description/proof of the chain rule for derivative of composition of \(C^1\), Euclidean-normed Euclidean vectors spaces maps
Topics
About: 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 map from open subset of Euclidean \(C^\infty\) manifold into subset of Euclidean \(C^\infty\) manifold \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\).
- The reader knows a definition of derivative of map from open subset of normed vectors space into subset of normed vectors space at point.
- The reader knows a definition of matrix norm induced by vector norms.
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: Structured Description
Here is the rules of Structured Description.
Entities:
\(\mathbb{R}^{d_1}\): \(= \text{ the Euclidean-normed Euclidean vectors space }\) also as the Euclidean \(C^\infty\) manifold
\(\mathbb{R}^{d_2}\): \(= \text{ the Euclidean-normed Euclidean vectors space }\) also as the Euclidean \(C^\infty\) manifold
\(\mathbb{R}^{d_3}\): \(= \text{ the Euclidean-normed Euclidean vectors space }\) also as the Euclidean \(C^\infty\) manifold
\(f_1\): \(: \mathbb{R}^{d_1} \to \mathbb{R}^{d_2}\), \(\in \{\text{ the } C^1 \text{ maps }\}\)
\(f_2\): \(: \mathbb{R}^{d_2} \to \mathbb{R}^{d_3}\), \(\in \{\text{ the } C^1 \text{ maps }\}\)
\(f_2 \circ f_1\): \(\mathbb{R}^{d_1} \to \mathbb{R}^{d_3}\)
\(v_1\): \(\in \mathbb{R}^{d_1}\)
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Statements:
\(D (f_2 \circ f_1)_{v_1} = D {f_2}_{f_1 (v_1)} \circ D {f_1}_{v_1}\)
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2: Proof
Whole Strategy: Step 1: see that \(lim_{v'_1 \to 0} \Vert f_2 \circ f_1 (v_1 + v'_1) - (f_2 (f_1 (v_1)) + D {f_2}_{f_1 (v_1)} D {f_1}_{v_1} v'_1) \Vert / \Vert v'_1 \Vert = 0\).
Step 1:
Let \(v_1, v'_1 \in \mathbb{R}^{d_1}\) be any.
\(f_2 \circ f_1 (v_1 + v'_1) = f_2 (f_1 (v_1 + v'_1)) = f_2 (f_1 (v_1) + D {f_1}_{v_1} v'_1 + r_1 (v_1, v'_1)) = f_2 (f_1 (v_1)) + D {f_2}_{f_1 (v_1)} (D {f_1}_{v_1} v'_1 + r_1 (v_1, v'_1)) + r_2 (f_1 (v_1), D {f_1}_{v_1} v'_1 + r_1 (v_1, v'_1)) := V1\).
As \(D {f_2}_{f_1 (v_1)}\) is linear, \(V1 = f_2 (f_1 (v_1)) + D {f_2}_{f_1 (v_1)} D {f_1}_{v_1} v'_1 + D {f_2}_{f_1 (v_1)} r_1 (v_1, v'_1) + r_2 (f_1 (v_1), D {f_1}_{v_1} v'_1 + r_1 (v_1, v'_1))\).
What we need to see is that \(lim_{v'_1 \to 0} \Vert D {f_2}_{f_1 (v_1)} r_1 (v_1, v'_1) + r_2 (f_1 (v_1), D {f_1}_{v_1} v'_1 + r_1 (v_1, v'_1)) \Vert / \Vert v'_1 \Vert = 0\).
As \(\Vert D {f_2}_{f_1 (v_1)} r_1 (v_1, v'_1) + r_2 (f_1 (v_1), D {f_1}_{v_1} v'_1 + r_1 (v_1, v'_1)) \Vert / \Vert v'_1 \Vert \le \Vert D {f_2}_{f_1 (v_1)} r_1 (v_1, v'_1) \Vert / \Vert v'_1 \Vert + \Vert r_2 (f_1 (v_1), D {f_1}_{v_1} v'_1 + r_1 (v_1, v'_1)) \Vert / \Vert v'_1 \Vert\), seeing that \(lim_{v'_1 \to 0} \Vert D {f_2}_{f_1 (v_1)} r_1 (v_1, v'_1) \Vert / \Vert v'_1 \Vert = 0\) and \(lim_{v'_1 \to 0} \Vert r_2 (f_1 (v_1), D {f_1}_{v_1} v'_1 + r_1 (v_1, v'_1)) \Vert / \Vert v'_1 \Vert = 0\) is enough.
