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Partition of an interval

Revision as of 00:35, 16 February 2008 by Shreyas patankar (talk | contribs)

A Partition of an interval is a way to formalise the intutive notion of 'infinitesimal parts' of an interval.

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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