# Difference between revisions of "Newton-Leibniz formula"

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− | + | The formula expressing the value of a definite integral of a given integrable function $f$ over an interval as the difference of the values at the endpoints of the interval of any primitive (cf. [[Integral calculus|Integral calculus]]) $F$ of the function $f$: | |

+ | \begin{equation}\label{eq:*} | ||

+ | \int\limits_a^bf(x)\,dx = F(b)-F(a). | ||

+ | \end{equation} | ||

+ | It is named after I. Newton and G. Leibniz, who both knew the rule expressed by \ref{eq:*}, although it was published later. It is also known as "Fundamental theorem of calculus". | ||

− | + | If $f$ is [[ Lebesgue integral | Lebesgue integrable]] over $[a,b]$ and $F$ is defined by | |

− | + | \begin{equation*} | |

− | If | + | F(x) = \int\limits_a^xf(t)\,dt + C, |

− | + | \end{equation*} | |

− | + | where $C$ is a constant, then $F$ is [[Absolute_continuity#Absolute_continuity_of_a_function | absolutely continuous]], $F'(x) = f(x)$ almost-everywhere on $[a,b]$ (everywhere if $f$ is continuous on $[a,b]$) and \ref{eq:*} is valid. | |

− | |||

− | where | ||

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

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====References==== | ====References==== | ||

− | + | {| | |

+ | |- | ||

+ | |valign="top"|{{Ref|Ru}}|| W. Rudin, "Real and complex analysis" , McGraw-Hill (1966). | ||

+ | |- | ||

+ | |valign="top"|{{Ref|St}}|| K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981). | ||

+ | |- | ||

+ | |} |

## Latest revision as of 20:49, 8 December 2013

2010 Mathematics Subject Classification: *Primary:* 26A06 *Secondary:* 26A46 [MSN][ZBL]

The formula expressing the value of a definite integral of a given integrable function $f$ over an interval as the difference of the values at the endpoints of the interval of any primitive (cf. Integral calculus) $F$ of the function $f$: \begin{equation}\label{eq:*} \int\limits_a^bf(x)\,dx = F(b)-F(a). \end{equation} It is named after I. Newton and G. Leibniz, who both knew the rule expressed by \ref{eq:*}, although it was published later. It is also known as "Fundamental theorem of calculus".

If $f$ is Lebesgue integrable over $[a,b]$ and $F$ is defined by \begin{equation*} F(x) = \int\limits_a^xf(t)\,dt + C, \end{equation*} where $C$ is a constant, then $F$ is absolutely continuous, $F'(x) = f(x)$ almost-everywhere on $[a,b]$ (everywhere if $f$ is continuous on $[a,b]$) and \ref{eq:*} is valid.

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

#### References

[Ru] | W. Rudin, "Real and complex analysis" , McGraw-Hill (1966). |

[St] | K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981). |

**How to Cite This Entry:**

Newton-Leibniz formula.

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