinate
independence stated. Divergence theorem motivated and stated.
Fri Apr 10:
GOOD FRIDAY
Mon Apr 13:
Proof sketch for Divergence Theorem.
Tue Apr 14:
Hwk: 52-55 due Fri, if possibleIntro to new hwk given (in particular
2dim version of flux integrals). Curve integrals of a vector field in the
plane and space defined and motivated as work in physics. If the vector field
is the gradient of a scalar field, they depend only on the endpoints of the
curve, not on the chosen route. Integrability conditions stated (as necessary
conditions)
Wed Apr 15:
Curl of a vector field; Stokes' Theorem stated.
Fri Apr 17:
Review and questions.
Mon Apr 20:
Review. Teaching Evaluation
Tue Apr 21:
day off (6:15pm question and answer session in AC113 for exam prep)
Wed Apr 22:
EXAM 3
Fri Apr 24:
MON APR 27: STUDY DAY
Fri May 01: FINAL EXAM 08:00-10:00
(scheduled by university policy)
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