ONT Re: Differential Geometry for Engineers
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Note 4
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| 2. Manifolds And Their Maps
|
| The first part of this chapter is devoted to the concept of a manifold.
| It is defined first by a projection then by a more useful though less
| intuitive definition. Finally, it is seen how implicitly defined
| functions give manifolds. Examples are considered both to enhance
| intuition and to bring out conceptual details. The idea of a
| manifold is brought out more clearly by considering mappings
| between manifolds. The properties of these mappings occupy
| the last part of this chapter.
|
| 2.1. Differentiable Manifolds
|
| Although the detailed global description of a manifold
| can be quite complicated, basically a differentiable
| manifold is just a topological space (X, !W!) that
| in the neighborhood of each point looks like an
| open subset of R^k. (In the notation (X, !W!),
| X is some set and !W! consists of all the sets
| defined as open in X and that characterize its
| topology. As to the notation R^k, each point
| in R^k is specified as an ordered set of k
| real numbers. These and other notions
| arising below are discussed in the
| appendix.) This description can
| be formalized into a definition:
|
| 2.1. Definition. A subset M of R^n is a k-dimensional manifold
| if for each x in M there are: open subsets U and V of R^n
| with x in U, and a diffeomorphism f from U to V such that:
|
| f(U |^| M) = {y in V : y_(k+1) = ... = y_n = 0}.
|
| Thus, a point y in the image of f has a representation like:
|
| y = (y_1 (x), y_2 (x), ..., y_k (x), 0, ..., 0).
|
| A straight line is a simple example of a one-dimensional manifold,
| a manifold in R^1. It is a manifold in R^1 even if it is given, for
| example, in R^2. There it might represent the surface of solutions of
| the equation of a particle of unit mass under no forces: x`` = 0 and
| with given initial momentum: x`(t=0) = a. [The (`) is a fluxion dot.]
| In the coordinate system y_1 = x, y_2 = x` - a, the manifold is given by
| the points (y_1, 0). To the particle, its whole world looks like part of
| R^1 though we see its tracks clearly as part of R^2. Any open subset of
| the straight line is also a one-dimensional manifold, but a closed subset
| of it is not.
|
| The sphere in R^3 is an example of a two-dimensional manifold.
| It is an example of a closed manifold and is often denoted as
| S^2. Thus, for a point P in R^3: P = (x_1, x_2, x_3), the
| manifold is given as the set:
|
| S^2 = {P in R^3 : (x_1)^2 + (x_2)^2 + (x_3)^2 - 1 = 0}.
|
| Its two-dimensional character is clear when a point in S^2 is
| given in terms of two variables, say, latitude and longitude.
|
| Doolin & Martin, DGFE, pages 5-7.
|
| Brian F. Doolin & Clyde F. Martin,
|'Introduction to Differential Geometry for Engineers',
| Marcel Dekker, New York, NY, 1990.
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