If you want to know more about the Principle of Least Action and Hamilton's Principle, the following famous physics book kicks off with those instead of Newton's Equations of Motion:
Note that the conference webpage
created by my friend and collaborator Lotfi Hermi (currently at Florida International Univ.) contains a lot of quite interesting and useful information about the
isoperimetric problems and their applications.
For the historical study of isoperimetric problems conducted by Euler and
Lagrange, see:
Also, there is an amusing article on dogs and calculus of variations (
not necessarily the same dog swimming problem I discussed in the class though)
as follows:
There are a vast amount of literature on spline interpolation, spline approximation, and scattered data interpolation, as you can easily imagine. Below, I list a few interesting articles discussing the variational aspects of such problems:
The books on splines and scattered data interpolation are too numerous to list them here. I only list the following books that provide comprehensive,
multidimensional, and modern aspects for your reference:
I stated the Lagrange polynomial for the equispaced
points on an interval leads to Runge's phenomenon.
Then you may have several naturally occurring questions regarding algebraic
polynomial interpolations, e.g.,
How about the other point configuration? How to design a good point
configuration that are less detrimental than the equispaced points?
What is a good way to measure the goodness of the Lagrange polynomial
interpolation for a given set of points that are not necessarily equispaced?
The so-called Lebesgue constant provides such a measure.
The following references discuss this concept and attempts to answer the above
questions:
This new book by Nick Trefethen is an excellent resource for interpolation and
approximation in terms of both theoretical and numerical aspects, but
it is restricted on 1D problems.
Lectures 7-8: Basics of PDEs; String/Wave and Heat Equations
Other than Chap. II of the textbook, I would recommend the following literature:
I also briefly mentioned anisotropic diffusion equations and their use for image enhancement. The following references discuss these interesting ideas:
The potential theory (as well as the heat equation) is deeply connected to the theory of Brownian motion and random walks.
This was started by the following ground-breaking paper of Shizuo Kakutani:
I also used Laplace's and Poisson's equations for image processing
applications (e.g., approximation and compression), and got even patents
in US and Japan. If you are interested, check the following articles:
The backward heat equation (a.k.a. antidiffusion) is a typical example of ill-posed problems, yet there have been interest in many applications. One of typical applications is image deblurring/sharpening:
This equivalence has a practical importance: if you have a good numerical solver for Poisson's equation with, say, homogeneous Dirichlet boundary condition,
then you can use that solver to solve Laplace's equation with
inhomogeneous Dirichlet boundary condition.
I highly recommend the following articles, which describe
one of the nicest (i.e., very accurate and fast) Poisson solvers on a 2D
rectangle or a 3D rectangular cuboid:
H. Dym & H. P. McKean: Fourier Series & Integrals, Academic Press, 1972.
The last two books require some basic knowledge of the measure theory.
Of course, any serious person who is interested in Fourier series cannot
miss the following Bible:
Importance of Fourier Analysis according to Cornelius Lanczos (1893-1974)
On the occasion of the 120th anniversary of Cornelius Lanczos's birth
the NA group at Manchester made available online a series of video tapes
produced in 1972: http://guettel.com/lanczos. I highly recommend you to watch this video series. In particular, in Tape 1, he talks about the most relevant subjects for this course. Don't miss the video segment around 24-30 minutes where he talks about the importance of Fourier analysis!