Lecture notes and
supporting material for
MATH 235
Mathematical
Models and Their Analysis
- Lecture 1 Elementary theory of
rainbows and related phenomena
- Les
Crowley's website about atmospheric optics phenomena
- Philip Laven's website
(it is much more technical than the previous one, but also contains
some remarkable pictures)
- In particular, the second link on this
website illustrates the effect of the size of raindrops on the
appearance of a rainbow. You may think that raindrops with a diameter
of 50 microns are a hundered times larger than the wavelength of light
(which is about 500 nm = 0.5 micron), and so should be large enough to
refract and reflect light according to the laws of geometric optics.
However, this is not at all the case. One needs raindrops that are more than a thousand times larger
than the wavelength (see the last plot in the above link) to correctly
account for the appearance of a rainbow!
- Lecture 2 Kepler's Laws
- Lecture 3 Which is faster, going
up or coming down?
- Lecture 4 Will the car fly off a
rollercoaster?
- A fun link to a simulator
that accounts for friction and bouncing. In the past, this
simulator was freely accessible. While its use remains free of charge,
the system now requires you to register an account and log in to watch
the simulation. There is probably no harm in registering, but it is an
unfortunate hassle :-(.
- Lecture 5 Markov Chain
models
- Lecture 6 Simple pendulum. Part
I: The basics
- Lecture 7 Pendulum. Part
II: More complicated situations
- A link to a collection of simulations of
complex pendula. You may need to donwload Java following the link
provided on that webpage.
- Type Kapitza pendulum (or/and Kapitsa pendulum) into Google to
see what diverse applications this problem has.
- Lecture 8 Pendulum. Part
III: Higher-order corrections and perturbation theory
- Lecture 9 Continuous limits of
difference balance equations
- Lecture 10 Inverse problems. Part
I: Why they may be unstable (and what this means)
- Lecture 11 Inverse problems. Part
II: Making the reconstructed solution smooth -- the idea behind
the Tikhonov regularization
- Lecture 12 Nondiagonalizable
matrices: What can matter beyond eigenvalues and
eigenvectors
- Lecture 13 Black-body radiation
and Planck's formula
- P.-M.
Robitaille,
"Kirchoff's law of thermal emission: 150 years," Progress in
Physics 4, 3 (2009)
- Physiological aspects of color vision are reviewed in this Wikipedia
entry.
- Table of
colors versus their wavelength and frequency. (Note: There are
different such tables; I have chosen the one that looked the most
appealing to me.)
- Here are two books (with links to Amazon) about color
whose references I came across:
-
The
Physics and Chemistry of Color, 2nd Ed., by Kurt Nassau (Wiley -
Interscience, 2001) This book has a comprehensive coverage, at a
serious level, of many aspects of the color and vision science.
-
Colour:
Why the World Isn't Grey, by Hazel Rossotti (Princeton Univ. Press,
1985) This is a more accessible (well, almost a leisurely)
reading about color and vision.
Acknowledgement:
Typesetting of these lectures was made possible by an Instructional
Incentive Grant from UVM.
The lectures in their original form were typed by Anna
Holt-Gosselin, Vasiliy Lakoba,
and Barbara Dewey.
The figures were prepared by Vasiliy Lakoba.
Last modified by T. Lakoba in April 2021.