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Difference between revisions of "Partition of an interval"

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A '''Partition of an interval''' is a way to formalise the intutive notion of 'infinitesimal parts' of an interval.  
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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==
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Let <math>[a,b]</math> be an interval of real numbers
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Let <math>[a,b]</math> be an interval of [[real number]]s.
  
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A '''Partition''' <math>\mathcal{P}</math> is defined as the ordered n-tuple of real numbers <math>\mathcal{P}=(x_0,x_1,\ldots,x_n)</math> such that
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A '''partition''' <math>\mathcal{P}</math> is defined as the ordered <math>n</math>-[[tuple]] of real numbers <math>\mathcal{P}=(x_0,x_1,\ldots,x_n)</math> such that
 
<math>a=x_0<x_1<\ldots<x_n=b</math>
 
<math>a=x_0<x_1<\ldots<x_n=b</math>
  
 
===Norm===
 
===Norm===
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The '''Norm''' of a partition <math>\mathcal{P}</math> is defined as <math>\|\mathcal{P}\|=\sup\{x_i-x_{i-1}\}_{i=1}^n</math>
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The '''norm''' of a partition <math>\mathcal{P}</math> is defined as <math>\|\mathcal{P}\|=\sup\{x_i-x_{i-1}\}_{i=1}^n</math>
  
 
===Tags===
 
===Tags===
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==See also==
 
==See also==
 
*[[Integral]]
 
*[[Integral]]
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*[[Reimann sum]]
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*[[Riemann sum]]
 
*[[Gauge]]
 
*[[Gauge]]
  
 
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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

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