# Difference between revisions of "Newton-Leibniz formula"

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− | + | The formula expressing the value of a definite integral of a given function $f$ over an interval as the difference of the values at the end points 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. | ||

− | + | 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. |

## Revision as of 08:22, 30 November 2012

The formula expressing the value of a definite integral of a given function $f$ over an interval as the difference of the values at the end points 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.

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.

#### 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://encyclopediaofmath.org/index.php?title=Newton-Leibniz_formula&oldid=22843