ONT Re: Differential Geometry for Engineers
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Note 6
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| 2. Manifolds And Their Maps
|
| 2.1. Differentiable Manifolds (cont.)
|
| We opened this discussion of differentiable manifolds with the remark
| that basically a differentiable manifold is a topological space that
| in the neighborhood of each point looks like an open subset of R^k.
| The first definition said that each neighborhood, even though
| expressed as a subset of R^n, was equivalent to R^k. That is,
| the space expressed in R^n really only had k, not n, degrees
| of freedom. Another way of saying this is by saying that a
| k-dimensional manifold can be expressed using n variables
| with n - k conditions imposed on them.
|
| These remarks are made because, in practice, manifolds are often given
| as the set of points where a certain function vanishes. The implicit
| function theorem gives conditions under which the vanishing of the
| function gives k constraints (exchanging the k and the n - k of the
| previous paragraph), so that only n - k of the variables are free,
| and the space is a manifold with dimension n - k if the theorem
| is satisfied everywhere. Then the manifold is said to be given
| implicitly, or by the implicit function theorem.
|
| Formalizing the above remarks, we consider a C^oo function F with
| domain A c R^n and range in R^k. That is, for every choice of n
| real numbers (x_1, ..., x_n) in A, the function F has the k real
| numbers F = (f_1, ..., f_k). Let M be the set:
|
| M = {x : F(x) = 0 = (0, 0, ..., 0)}.
|
| If the rank of the Jacobian matrix F' is equal to k for all x in M,
| then M is an (n-k)-dimensional manifold.
|
| Under the conditions stated, the implicit function theorem says that k of
| the variables can be expressed in terms of the other n - k, and the latter
| can be given values arbitrarily. Another statement of the implicit function
| theorem (see [4, p. 43]) shows that a coordinate transformation can be found
| that assigns the value zero to the k explicit functions. In other words, the
| conditions of the first definition of a manifold are satisfied.
|
| Doolin & Martin, DGFE, pages 10-12.
|
| 4. M. Spivak,
| 'Calculus on Manifolds',
| W.A. Benjamin, New York, NY, 1965.
|
| Brian F. Doolin & Clyde F. Martin,
|'Introduction to Differential Geometry for Engineers',
| Marcel Dekker, New York, NY, 1990.
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