definition of derivative of real-1-parameter family of vectors in finite-dimensional real vectors space
Topics
About: vectors space
About: derivative
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of %field name% vectors space.
Target Context
- The reader will have a definition of derivative of real-1-parameter family of vectors in finite-dimensional real vectors space.
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:
\( V\): \(\in \{\text{ the finite-dimensional real vectors spaces }\}\)
\( K\): \(\in \{\text{ the finite index sets }\}\)
\( \{e_j\}\): \(\in \{\text{ the bases of } V\}\), \(j \in K\), \(e_j \in V\),
\( J\): \(= (t_1, t_2) \subseteq \mathbb{R}\)
\( v\): \(: J \to V\), \(= v^j (t) e_j\)
\(*\frac{d v (t)}{d t}\): \(= \frac{d v^j (t)}{d t} e_j\)
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Conditions:
\(\frac{d v^j (t)}{d t}\) s exist.
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2: Natural Language Description
For any finite-dimensional real vectors space, \(V\), any interval, \(J = (t_1, t_2) \subseteq \mathbb{R}\), and any 1-parameter family of vectors, \(v: J \to V\), \(\frac{d v (t)}{d t} = \frac{d v^j (t)}{d t} e_j\),if it exists, where \({e_j \in V \vert j \in K}\) is any basis of \(V\) and \(v (t) = v^j e_j\)
3: Note
The definition does not really depend on the choice of the basis: for any bases, \({e_{1, j}}\) and \({e_{2, j}}\), where \(e_{1, j} = e_{2, k} T^k_j\) with a matrix, \(T\), \(v (t) = v_1^j (t) e_{1, j} = v_2^j (t) e_{2, j}\), where \(v_2^j (t) = T^j_k v_1^k (t)\), because \(v (t) = v_1^j (t) e_{1, j} = v_1^j (t) e_{2, k} T^k_j = v_1^k (t) e_{2, j} T^j_k\), and \(\frac{d v_2^j (t)}{d t} e_{2, j} = \frac{d T^j_k v_1^k (t)}{d t} e_{2, j} = T^j_k \frac{d v_1^k (t)}{d t} e_{2, j} = \frac{d v_1^k (t)}{d t} e_{2, j} T^j_k = \frac{d v_1^k (t)}{d t} e_{1, k} = \frac{d v_1^j (t)}{d t} e_{1, j}\).
