Microworld: Calculus
in Action: Chapter 3, Section 1: Plane Curvature
Click the Hyperlink
above to visit the Microworld.
Author:James
E. White
In
the previous Chapter, we described satellite orbits, following Newton, with
plane curves. And in the next Chapter we will see that those curves are ellipses
for the planets, and conic sections in general, if the satellites satisfy
Newton's law of gravitation.
But
Science does not move in a straight line. We have come to understand, through
the work of Gauss, Riemann, and Einstein (among others) that curves can have
intrinsic geometric properties. One of those properties is the curvature that
defines at each point the osculating circle. In order for such an idea to
have meaning, we must assume that there is an absolute meaning to distance.
Once we accept, as Gauss counseled us to do, that distance has an absolute
meaning, then we discover paradoxically strange new Geometries, such as the
Riemannian Geometry of a sphere, or the Einsteinian Geometry of spacetime.
This absolute notion of distance was named by Gauss. He called it the metric.
The physical metric in spacetime is in our view today the simplest description
of gravitation.
The
first topic we take up is the notion of invariance under parameterization.
Intrinsic geometric properties of curves and surfaces must be independent
of the manner in which they are described (or parameterized). One of the first
invariants that we will study is the plane curvature of a curve.
In
the next section, we revisit this topic with a vivid illustration of the meaning
of this curvature using a special parameterization for the curve called the
arc-length parameterization. This topic gives us the occasion to discuss the
second great "art of approximation," the method of Integration.
To
this end, we state and prove the Fundamental Theorem of Calculus in the context
of our discussion of arc length parameterization. We use it to guarantee the
existence of, and to characterize, this important way of parametrizing curves.
This
will bring us in the final section to the threshold of the Gaussian Curvature
of Surfaces, the topic that opened the door in the hands of Riemann, to an
infinite variety of geometries, called "non-Euclidean." These geometries
led in turn to a new view of the physical world with Einstein's General Theory
of Relativity. While we cannot visit those subjects here, we will present
the plane curvature of curves using the Gauss map, and will see in it the
simplest interpretation of that invariant. It is a rich and beautiful construction,
and an appropriate note on which to close our discussion of plane curvature.
The
following is the Table of Contents for this 5 page Microworld
- Velocity Moving Frames
- Arc Length and the Fundamental Theorem of Calculus
- The Gauss Map
- Symbolic Calculator
Each page
of the Microworld, including the Calculator page has the story for that page
under the 
icon.
Just click on this icon to read the story for the page. The Calculator is
quite versatile, and so we recommend you read through the instructions there
to become familiar with it.
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| - James E. White, Ph.D. , Library Director, | ||
| author of this website, Mathwright 2000, MindScapes, | ||
| MathwrightWeb, and Mathwright32 |
