26: Fundamental Theorem of Calculus for Euclidean-Normed Spaces Map

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description/proof of the fundamental theorem of calculus for $C^1$, Euclidean-normed spaces map

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

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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.

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Target Context

• The reader will have a description and a proof of the fundamental theorem of calculus for $C^1$, Euclidean-normed spaces map.

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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.

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Main Body

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1: Description

For any Euclidean-normed spaces, ${\mathbb{R}}^{d1}$ and ${\mathbb{R}}^{d2}$, and any $C^1$ map, $f:{\mathbb{R}}^{d1}\to {\mathbb{R}}^{d2}$, the derivative of the map, $Df$, is related with integral as $f(v_{11}+t{v}_{12})=f(v_{11})+{\int }_0^{t}Df(v_{11}+s{v}_{12})v_{12}ds$ where $v_{11}$ and $v_{12}$ are any vectors on ${\mathbb{R}}^{d1}$ and t and s are any real numbers.

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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, $$f_i(v_{11}+t{v}_{12})=f_i(v_{11})+{\int }_0^t\frac{\partial f_i}{\partial v_1^j}(v_{11}+s{v}_{12})v_{12}^{j}ds.$$ That is $$f(v_{11}+t{v}_{12})=f(v_{11})+{\int }_0^{t}Df(v_{11}+s{v}_{12})v_{12}ds,$$ because the derivative of the map is the Jacobian.

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References

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