description/proof of the chain rule for derivative of composition of \(C^1\), Euclidean-normed Euclidean vectors spaces maps
Topics
About: normed vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of Euclidean-normed Euclidean vectors space.
- The reader knows a definition of matrix norm induced by vector norm.
- The reader knows a definition of map.
- The reader knows a definition of derivative of normed vectors spaces map.
- The reader admits some properties of matrix norm induced by vector norm.
Target Context
- The reader will have a description and a proof of the chain rule for derivative of composition map of \(C^1\), Euclidean-normed Euclidean vectors spaces maps.
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: Description
For any Euclidean-normed Euclidean vectors spaces, \(\mathbb{R}^{d_1}\), \(\mathbb{R}^{d_2}\), and \(\mathbb{R}^{d_3}\), any \(C^1\) maps, \(f_1: \mathbb{R}^{d_1} \to \mathbb{R}^{d_2}\) and \(f_2: \mathbb{R}^{d_2} \to \mathbb{R}^{d_3}\), and the composition map, \(f_3: \mathbb{R}^{d_1} \to \mathbb{R}^{d_3}\), such that \(f_3 = f_2 (f_1)\), the derivative of \(f_3\), \(D f_3\), is \(D f_2 D f_1\).
2: Proof
For any vectors, \(v_{1, 1}, v_{1, 2} \in \mathbb{R}^{d_1}\), \(f_3 (v_{1, 1} + v_{1, 2}) = f_2 (f_1 (v_{1, 1} + v_{1, 2})) = f_2 (f_1 (v_{1, 1}) + f_1' (v_{1, 1}) v_{1, 2} + r_1 (v_{1, 1}, v_{1, 2})) = f_2 (f_1 (v_{1, 1})) + f_2' (f_1 (v_{1, 1})) (f_1' (v_{1, 1}) v_{1, 2} + r_1 (v_{1, 1}, v_{1, 2})) + r_2 (f_1 (v_{1, 1}), f_1' (v_{1, 1}) v_{1, 2} + r_1 (v_{1, 1}, v_{1, 2})) := V_1\). As \(f_2'\) is linear, \(V_1 = f_2 (f_1 (v_{1, 1})) + f_2' (f_1 (v_{1, 1})) f_1' (v_{1, 1}) v_{1, 2} + f_2' (f_1 (v_{1, 1})) r_1 (v_{1, 1}, v_{1, 2}) + r_2 (f_1 (v_{1, 1}), f_1' (v_{1, 1}) v_{1, 2} + r_1 (v_{1, 1}, v_{1, 2}))\).
But as \(\Vert f_2' (f_1 (v_{1, 1})) r_1 (v_{1, 1}, v_{1, 2}) \Vert \le \Vert f_2' (f_1 (v_{1, 1})) \Vert \Vert r_1 (v_{1, 1}, v_{1, 2})\Vert\), \(lim_{v_{1, 2} \to 0} \frac{\Vert f_2' (f_1 (v_{1, 1})) r_1 (v_{1, 1}, v_{1, 2}) \Vert}{\Vert v_{1, 2} \Vert} = 0\), and \(\frac{\Vert r_2 (f_1 (v_{1, 1}), f_1' (v_{1, 1}) v_{1, 2} + r_1 (v_{1, 1}, v_{1, 2})) \Vert}{\Vert f_1' (v_{1, 1}) v_{1, 2} + r_1 (v_{1, 1}, v_{1, 2}) \Vert} = \frac{\Vert r_2 (f_1 (v_{1, 1}), f_1' (v_{1, 1}) v_{1, 2} + r_1 (v_{1, 1}, v_{1, 2})) \Vert}{\Vert v_{1, 2} \Vert} \frac{\Vert v_{1, 2} \Vert}{\Vert f_1' (v_{1, 1}) v_{1, 2} + r_1 (v_{1, 1}, v_{1, 2})\Vert} := V_2\).
But \(\frac{\Vert f_1' (v_{1, 1}) v_{1, 2} + r_1 (v_{1, 1}, v_{1, 2}) \Vert}{\Vert v_{1, 2} \Vert} \le \frac{\Vert f_1' (v_{1, 1}) v_{1, 2} \Vert}{\Vert v_{1, 2} \Vert} + \frac{\Vert r_1 (v_{1, 1}, v_{1, 2}) \Vert}{\Vert v_{1, 2} \Vert}\) where \(\lim_{v_{1, 2} \to 0} \frac{\Vert r_1 (v_{1, 1}, v_{1, 2}) \Vert}{\Vert v_{12} \Vert} = 0\) and \(\frac{\Vert f_1' (v_{1, 1}) v_{1, 2} \Vert}{\Vert v_{1, 2}\Vert} \le \frac{\Vert f_1' (v_{1, 1}) \Vert \Vert v_{1, 2} \Vert}{\Vert v_{1, 2} \Vert} = \Vert f_1' (v_{1, 1})\Vert\). So, as \(v_{1, 2} \to 0\), \(f_1' (v_{1, 1}) v_{1, 2} + r_1 (v_{1, 1}, v_{1, 2}) \to 0\), and so \(V_2 \to 0\), but as \(\frac{\Vert v_{1, 2}\Vert}{\Vert f_1' (v_{1, 1}) v_{1, 2} + r_1 (v_{1, 1}, v_{1, 2})\Vert}\) does not approach 0, \(lim_{v_{1, 2} \to 0} \frac{\Vert r_2 (f_1 (v_{1, 1}), f_1' (v_{1, 1}) v_{1, 2} + r_1 (v_{1, 1}, v_{1, 2}))\Vert}{\Vert v_{1, 2} \Vert} = 0\).
