Difference between revisions of "Partition of an interval"
(New page: A '''Partition of an interval''' is a way to formalise the intutive notion of 'infinetesimal parts' of an interval. ==Definition== Let <math>[a,b]</math> be an interval of real numbers ...) |
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| − | A ''' | + | 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== | ||
| − | Let <math>[a,b]</math> be an interval of real | + | Let <math>[a,b]</math> be an interval of [[real number]]s. |
| − | A ''' | + | 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=== | ||
| − | The ''' | + | 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=== | ||
Let <math>\mathcal{P}=\{x_0,x_1,\ldots,x_n\}</math> be a partition. | Let <math>\mathcal{P}=\{x_0,x_1,\ldots,x_n\}</math> be a partition. | ||
| − | A '''Tagged partition''' <math>\mathcal\dot{P}}</math> is defined as the set of ordered pairs <math>\mathcal\dot{P}}=\{([x_{i-1},x_i],t_i)\}_{i=1}^n</math>. The points <math>t_i</math> are called the '''Tags'''. | + | A '''Tagged partition''' <math>\mathcal{\dot{P}}</math> is defined as the set of ordered pairs <math>\mathcal{\dot{P}}=\{([x_{i-1},x_i],t_i)\}_{i=1}^n</math>. |
| + | |||
| + | Where <math>x_{i-1}<t_i<x_i\forall t_i</math>. The points <math>t_i</math> are called the '''Tags'''. | ||
==See also== | ==See also== | ||
*[[Integral]] | *[[Integral]] | ||
| − | *[[ | + | *[[Riemann sum]] |
*[[Gauge]] | *[[Gauge]] | ||
{{stub}} | {{stub}} | ||
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.
Contents
Definition
Let ![$[a,b]$](http://latex.artofproblemsolving.com/8/e/c/8ecbd1ba3da8f2adef66a63f2ab32c47e63fa734.png)
be an interval of real numbers.
A partition 
is defined as the ordered 
-tuple of real numbers 
such that
Norm
The norm of a partition is defined as 
Tags
Let 
be a partition.
A Tagged partition 
is defined as the set of ordered pairs ![$\mathcal{\dot{P}}=\{([x_{i-1},x_i],t_i)\}_{i=1}^n$](http://latex.artofproblemsolving.com/1/e/4/1e4a621081b8c5c6b18dfd7163da5adfe7dcb1a2.png)
.
Where 
. The points 
are called the Tags.
See also
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