description/proof of the fundamental theorem of calculus for \(C^1\), Euclidean-normed spaces map
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of Euclidean-normed space.
- The reader knows a definition of map.
- The reader knows a definition of derivative of normed spaces map.
- The reader admits the fundamental theorem of calculus for 1-argument 1-value function.
- The reader admits the propositions that derivative of \(C^1\), Euclidean-normed spaces map is the Jacobian.
Target Context
- The reader will have a description and a proof of the fundamental theorem of calculus for \(C^1\), Euclidean-normed spaces map.
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 spaces, \(\mathbb{R}^{d1}\) and \(\mathbb{R}^{d2}\), and any \(C^1\) map, \(f: \mathbb{R}^{d1} \rightarrow \mathbb{R}^{d2}\), the derivative of the map, \(Df\), is related with integral as \(f (v_{11} + t v_{12}) = f (v_{11}) + \int^t_0 Df (v_{11} + s v_{12}) v_{12} ds\) where \(v_{11}\) and \(v_{12}\) are any vectors on \(\mathbb{R}^{d1}\) and t and s are any real numbers.
2: Proof
As \(f_i (v_{11} + t v_{12})\) can be regarded as a 1-argument 1-value function from \(\mathbb{R}\) to \(\mathbb{R}\) with respect to t, by the fundamental theorem of calculus for 1-argument 1-value function, $$f_i (v_{11} + t v_{12}) = f_i (v_{11}) + \int^t_0 \frac{\partial f_i}{\partial v_1^j} (v_{11} + s v_{12}) v_{12}^j ds.$$ That is $$f (v_{11} + t v_{12}) = f (v_{11}) + \int^t_0 Df (v_{11} + s v_{12}) v_{12} ds,$$ because the derivative of the map is the Jacobian.
