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Newton-Leibniz formula

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The formula expressing the value of a definite integral of a given function over an interval as the difference of the values at the end points of the interval of any primitive (cf. Integral calculus) of the function :

(*)

It is named after I. Newton and G. Leibniz, who both knew the rule expressed by (*), although it was published later.

If is Lebesgue integrable over and is defined by

where is a constant, then is absolutely continuous, almost-everywhere on (everywhere if is continuous on ) and (*) is valid.

A generalization of the Newton–Leibniz formula is the Stokes formula for orientable manifolds with a boundary.


Comments

The theorem expressed by the Newton–Leibniz formula is called the fundamental theorem of calculus, cf. e.g. [a1].

References

[a1] K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) pp. 318ff
[a2] W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 165ff
How to Cite This Entry:
Newton-Leibniz formula. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Newton-Leibniz_formula&oldid=17006
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article