Mesh of a partition

Mesh of a partition
Tag	`04GM4NYXJB`
URI	`m2://04GM4NYXJB`
Aliases	`norm-of-a-partition`
Kind	definition
Writing	W0 stub
MSC	26A42
Defines	Mesh of a partition
Uses	Partition of an interval

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Definition

Let $P = (x_0, \dots, x_n)$ be a partition of $[a, b]$ . The mesh of $P$ is $\| P \| = \max_{1 \le i \le n} (x_i - x_{i-1}),$ the length of the longest subinterval of $P$ .

Remarks

Refining a partition does not increase its mesh. The Riemann integral is a limit of Riemann sums as the mesh tends to $0$ . For the partition of $[0, 1]$ into $n$ equal pieces, the mesh is $1/n$ .

Explanations

L1 human

The mesh measures how fine a partition is. A small mesh means that all the pieces are short.

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Partition of an interval definition
Mesh of a partition definition

What depends on this

48 nodes use this node, directly or through other nodes.

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Comparison test for improper integrals 0FR1BDKK10
Area under a curve 0GXA8Z205E
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Stirling's formula 0KC0F51T6A
Monotonicity of the integral 0MBWQKM8QR
Riemann integrable function 0N1JQXZXHC
Mean value theorem for integrals 0N91N98EG5
The integral of a continuous function is an antiderivative 0NC2A3HWGT
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Triangle inequality for integrals 0NR82KMYKD
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Tangent half-angle substitution 0W8ETY2AZ6
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