DerivedDistribution - Maple Help
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DerivedDistribution

find derived distribution of a distribution

 Calling Sequence DerivedDistribution(dist)

Parameters

 dist - a Distribution object

Description

 • The DerivedDistribution method returns a Distribution object spanned by all the commutators of vector fields in dist.
 • This method is of little interest if the input Distribution dist is involutive, since in that case DerivedDistribution(dist) will simply return dist itself.
 • This method is associated with the Distribution object. For more detail see Overview of the Distribution object.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$

Build vector fields...

 > $\mathrm{V1}≔\mathrm{VectorField}\left({\mathrm{D}}_{x},\mathrm{space}=\left[x,y,z,w\right]\right)$
 ${\mathrm{V1}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}$ (1)
 > $\mathrm{V2}≔\mathrm{VectorField}\left({\mathrm{D}}_{y}+x{\mathrm{D}}_{z}+z{\mathrm{D}}_{w},\mathrm{space}=\left[x,y,z,w\right]\right)$
 ${\mathrm{V2}}{≔}\frac{{ⅆ}}{{ⅆ}{y}}{+}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{z}}\right){+}{z}{}\left(\frac{{ⅆ}}{{ⅆ}{w}}\right)$ (2)

Construct the associated distribution...

 > $\mathrm{Σ}≔\mathrm{Distribution}\left(\mathrm{V1},\mathrm{V2}\right)$
 ${\mathrm{\Sigma }}{≔}\left\{\frac{{ⅆ}}{{ⅆ}{x}}{,}\frac{\frac{{ⅆ}}{{ⅆ}{y}}}{{z}}{+}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{z}}\right)}{{z}}{+}\frac{{ⅆ}}{{ⅆ}{w}}\right\}$ (3)

Construct derived distribution...

 > $\mathrm{DerivedDistribution}\left(\mathrm{Σ}\right)$
 $\left\{\frac{\frac{{ⅆ}}{{ⅆ}{y}}}{{z}}{+}\frac{{ⅆ}}{{ⅆ}{w}}{,}\frac{{ⅆ}}{{ⅆ}{x}}{,}\frac{{ⅆ}}{{ⅆ}{z}}\right\}$ (4)
 > $\mathrm{IsInvolutive}\left(\mathrm{Σ}\right)$
 ${\mathrm{false}}$ (5)

Compatibility

 • The DerivedDistribution command was introduced in Maple 2020.
 • For more information on Maple 2020 changes, see Updates in Maple 2020.
The text passage describes a Maple help page for the `DerivedDistribution` method, which is used to find the derived distribution of a given distribution object in the context of Lie algebras of vector fields. The passage includes examples of how to use the method and a check for involutivity.
no_question

the text does not contain any mathematical questions or answers to such questions. it is purely descriptive and provides examples of how to use the `deriveddistribution` method.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
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- MULTIPLE_QUESTIONS
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system
DerivedDistribution - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

DerivedDistribution

find derived distribution of a distribution

 Calling Sequence DerivedDistribution(dist)

Parameters

 dist - a Distribution object

Description

 • The DerivedDistribution method returns a Distribution object spanned by all the commutators of vector fields in dist.
 • This method is of little interest if the input Distribution dist is involutive, since in that case DerivedDistribution(dist) will simply return dist itself.
 • This method is associated with the Distribution object. For more detail see Overview of the Distribution object.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$

Build vector fields...

 > $\mathrm{V1}≔\mathrm{VectorField}\left({\mathrm{D}}_{x},\mathrm{space}=\left[x,y,z,w\right]\right)$
 ${\mathrm{V1}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}$ (1)
 > $\mathrm{V2}≔\mathrm{VectorField}\left({\mathrm{D}}_{y}+x{\mathrm{D}}_{z}+z{\mathrm{D}}_{w},\mathrm{space}=\left[x,y,z,w\right]\right)$
 ${\mathrm{V2}}{≔}\frac{{ⅆ}}{{ⅆ}{y}}{+}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{z}}\right){+}{z}{}\left(\frac{{ⅆ}}{{ⅆ}{w}}\right)$ (2)

Construct the associated distribution...

 > $\mathrm{Σ}≔\mathrm{Distribution}\left(\mathrm{V1},\mathrm{V2}\right)$
 ${\mathrm{\Sigma }}{≔}\left\{\frac{{ⅆ}}{{ⅆ}{x}}{,}\frac{\frac{{ⅆ}}{{ⅆ}{y}}}{{z}}{+}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{z}}\right)}{{z}}{+}\frac{{ⅆ}}{{ⅆ}{w}}\right\}$ (3)

Construct derived distribution...

 > $\mathrm{DerivedDistribution}\left(\mathrm{Σ}\right)$
 $\left\{\frac{\frac{{ⅆ}}{{ⅆ}{y}}}{{z}}{+}\frac{{ⅆ}}{{ⅆ}{w}}{,}\frac{{ⅆ}}{{ⅆ}{x}}{,}\frac{{ⅆ}}{{ⅆ}{z}}\right\}$ (4)
 > $\mathrm{IsInvolutive}\left(\mathrm{Σ}\right)$
 ${\mathrm{false}}$ (5)

Compatibility

 • The DerivedDistribution command was introduced in Maple 2020.
 • For more information on Maple 2020 changes, see Updates in Maple 2020.
user
The text passage describes a Maple help page for the `DerivedDistribution` method, which is used to find the derived distribution of a given distribution object in the context of Lie algebras of vector fields. The passage includes examples of how to use the method and a check for involutivity.

TYPE: NO_QUESTION

The text does not contain any mathematical questions or answers to such questions. It is purely descriptive and provides examples of how to use the `DerivedDistribution` method.