Timewise-average function - Revision history

Vipul: 5 revisions

2008-05-18T20:10:52Z

5 revisions

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Vipul at 11:47, 8 April 2007

2007-04-08T11:47:50Z

← Older revision		Revision as of 11:47, 8 April 2007
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	~~{{scalar function}}~~

	==Definition==		==Definition==

Vipul: /* Definition */

2007-04-08T11:37:04Z

Definition

← Older revision		Revision as of 11:37, 8 April 2007
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	<math>f(t) = \frac{\int_M u(t,x) dx}{\int_M dx}</math>		<math>f(t) = \frac{\int_M u(t,x) dx}{\int_M dx}</math>

	For instance, the above notion makes sense if we embed the manifold in ~~Euclidena~~ space and look at the induced measure.		For instance, the above notion makes sense if we embed the manifold in Euclidean space and look at the induced measure, or in a [[Riemannian manifold]] where we take the naturally induced measure.

	==Related notions==		==Related notions==

Vipul at 11:36, 8 April 2007

2007-04-08T11:36:07Z

← Older revision		Revision as of 11:36, 8 April 2007
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			{{scalar function}}

	==Definition==		==Definition==

Vipul: /* Definition */

2007-04-08T10:59:51Z

Definition

← Older revision		Revision as of 10:59, 8 April 2007
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	Let <math>u:\R \times M \to \R</math> be any continuous function. Then the timewise-average function of <math>u</math> is a map <math>f:\R \to \R</math> defined by:		Let <math>u:\R \times M \to \R</math> be any continuous function. Then the timewise-average function of <math>u</math> is a map <math>f:\R \to \R</math> defined by:

	<math>f(t) = \frac{\int_M u(t,x) dm}{\int_M dm}</math>		<math>f(t) = \frac{\int_M u(t,x) dx}{\int_M dx}</math>

	For instance, the above notion makes sense if we embed the manifold in Euclidena space and look at the induced measure.		For instance, the above notion makes sense if we embed the manifold in Euclidena space and look at the induced measure.

Vipul at 10:59, 8 April 2007

2007-04-08T10:59:10Z

New page

==Definition==

Let <math>M</math> be a differential manifold equipped with a measure over which we can integrate.

Let <math>u:\R \times M \to \R</math> be any continuous function. Then the timewise-average function of <math>u</math> is a map <math>f:\R \to \R</math> defined by:

<math>f(t) = \frac{\int_M u(t,x) dm}{\int_M dm}</math>

For instance, the above notion makes sense if we embed the manifold in Euclidena space and look at the induced measure.

==Related notions==

* [[Timewise-max function]]
* [[Timewise-min function]]