Let us see the former.
As \(\Vert D {f_2}_{f_1 (v_1)} r_1 (v_1, v'_1) \Vert \le \Vert D {f_2}_{f_1 (v_1)} \Vert \Vert r_1 (v_1, v'_1) \Vert\), where \(\Vert D {f_2}_{f_1 (v_1)} \Vert\) is the matrix norm induced by vector norms, \(\Vert D {f_2}_{f_1 (v_1)} r_1 (v_1, v'_1) \Vert / \Vert v'_1 \Vert \le \Vert D {f_2}_{f_1 (v_1)} \Vert \Vert r_1 (v_1, v'_1) \Vert / \Vert v'_1 \Vert\), and \(lim_{v'_1 \to 0} \Vert D {f_2}_{f_1 (v_1)} r_1 (v_1, v'_1) \Vert / \Vert v'_1 \Vert = 0\), because \(lim_{v'_1 \to 0} \Vert r_1 (v_1, v'_1) \Vert / \Vert v'_1 \Vert = 0\).
Let us see the latter.
\(\Vert r_2 (f_1 (v_1), D {f_1}_{v_1} v'_1 + r_1 (v_1, v'_1)) \Vert / \Vert D {f_1}_{v_1} v'_1 + r_1 (v_1, v'_1) \Vert = \Vert r_2 (f_1 (v_1), D {f_1}_{v_1} v'_1 + r_1 (v_1, v'_1)) \Vert / \Vert v'_1 \Vert \Vert v'_1 \Vert / \Vert D {f_1}_{v_1} v'_1 + r_1 (v_1, v'_1) \Vert := V2\).
But \(\Vert D {f_1}_{v_1} v'_1 + r_1 (v_1, v'_1) \Vert / \Vert v'_1 \Vert \le \Vert D {f_1}_{v_1} v'_1 \Vert / \Vert v'_1 \Vert + \Vert r_1 (v_1, v'_1) \Vert / \Vert v'_1 \Vert \le \Vert D {f_1}_{v_1} \Vert \Vert v'_1 \Vert / \Vert v'_1 \Vert + \Vert r_1 (v_1, v'_1) \Vert / \Vert v'_1 \Vert = \Vert D {f_1}_{v_1} \Vert + \Vert r_1 (v_1, v'_1) \Vert / \Vert v'_1 \Vert\), where \(\lim_{v'_1 \to 0} \Vert r_1 (v_1, v'_1) \Vert / \Vert v'_1 \Vert = 0\), so, \(lim_{v'_1 \to 0} \Vert D {f_1}_{v_1} v'_1 + r_1 (v_1, v'_1) \Vert / \Vert v'_1 \Vert \le \Vert D {f_1}_{v_1} \Vert\ \lt \infty\).
So, as \(v'_1 \to 0\), \(D {f_1}_{v_1} v'_1 + r_1 (v_1, v'_1) \to 0\), and so \(V2 \to 0\), because of the property of \(r_2\), but as \(\Vert v'_1 \Vert / \Vert D {f_1}_{v_1} v'_1 + r_1 (v_1, v'_1) \Vert\) does not approach \(0\), \(lim_{v'_1 \to 0} \Vert r_2 (f_1 (v_1), D {f_1} (v_1) v'_1 + r_1 (v_1, v'_1)) \Vert / \Vert v'_1 \Vert = 0\).
So, \(D (f_2 \circ f_1)_{v_1} = D {f_2}_{f_1 (v_1)} \circ D {f_1}_{v_1}\).
