# Generalised N-dimensional Riemann Sum

Let $I=[{a}_{1},{b}_{1}]\times \mathrm{\cdots}\times [{a}_{N},{b}_{N}]$ be an $N$-cell in ${\mathbb{R}}^{N}$. For each $j=1,\mathrm{\dots},N$, let $$ be a partition^{} ${P}_{j}$ of $[{a}_{j},{b}_{j}]$. We define a partition $P$ of $I$ as

$$P:={P}_{1}\times \mathrm{\cdots}\times {P}_{N}$$ |

Each partition $P$ of $I$ generates a subdivision of $I$ (denoted by ${({I}_{\nu})}_{\nu}$) of the form

$${I}_{\nu}=[{t}_{1,j},{t}_{1,j+1}]\times \mathrm{\cdots}\times [{t}_{N,k},{t}_{N,k+1}]$$ |

Let $f:U\to {\mathbb{R}}^{M}$ be such that $I\subset U$, and let ${({I}_{\nu})}_{\nu}$ be the corresponding subdivision of a partition $P$ of $I$. For each $\nu $, choose ${x}_{\nu}\in {I}_{\nu}$. Define

$$S(f,P):=\sum _{\nu}f({x}_{\nu})\mu (I\nu )$$ |

As the Riemann sum of $f$ corresponding to the partition $P$.

A partition $Q$ of $I$ is called a refinement of $P$ if $P\subset Q$.

Title | Generalised N-dimensional Riemann Sum |
---|---|

Canonical name | GeneralisedNdimensionalRiemannSum |

Date of creation | 2013-03-22 13:37:40 |

Last modified on | 2013-03-22 13:37:40 |

Owner | vernondalhart (2191) |

Last modified by | vernondalhart (2191) |

Numerical id | 4 |

Author | vernondalhart (2191) |

Entry type | Definition |

Classification | msc 26B12 |