description/proof of mean-value theorem for differentiable map from closed interval into \(1\)-dimensional Euclidean \(C^\infty\) manifold
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of map between arbitrary subsets of Euclidean \(C^\infty\) manifolds differentiable at point.
- The reader admits the proposition that the image of any continuous map from any compact topological space into the \(\mathbb{R}\) Euclidean topological space has the minimum and the maximum.
- The reader admits Fermat's theorem for partial derivatives at local maximum or minimum point: for any differentiable map from any open subset of any Euclidean \(C^\infty\) manifold into the \(1\)-dimensional Euclidean \(C^\infty\) manifold, if the map has the local maximum or the local minimum at any point, the partial derivatives at the point are \(0\).
Target Context
- The reader will have a description and a proof of the mean-value theorem for any differentiable map from any closed interval into the \(1\)-dimensional Euclidean \(C^\infty\) manifold.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(\mathbb{R}\): \(= \text{ the Euclidean } C^\infty \text{ manifold }\)
\(J\): \(= [r_1, r_2] \subseteq \mathbb{R}\) such that \(r_1 \lt r_2\)
\(f\): \(: J \to \mathbb{R}\), \(\in \{\text{ the differentiable maps }\}\)
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Statements:
\(\exists r \in (r_1, r_2) ({\partial_1 f}_r = (f (r_2) - f (r_1)) / (r_2 - r_1))\)
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2: Proof
Whole Strategy: Step 1: take \(f' : J \to \mathbb{R}, x \mapsto f (x) - (f (r_2) - f (r_1)) / (r_2 - r_1) x\); Step 2: see that \(f'\) has the maximum or the minimum at an \(r \in (r_1, r_2)\); Step 3: see that \({\partial_1 f'}_r = 0\).
Step 1:
Let us take \(f' : J \to \mathbb{R}, x \mapsto f (x) - (f (r_2) - f (r_1)) / (r_2 - r_1) x\).
Step 2:
\(f' (r_1) = f (r_1) - (f (r_2) - f (r_1)) / (r_2 - r_1) r_1 = (f (r_1) (r_2 - r_1) - (f (r_2) - f (r_1)) r_1) / (r_2 - r_1) = (f (r_1) r_2 - f (r_2) r_1) / (r_2 - r_1)\); \(f' (r_2) = f (r_2) - (f (r_2) - f (r_1)) / (r_2 - r_1) r_2 = (f (r_2) (r_2 - r_1) - (f (r_2) - f (r_1)) r_2) / (r_2 - r_1) = (f (r_2) (- r_1) - (- f (r_1)) r_2) / (r_2 - r_1) = (f (r_1) r_2 - f (r_2) r_1) / (r_2 - r_1)\), so, \(f' (r_1) = f' (r_2)\).
\(f'\) is continuous and \([r_1, r_2]\) is compact, so, \(f' ([r_1, r_2])\) has the maximum and the minimum, by the proposition that the image of any continuous map from any compact topological space into the \(\mathbb{R}\) Euclidean topological space has the minimum and the maximum.
If \(f'\) takes the both of the maximum and the minimum at \(r_1\) or \(r_2\), as \(f' (r_1) = f' (r_2)\), \(f'\) will be constant, so, at any \(r \in (r_1, r_2)\), \(f'\) will have the maximum.
Otherwise, \(f'\) will have the maximum or the minimum at an \(r \in (r_1, r_2)\).
So, anyway, \(f'\) has the maximum or the minimum at an \(r \in (r_1, r_2)\).
Step 3:
\({\partial_1 f'}_r = 0\), by Fermat's theorem for partial derivatives at local maximum or minimum point: for any differentiable map from any open subset of any Euclidean \(C^\infty\) manifold into the \(1\)-dimensional Euclidean \(C^\infty\) manifold, if the map has the local maximum or the local minimum at any point, the partial derivatives at the point are \(0\).
\({\partial_1 f'}_r = {\partial_1 f}_r - (f (r_2) - f (r_1)) / (r_2 - r_1)\).
So, \({\partial_1 f}_r = (f (r_2) - f (r_1)) / (r_2 - r_1)\).
