A definition of derivative of real-1-parameter family of vectors
Topics
About: vectors space
About: derivative
The table of contents of this article
Starting Context
- The reader knows a definition of vectors space.
Target Context
- The reader will have a definition of the derivative of real-1-parameter family of vectors.
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: Definition
For any vectors space, V, and any 1-parameter family of vectors, \(v (t) \in V\), where t is an \(\mathbb{R}\) interval, \(\frac{dv (t)}{dt} = \frac{dv^i (t)}{dt} e_i\) where \({e_i}\) is any basis of V and \({v^i}\) are the vector components with respect to the basis. The definition does not depend on the choice of the basis, because for bases, \({e_{1i}}\) and \({e_{2i}}\), and the components, \({v_1^i}\) and \({v_2^i}\) of the same vector, v, with respect to the bases, \(e_{1i} = T^j_ie_{2j}\) with a matrix, T, and \(v_2^i = T^i_jv_1^j\), because \(v (t) = v_1^ie_{1i} = v_1^iT^j_ie_{2j}\), and \(\frac{d v_2^i}{d t}e_{2i} = \frac{d T^i_jv_1^j}{d t}e_{2i} = T^i_j\frac{d v_1^j}{d t}e_{2i} = \frac{d v_1^j}{d t}e_{1j}\).