Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Partition of an interval"

m
m (Fixed link to Riemann integral.)
 
Line 1: Line 1:
−
A '''partition of an interval''' is a division of an [[interval]] into several disjoint sub-intervals.  Partitions of intervals arise in [[calculus]] in the context of [[Riemann integral]]s.
+
A '''partition of an interval''' is a division of an [[interval]] into several disjoint sub-intervals.  Partitions of intervals arise in [[calculus]] in the context of [[Integral#Riemann Integral|Riemann integral]]s.
  
 
==Definition==
 
==Definition==

Latest revision as of 19:34, 6 March 2022

A partition of an interval is a division of an interval into several disjoint sub-intervals. Partitions of intervals arise in calculus in the context of Riemann integrals.

Definition

Let $[a,b]$ be an interval of real numbers.

A partition $\mathcal{P}$ is defined as the ordered $n$-tuple of real numbers $\mathcal{P}=(x_0,x_1,\ldots,x_n)$ such that $a=x_0<x_1<\ldots<x_n=b$

Norm

The norm of a partition $\mathcal{P}$ is defined as $\|\mathcal{P}\|=\sup\{x_i-x_{i-1}\}_{i=1}^n$

Tags

Let $\mathcal{P}=\{x_0,x_1,\ldots,x_n\}$ be a partition.

A Tagged partition $\mathcal{\dot{P}}$ is defined as the set of ordered pairs $\mathcal{\dot{P}}=\{([x_{i-1},x_i],t_i)\}_{i=1}^n$.

Where $x_{i-1}<t_i<x_i\forall t_i$. The points $t_i$ are called the Tags.

See also

This article is a stub. Help us out by expanding it.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Partition_of_an_interval&oldid=172408"