Increment theorem

Increment theorem

In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function "f" is differentiable at "x" and that Δ"x" is infinitesimal. Then

:Delta f = f'(x),Delta x + varepsilon, Delta x,

for some infinitesimal ε, where

: Delta f=f(x+Delta x)-f(x).,

If scriptstyleDelta x ot=0 then we may write

: frac{Delta f}{Delta x} = f'(x)+varepsilon,

which implies that scriptstylefrac{Delta f}{Delta x}approx f'(x), or in other words that scriptstyle frac{Delta f}{Delta x} is infinitely close to scriptstyle f'(x),.

See also

*Nonstandard calculus
*Calculus
*Abraham Robinson

References

* H. Jerome Keisler: "Elementary Calculus: An Approach Using Infinitesimals". First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html
*cite book|last=Robinson|first=Abraham| title=Non-standard analysis|year=1996| edition=Revised edition | publisher=Princeton University Press| id=ISBN 0-691-04490-2


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • List of theorems — This is a list of theorems, by Wikipedia page. See also *list of fundamental theorems *list of lemmas *list of conjectures *list of inequalities *list of mathematical proofs *list of misnamed theorems *Existence theorem *Classification of finite… …   Wikipedia

  • List of mathematics articles (I) — NOTOC Ia IA automorphism ICER Icosagon Icosahedral 120 cell Icosahedral prism Icosahedral symmetry Icosahedron Icosian Calculus Icosian game Icosidodecadodecahedron Icosidodecahedron Icositetrachoric honeycomb Icositruncated dodecadodecahedron… …   Wikipedia

  • Non-standard calculus — Abraham Robinson Contents 1 Motivation …   Wikipedia

  • Pierre de Fermat — Born August 17, 1601( …   Wikipedia

  • Non-standard analysis — Abraham Robinson Gottfried Wilhelm Leibniz argued tha …   Wikipedia

  • Dirac delta function — Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually used to specify the value of any multiplicative constant, which will give the area under the function. The other convention… …   Wikipedia

  • Differential (infinitesimal) — For other uses of differential in calculus, see differential (calculus), and for more general meanings, see differential. In calculus, a differential is traditionally an infinitesimally small change in a variable. For example, if x is a variable …   Wikipedia

  • Cavalieri's principle — Two stacks of coins with the same volume In geometry, Cavalieri s principle, sometimes called the method of indivisibles, named after Bonaventura Cavalieri, is as follows:[1] 2 dimensional case: Suppose two regions in a plane are included between …   Wikipedia

  • Surreal number — In mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share… …   Wikipedia

  • Hyperreal number — *R redirects here. For R*, see Rockstar Games. The system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”