Re: closed curve question

On Sun, 10 Jul 2005 03:02:13 GMT, "Dave Eberly"
<dNOSPAMeberly@xxxxxxxxxxxxxxx> wrote:
> Topology: A First Course, by James R. Munkres
> Prentice-Hall, 1975

Munkres does tie issues of calculus to topology, as is only natural.
Long ago topology was called "analysis situs", which hints at its
origins. For example, as soon as we talk about a continuous function
we step onto topology's home turf. Or consider that some questions
about integrals have answers that depend purely on topology. In the
beginning, it was these questions raised in the context of calculus,
differential and integral, that provoked closer study, formalization,
and isolation of topology as a topic in itself.

Working with real numbers, we naturally discuss neighborhoods based on
distance. This is the idea of a metric space. As topology advanced,
other situations required a more flexible way to discuss "what's near
what". This is formalized in the "open set" definition topology uses
today. Graph theory, the idea of nodes and edges, fits comfortably in
modern topology; it would be an alien presence in calculus. Computer
science without graphs is hard to imagine, so we're grateful for the
evolution.

To illustrate the power of the abstractions, consider integrals on two
surfaces, a sphere and a torus. We can distill the topological essence
of each surface by an arrangement of vertices, edges, and faces. By no
more complicated a test than counting, V-E+F, we can see which surface
is which. It doesn't matter how many faces we use! And with this data
we know that some integrals on the torus will cause problems, we will
not be able to simplify them as we might like. (Some buzz words are:
de Rham cohomology, Betti numbers, Euler characteristic, fundamental
group, exact differential.)

Another example, of practical significance to computer graphics: We
learn from topology that there can be no formula to take the aim of a
camera and make an orientation (control for roll) that succeeds for
all directions. Somewhere we must make an exception.

.

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