Talk:PlanetPhysics/Fundamental Theorem of Integral Calculus
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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: fundamental theorem of integral calculus %%% Primary Category Code: 02.30.-f %%% Filename: FundamentalTheoremOfIntegralCalculus.tex %%% Version: 1 %%% Owner: pahio %%% Author(s): pahio %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}
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The derivative of a real \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html}, which has on a whole interval a constant value $c$, vanishes in every point of this interval:
$$\frac{d}{dx}c \;=\; 0$$\\
The converse \htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html} of this is also true.\, Ernst Lindel\"of calls it the \emph{fundamental theorem of integral calculus} (in Finnish \emph{integraalilaskun peruslause}).\, It can be formulated as
\textbf{Theorem.}\, If a real function in continuous and its derivative vanishes in all points of an interval, the value of this function does not change on this interval.
\emph{Proof.}\, We make the antithesis that there were on the interval two distinct points $x_1$ and $x_2$ with\, $f(x_1) \neq f(x_2)$.\, Then the mean-value theorem guarantees a point $\xi$ between $x_1$ and $x_2$ such that $$f'(\xi) \;=\; \frac{f(x_1)\!-\!f(x_2)}{x_1\!-\!x_2},$$ which value is distinct from zero.\, This is, however, impossible by the assumption of the theorem.\, So the antithesis is wrong and the theorem right.\\
The contents of the theorem may be expressed also such that if two functions have the same derivative on a whole interval, then the difference of the functions is constant on this interval.\, Accordingly, if $F$ is an antiderivative of a function $f$, then any other antiderivative of $f$ has the form $x \mapsto F(x)\!+\!C$, where $C$ is a constant.
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