This advertises Math 534, Calculus of Variations, which I am scheduled
to teach in the spring semester.
 
The course should be appropriate for advanced undergraduates
and graduates in Mathematics and Physics, and graduate students in
Engineering.
 
Calculus of Variations deals with the task of finding minima of functionals
(typically integral expressions depending on a function).
Classical examples are the brachystochrone problem (find a curve from A to B
along which a ball rolls fastest from A to B, under the influence of
gravity), the catenary (find a curve of given length connecting two points,
such that its center of mass is lowest possible), geodesics (find the shortest
connection between two points, on a curved surface), the Lagrangian variant
of Newtonian mechanics, and electrostatic fields  (which minimize the field
energy).
 
I intend to make a compromise between classical and modern material,
thus keeping the course accessible as far as prereq's are concerned,
but still having enough breadth and depth to blend in with contemporary needs.
 
The single integral case, which leads to Ordinary Differential Equations
(ODEs) will be covered in more detail, including the classical examples. The
issue of sufficient conditions for minima, which looks more tricky in the
classical textbooks than it actually is, can be conceptually clarified very
early in the theory, since each relevant phenomenon has its intuitive
representative example.
 
The case of multivariable integrals leads to Partial Differential Equations
(PDEs) and has important applications in this area. Calculus of Variations
sheds light on Laplace's equation and related PDEs, and also on finite element
methods used in numerical solutions for them. It can easily prove nice facts
about vibrating membranes. A quick introduction to relevant areas of PDEs
will be included.
In this part, some technical details will be sacrificed, such as to convey
a conceptually clear picture of the mathematically more advanced material.
This approach should be amply sufficient for Engineering and Physics students.
Mathematics audience should perceive it as a conceptual framework in
which they can later fill in technical details from more advanced courses.
                                                                                
Prereq's:
- Multivariable Calculus
- Math 251 or 257
- Math 231
 
- one out of Math {431, 435, 453, 445, 447 ...}
  (each of these provides a small benefit for Calculus of Variations,
   but material needed from them will be re-introduced. Requiring *one* of
   them as prereq' is meant to require the mathematical maturity that comes
   with such courses, not individual fact knowledge thereof)
 
Jochen Denzler