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.
The theorem expressed by the Newton–Leibniz formula is called the fundamental theorem of calculus, cf. e.g. [a1].
|[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://www.encyclopediaofmath.org/index.php?title=Newton-Leibniz_formula&oldid=28964