38: Derivative of Real-1-Parameter Family of Vectors in Finite-Dimensional Real Vectors Space

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definition of derivative of real-1-parameter family of vectors in finite-dimensional real vectors space

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of canonical topology for finite-dimensional real vectors space.
• The reader knows a definition of convergence of map from topological space minus point into topological space with respect to point.

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Target Context

• The reader will have a definition of derivative of real-1-parameter family of vectors in finite-dimensional real vectors space.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$T$: $=(r_1,r_2)\subseteq \mathbb{R}$, with the subspace topology with $\mathbb{R}$ as the Euclidean topological space
$r$: $\in T$
$V$: $\in \{\text{ the finite-dimensional real vectors spaces }\}$, with the canonical topology
$v$: $:T\to V$
$\ast dv/dr$: $=\underset{r^{\prime }\to r}{lim}(v(r^{\prime })-v(r))/(r^{\prime }-r)$
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Conditions:
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2: Note

For any basis, $B=\{b_1,...,b_d\}$, for $V$, $v(r^{\prime })=v^j(r^{\prime })b_j$, and if and only if $d{v}^j/dr$ exists, $dv/dr$ exists and $dv/dr=d{v}^j/dr{b}_j$, by the proposition that for any map from any topological space minus any point into any finite-dimensional real vectors space with the canonical topology, the convergence of the map with respect to the point exists if and only if the convergences of the component maps (with respect to any basis) with respect to the point exist, and then, the convergence Is expressed with the convergences: $dv/dr=\underset{r^{\prime }\to r}{lim}(v(r^{\prime })-v(r))/(r^{\prime }-r)=\underset{r^{\prime }\to r}{lim}(v^j(r^{\prime })b_j-v^j(r)b_j)/(r^{\prime }-r)=\underset{r^{\prime }\to r}{lim}(v^j(r^{\prime })-v^j(r))/(r^{\prime }-r)b_j=d{v}^j/dr{b}_j$.

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References

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