View source for Newton-Leibniz formula

The formula expressing the value of a definite integral of a given function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n0665501.png" /> over an interval as the difference of the values at the end points of the interval of any primitive (cf. [[Integral calculus|Integral calculus]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n0665502.png" /> of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n0665503.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n0665504.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

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

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n0665505.png" /> is Lebesgue integrable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n0665506.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n0665507.png" /> is defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n0665508.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n0665509.png" /> is a constant, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n06655010.png" /> is absolutely continuous, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n06655011.png" /> almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n06655012.png" /> (everywhere if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n06655013.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n06655014.png" />) and (*) is valid.

A generalization of the Newton–Leibniz formula is the [[Stokes formula|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. [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.R. Stromberg,   "Introduction to classical real analysis" , Wadsworth  (1981)  pp. 318ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rudin,   "Real and complex analysis" , McGraw-Hill  (1966)  pp. 165ff</TD></TR></table>