Physics Library
 An open source physics library
Encyclopedia | Forums | Docs | Random | Template Test |  
Login
create new user
Username:
Password:
forget your password?
Main Menu
Sections

Talkback

Downloads

Information
fundamental theorem of integral calculus (Theorem)

The derivative of a real function, which has on a whole interval a constant value $c$, vanishes in every point of this interval:

$\displaystyle \frac{d}{dx}c \;=\; 0$

The converse theorem of this is also true.  Ernst Lindelöf calls it the fundamental theorem of integral calculus (in Finnish integraalilaskun peruslause).  It can be formulated as

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.

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

$\displaystyle f'(\xi) \;=\; \frac{f(x_1)\!-\!f(x_2)}{x_1\!-\!x_2},$
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.



"fundamental theorem of integral calculus" is owned by pahio.

View style:


Cross-references: theorem, function

This is version 1 of fundamental theorem of integral calculus, born on 2009-05-01.
Object id is 712, canonical name is FundamentalTheoremOfIntegralCalculus.
Accessed 262 times total.

Classification:
Physics Classification: 02.30.-f (Function theory, analysis)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:

No messages.

Testing some escape charachters for html category with a generator has an injective cogenerator" now escape ” with "