The point of this book is to mix together
differential geometry, the calculus of variations and some applications
(e.g. soap film formation, constrained particle motion, Foucault's
pendulum) to see how geometry fits into science and mathematics.
The book includes many Maple procedures that allow students to
view geometry and calculate things such as Euler-Lagrange equations.
In particular, Chapter 5 on geodesics contains a procedure to
plot geodesics on surfaces and this procedure gives beautiful
illustrations of the Clairaut relation for example. The same type
of procedure also allows students to visualize the motion of
a particle constrained to move in bowls (of various shapes) under
gravity. These are the kinds of connections between geometry and
applications which I like and which I think are important for
students to see. Here is an example of a geodesic on the surface
of revolution obtained by revolving the Witch of Agnesi about the
x-axis. Notice how the geodesic is bounded between two parallels.
This is the Clairaut relation in action. By the way, the following
picture is only a first attempt at using Maple to create a JPEG file
for the web --- better things will surely come later!
The picture above was created by a procedure called `plotgeo' which
may be found in Chapter 5 of the book. Here are a few other examples
of geodesics on surfaces constructed from this procedure.
Geodesic on a
Torus
Geodesic
on a Cylinder
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