Newton-Leibniz formula
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 |
Newton-Leibniz formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Newton-Leibniz_formula&oldid=17006