Class Diary for M247, Fall 2015, Jochen Denzler
Wed Aug 19:
Intro material. Examples of multi-variable functions; different notation
cultures distinguished. Points and their coordinates.
Thu Aug 20:
Vectors; geometric and algebraic aspects; dot product and angle measurement.
spherical and cylindrical coordinates introduced.
Fri Aug 21:
Cauchy Schwarz inequality explained and proved. Vector valued functions of a
single variable (example of a spiral staircase). Hwk
1-5 due Wed
Mon Aug 24:
clarification hwk 4; vector valued functions of a single variable: path of a
particle, velocity (vs speed), acceleration; parametrized curve and arclength
Wed Aug 26:
Help with hwk 5 (extended till tomorrow); example of cycloid and its
arclength. Hwk 6-9 due Mon.
Thu Aug 27:
Example: evolvent of cycloid (ie unrolling twine from a cycloid).
Unit tangent vector; its deriv is orthogonal to the tangent vector itself.
Definition of curvature
Fri Aug 28:
graph y=f(x) and its curvature. Intro to cross product (geometric definition).
Mon Aug 31:
Properties of cross product and formula for it.
Hwk 10-13 due Fri.
Wed Sep 02:
Curvature in terms of cross product; Discussion of returned homework 6-8.
Thu Sep 03:
multi-variable functions; their graphs and level sets. Definition of a limit
explained.
Fri Sep 04:
>Hwk 14-18 due next Fri. Class discussion
mainly guidance through these hwk problems.
Mon Sep 07:
LABOR DAY
Wed Sep 09:
some Q re hwk. Refinement of def. of limit, when function is not the entire
R^n. Open sets
Thu Sep 10:
Open sets; Boundary; closed sets. Partial derivatives of a function (that we
assume to be defined on an open set). Directional derivative.
Fri Sep 11:
Intro to total derivative; tangent planes to a graph; what is a matrix?
Mon Sep 14:
Discussion of old and new homework Hwk 19-22 due Thu if possible
Tangent lines from directional derivatives may or may not assemble into a
tangent plane. Matrix multiplication defined.
Wed Sep 16:
Hwk deadline extended till Fri More on matrix multiplication.
Differentiability (and total derivative) in terms of tangential planes
and by a limit reminiscent of single variable difference quotients.
Thu Sep 17:
Total derivative: an example calculation for working with the definition.
Continuity of partial derivatives on an open set guarantees total
differentiability.
Fri Sep 18:
Total derivative also for vector valuaed functions; its relation with
directional derivatives.
Mon Sep 21:
Hwk discussion
Wed Sep 23:
EXAM 1 ; New Hwk #22-27 due next Wed
Thu Sep 24:
Gradient; chain rule just started
Fri Sep 25:
Chain rule
Mon Sep 28:
Applications of chain rule
Wed Sep 30:
Exam return and discussion
Thu Oct 01:
2nd partial derivatives; Hessian
Fri Oct 02:
The Hessian. Two theorems about minimax: existence, and 1st derivative test.
Mon Oct 05:
2nd derivative test. 2nd order directional derivatives are v^T H v.
Wed Oct 07:
Positive definiteness and Hurwitz test (determinants up to 2x2)
Hwk 32-34 due next Wed
Thu Oct 08:
3x3 determinants (for the sake of Hurwitz test for positive definiteness);
Gershgorin test. An example of a minimax problem in practice
Fri Oct 09:
Example of minimax problem finished
Mon Oct 12:
Constrained minimax problems; an example solved by elimination; Lagrange
multipliers started
Wed Oct 14:
Lagrange multipliers finished, and example.
Thu Oct 15:
FALL BREAK
Fri Oct 16:
FALL BREAK
Mon Oct 19:
Hwk 35-39 due Mon. Q&A. Implicit function theorem in 2 variables.
Wed Oct 21:
Intro to multi-variable integration.
Thu Oct 22:
Riemann-Integral of piecewise cont functions defined. Its evaluation by
repeated integrals. Example where inner integration limits are variable.
Fri Oct 23:
Example finished. Area element in polar coordinates.
Mon Oct 26:
Transformation formula in 2 dim; New Hwk 40-42 due
Mon
Wed Oct 28:
Example for transformation formula in 3d: torus coordinates
Thu Oct 29:
EXAM 2
Fri Oct 30:
Applications: Moment of inertia; center of mass. Started: integration of a
scalar function over a curve
Mon Nov 02:
integration of a scalar function over a curve; e.g., Center of mass of a
cardioid.
Wed Nov 04:
Integrals over curves: vector functions; work New Hwk
43-47 due next Wed
Thu Nov 05:
Curve integrals over vector fields, and path independence.
Fri Nov 06:
(Substituted by AF) Conservative vector fields
Mon Nov 09:
Parametrized surfaces and area.
Wed Nov 11:
center of mass of a surface, briefly. Flux integrals: definition
and interpretation
Thu Nov 12:
A calc' exaple of flux integrals. Flux integrals over closed surfaces can be
converted into volume integrals: divergence of a vector field defined.
[Result stated, no explanation yet.]
Fri Nov 13:
Statement of Gauss' theorem without proof yet. Green's theorem for `very nice'
domains (type 3 in Marsden-Tromba).
Mon Nov 16:
decomposition of domains into type 3 pieces; Green's thm in general;
application to area inside a curve.
Wed Nov 18:
Curl of a vector field. Stokes' theorem stated and explained.
Thu Nov 19:
EXAM 3
Fri Nov 20:
Stokes' theorem proved.
Mon Nov 23:
Return to conservative vector fields; domains with `holes' (not simply
connected). 2D Gauss theorem as a consequence from Green.
Wed Nov 25:
Fun material - not exam relevant. If you leave early: have a safe trip.
If you travel after class: have a safe trip, too.
Thu Nov 26:
THANKSGIVING
Fri Nov 27:
THANKSGIVING BREAK
Mon Nov 30:
Wed Dec 02: STUDY DAY
Tue Dec 08: FINAL EXAM 02:45-04:45
(scheduled by university policy), see
exam schedule by class.
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