Read this:
Central ideas of calculus: straightness and curves, zooming in to straighten out in order to quantify rates of change, measuring curvature by changing linear zoom line.
In class:
Hands on play with the idea of curvature and parallel transport on a cone by itself, then tangent to a sphere, then tangent to a planar circular orbit around a central body deformed by the presence of that body (EEK! wormholes! general relativity already?!)
[pdf, printed at 115%, paper handout in class, cut along positive x-axis to allow fold to a cone of your choice by lining up the cut edge with a ray for polar angle sliding from between say 7Pi/6 and 5Pi/3]
Discussion [polargrid.mw, terrible export from Maple to pdf, answers now included in Maple worksheet].
I meant it when I said we were going to have fun in this class, but can we keep this up? [Everything is relative, no...? yuck, yuck]
Read the Preface and Chapter 0 of dr bob's Differential Geometry book up to but not including the final Geometry subsection unless you can' t stop yourself.
[Excerpt file pages 1-19]
Make sure you have Maple 16 on your laptop. It is easy to download and install. If you have problems, ask bob for help.
We will revisit this geodetic effect later on when we understand better the geometry. For now it is just a motivator.
Will the dr bob show classroom automatic recording be useful or not?
Please send me an email (or reply to the email I sent you) sometime this week telling me a bit about yourself, your previous math classes, your previous experience with Maple if any, your comfort level with computers and graphing calculators (we won't use the latter) and why you chose this course.
bob talks about Chapter 0. Finish reading Chapter 0 and try the problems at the end.
Read Appendix A.1 on the hyperbolic functions. If you have a calculus book, check out the section on hyperbolic functions [Stewart: Section 3.11].
[hyperbolic-trigonometric.mw, hyperbolicrotations.htm, hypaddidentities.mw]
Begin reading Chapter 1: sections 1 and 2. Try some of the problems. These also review solving linear systems and the eigenvalue problem in our notation, but not a problem if you have not done this before. It is not essential to our applications.
WEEK 2:
M: 1/21: MLK day, no class. Class info sheet. Read this.
Today I look at Exercises 1.2.1-3 to illustrate some things about vector spaces and how we get interesting structures on them [ 3: asymmatrices-crossprod.mw, quadratics.mw].
I have no one to proof my book edits, so glitches occur like in Exercise 1.2.2 where I forgot to put the usual quadratic function representation
Q(x) = a x2 +b x +c at the beginning of the problem.
The subsections on solving linear systems and eigenvectors summarize material from MAT2705 but are a bit heavy and we don't need them at so serious a level: we will solve linear systems with Maple when we need them and they will usually have a unique solution using matrix inverses, and eigenvectors are just a nice example for us but not crucial.
Read Section 1.3 on linear functions. Try some problems.
Today we saw through 3 examples that 3 different vector spaces whose elements ("vectors") are polynomials or matrices, once we introduced an obvious basis, put us into an Rn space, so essentially every n-dimensional vector space is like one of those simple spaces once we introduce a basis and express the coordinate vector or component vector of the abstract vectors in the space, whatever their mathematical origin. We will build all of our structure on a vector space V, and its space of linear functions, the dual space. These lead to vectors and covectors with a natural pairing to produce scalar invariants (the value of a linear function on a vector). Index notation helps us distinguish these two basic spaces with upper and lower indices, from which we built more complicated objects (multilinear functions on vectors = tensors). Much of the structure we use already in multivariable calculus will then be put into context in this new light: dot product, cross product, determinant, matrices and inverses, linear transformations like rotations,etc. This is a slow process so we need patience to get it right, rather than just rushing to the action (exciting geometry).
WEEK 3[-1]:
Begin reading Section 1.6 up to a "Geometric interpretation of index shifting" (page 93). Try some of the exercises, your choice.
questions and answers? Monday?;
Finish section 1.6, "Cute fact" section not essential but fun. There are a number of interesting problems for the weekend. We definitely will talk about these Monday.
Read Appendix A.2 on special relativity. [Note we have to wait to do Exercise 1.6.5 till after we cover Appendix A.3.]
WEEK 4[-1]:
Read 1.7. Try problems. More specific Tuesday.
example of exponential and eigenvector diagonalization matrixexponential2by2.mw ;
[Maple matrix examples rotationmatrices.mw (1.7.11 spin calculations done a bit clumsily), asymmatrices-crossprod.mw, GLplusT.mw Exercise 1.7.3, for more see list]
Read 2.1-3.
We don't need multivariable Taylor series (last section), but it is worth being exposed to. Don't worry about related exercises unless you are interested.
Note: multitaylor.mw, momentsofinertia.mw, which show how one can use Maple to evaluate easily these kinds of calculations.
WEEK 5[-1]:
Read 4.1 (easy: chatter about determinants) and 4.2: the wedge product. Do 4.2 problems.
Appendix A.3 curves in space review:
geometry of space curves curvature and acceleration;
[example Maple worksheets: osculating_circle_cubic.mw, osculating_circle_helix.mw]
helixTNB.mw, helix-lorentz.mw (comparing the Euclidean and Lorentz helices)
> with(Student[VectorCalculus]):
SpaceCurveTutor(<t,t2,0>,t=-1..1) from the Tools Menu, Tutors, Vector Calculus, Space Curves [choose animate osculating circles]
WEEK 6[-1]:
bob talks about A.3.
Part 2! Chapter 5. Calculus.
Read? I am confused about what I said to do, problems with web edit interface!
Read 5.1-2. Do some problems. These are pretty routine.
WEEK 7[-1]:
Tempting look forward
Added Monday night: problem 5.3.6 on the equiangular = logarithmic spiral curve vector field in polar coordinates. Added Tuesday new Problem part 5.3.5.c).
Spring Break.
enjoy and be safe.
[Don't work too hard.]Week 8[-1]:
bob goes on about covariant derivatives, independent of particular constant metric.
Read 6.7-6.8. [6.9 is optional if you are interested in rigid body motion.]
WEEK 9 [-1]:
For fun I added a problem on the angular momentum ladder operators and a followup remark on the gluon strong interaction SU(3) Lie algebra matrices at the end of section 6.8, not for you to do necessarily but to show you how close we are to being able to understand much more that you might have dreamed of... We will return to this in chapter 11 differential forms.
I also added an exercise (in the same spirit) at the end of Chapter 4 on the quaternions, for those of you who might have encountered these fascinating mathematical objects at some point in your lives. It is exactly the SU(2)-SO(3,R) connection brought out in a previous exercise just before it.
Parallel transport finally arrives! Read 8.1.
WEEK 10 : http://www34.homepage.villanova.edu/robert.jantzen/notes/torus/2011-11-talk/2011-11-18.htm [on geodesics, gravitational orbits]
[fun video (NYTimes) from bob's talk on GR, geodesics and pasta]
[see surfaces of revolution worksheet (the animation is in the monster worksheet torus_geodesics.mw, we will get to that soon enough)]
Read 8.5-8.6 or just catch up. I will talk about these when you return.
Easter Break.
WEEK 10 continued:
WEEK 11:
[spherical pendulum problem origin: http://www34.homepage.villanova.edu/robert.jantzen/gem/centalk.htm]
[connection and curvature working in a frame with Maple: blackholetensorcalcs.mw]
WEEK 12:
interpretation of curvature in terms of parallel transport around a shrinking loop in a plane. Read 9.2-9.3.
WEEK 13:
WEEK 14:
Friday, May 10: 8:30-11:00. final exam slot for short low key presentations on any topic you choose. email me about it if you cannot stop by in person?