Definition:Upper Darboux Sum

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Definition

Let $\closedint a b$ be a closed real interval.

Let $f: \closedint a b \to \R$ be a bounded real function.


Let $P = \set {x_0, x_1, x_2, \ldots, x_n}$ be a finite subdivision of $\closedint a b$.

For all $\nu \in \set {1, 2, \ldots, n}$, let $M_\nu^{\paren f}$ be the supremum of $f$ on the interval $\closedint {x_{\nu - 1} } {x_\nu}$.


Then:

$\ds \map {U^{\paren f} } P = \sum_{\nu \mathop = 1}^n M_\nu^{\paren f} \paren {x_\nu - x_{\nu - 1} }$

is called the upper Darboux sum of $f$ on $\closedint a b$ belonging (or with respect) to (the subdivision) $P$.


If there is no ambiguity as to what function is under discussion, $M_\nu$ and $\map U P$ are often seen.


Closed Rectangle

Let $R$ be a closed rectangle in $\R^n$.

Let $f : R \to \R$ be a bounded real-valued function.


Let $P$ be a finite subdivision of $R$.

Let $S$ be the set of subrectangles of $P$.

For every:

$r = \closedint {a_1} {b_1} \times \dotso \times \closedint {a_n} {b_n} \in S$

define:

$\ds M_r^{\paren f} = \sup_{x \mathop \in r} \map f x$
$\ds \map v r = \prod_{1 \mathop \le i \mathop \le n} \paren {b_i - a_i}$


Then:

$\ds \map {U^{\paren f}} P = \sum_{r \mathop \in S} M_r^{\paren f} \map v r$

is called the upper Darboux sum of $f$ on $R$ with respect to $P$.


Also known as

The notation $\map U {f, P}$ or $\map U {P, f}$ can be used in place of $\map {U^{\paren f} } P$.


The upper Darboux sum is also known as:

the upper Riemann sum (for Georg Friedrich Bernhard Riemann)
the upper sum.


Also see

  • Results about the upper Darboux sum can be found here.


Source of Name

This entry was named for Jean-Gaston Darboux.


Sources