# Generalised N-dimensional Riemann Sum

Let $I=[a_{1},b_{1}]\times\cdots\times[a_{N},b_{N}]$ be an $N$-cell in $\mathbb{R}^{N}$. For each $j=1,\ldots,N$, let $a_{j}=t_{j,0}<\ldots be a partition   $P_{j}$ of $[a_{j},b_{j}]$. We define a partition $P$ of $I$ as

 $P:=P_{1}\times\cdots\times P_{N}$

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

 $I_{\nu}=[t_{1,j},t_{1,j+1}]\times\cdots\times[t_{N,k},t_{N,k+1}]$

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 GeneralisedNdimensionalRiemannSum 2013-03-22 13:37:40 2013-03-22 13:37:40 vernondalhart (2191) vernondalhart (2191) 4 vernondalhart (2191) Definition msc 26B12