16 April 2004

The Geometry of Differential Equations

George E. Hrabovsky, President, MAST

What We Did Last Time

We briefly studied arc length.

Where We Will Go This Time

We will discuss the idea of a phase plane.

Unit Vectors

We must add another word to our dictionary.  If you have been following, The Mind of a Theorist, you will already be familiar with them.  Any time we divide the components of a vector by the norm of the vector we have what is called a unit vector.

If we define a unit vector as being along the first axis of a coordinate system (for example the axis of a Cartesian plane, or the axis of a polar plane), then we write .

Phase Portraits

We have seen that we can characterize a curve by its arc length and its curvature.  If we take the example from the column, "More About Integral Curves," where the differential equation is

and the solution turns out to be,

We can use the chain rule to write the differential equation in terms of the parameter

If we choose,

then

We can define a set of functions,

and

and thus define a vector field

We can plot this vector field,

We can use the differential geometry of previous columns to study individual trajectories of this vector field.  Such a vector field is called a phase portrait and it can tell us a lot about the behavior of the solutions of differential equations.  We will study such curves in more depth in the future.

Mathematics Challenge from Last Time

We have discussed last week's Mathematics Challenge in this article and will continue to do so over the coming weeks.

Mathematics Challenge

How do we find the curvature and arc length of an integral curve.

Sources About Differential Equations

On-Line:

The ultimate resource for those interested in research related to differential equations is the Electronic Journal of Differential Equations, which is free:

http://ejde.math.swt.edu/

Another nice site that covers lots of topics, though with only a few examples, is the SOS Mathematics page for differential equations:

http://www.sosmath.com/diffeq/diffeq.html

Another free online journal is the Electronic Journal of Qualitative Theory of Differential Equations:

http://www.math.u-szeged.hu/ejqtde/

An online course in differential equations is given at this site:

http://math.stcc.edu/DiffEq/DiffEq.html

Thanks to Dan Lasly for pointing out this site!

http://www.mathforum.com/dr.math/

Books:

Martin Braun (1993), Differential Equations and Their Applications, Springer-Verlag.  This is a nice book crammed full of applications.

George F. Simmons (1991), Differential Equations with Applications and Historical Notes, McGraw-Hill.  This is one of my favorite differential equations books, it starts out at an elementary level and includes nonlinear equations and partial differential equations.  My columns will loosely follow this book up to a point and will then take a more geometrical viewpoint.

John W. Dettman (1974), Introduction to Linear Algebra and Differential Equations, McGraw-Hill (reprinted in 1986 by Dover Publications). This is a wonderful book that is also inexpensive; it begins with a discussion of complex numbers, then linear algebra, and then differential equations.

Morris Tenenbaum and Harry Pollard (1963), Ordinary Differential Equations, Harper and Row (reprinted in 1985 by Dover Publications). This is an inexpensive and encyclopedic treatment of the subject.

Vladimir I. Arnold (1973), Ordinary Differential Equations, MIT Press (as of 1991 in its 8th printing, translated from the Russian by Richard A. Silverman).  This is my favorite book of differential equations at a fairly elementary level.  What makes this book so remarkable is the use of geometric methods.  My columns will follow this book after a firm foundation has been laid with traditional methods.

Sources About Differential Geometry of Curves

On-line:

A nice set of lectures is located here,

http://noodle.med.yale.edu/seminar/shi/lecture1.pdf

Wikipedia is very nice and free,

http://en.wikipedia.org/wiki/Differential_geometry

Here is a free textbook online

http://people.uncw.edu/lugo/COURSES/DiffGeom/dg1.htm

Books:

Heinrich W. Guggenheimer (1963), Differential Geometry, McGraw-Hill Book Company (reprinted in 1977 by Dover Publications).  This is a very complete book, with lots of useful information.

Dirk J. Struik (1961), Lectures on Classical Differential Geometry, Addison Wesley (reprinted in 1988 by Dover Publications).  This is a nice set of lectures on classical differential geometry.

Alfred Gray (1998), Modern Differential Geometry of Curves and Surfaces with Mathematica 2nd Edition, CRC Press. This is a huge tome that not only covers the basic theory of differential geometry, it also covers how to develop Mathematica programs for it.

Created by Mathematica  (April 9, 2004)