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LieAlgebrasOfVectorFields

 Distribution
 Construct a Distribution object

 Calling Sequence Distribution(vf1,..., 'space'=vars) Distribution(str, 'space'=vars) Distribution(L)

Parameters

 vf1, ..., - VectorField object, or sequence of VectorField objects vars - (optional) list of names of coordinates of the space str - a string: either "trivial" or "universal" L - an LAVF object.

Description

 • The Distribution command constructs and returns a Distribution object (see Overview of the Distribution object).
 • In the first calling sequence, a sequence of VectorField objects must be provided which span the distribution at each point. The input vector fields are further processed by the constructor, so that the Distribution may be stored internally and displayed with a different basis.  The space= option is not necessary, since the space can be inferred from the vector fields.  For completeness, an empty sequence of vector fields is permitted: in this case, the space= option is required, and a trivial (0-dimensional) distribution is constructed on the specified space.
 • For convenience the second calling sequence construct the trivial (0-dimensional) distribution and 'universal' distribution (which spans the whole of tangent space at each point). In these sequences, the space= option must be specified.
 • In the final calling sequence, L is a Lie algebra of vector fields LAVF object whose determining system is of order 0. An exception is raised if the determining system of L is not 0th order.
 • These commands are part of the LieAlgebrasOfVectorFields package, for more detail see Overview of the LieAlgebrasOfVectorFields package.

Examples

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

Build vector fields associated with 3-d spatial rotations...

 > $R\left[x\right]≔\mathrm{VectorField}\left(-z\mathrm{D}\left[y\right]+y\mathrm{D}\left[z\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${{R}}_{{x}}{≔}{-}{z}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{y}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (1)
 > $R\left[y\right]≔\mathrm{VectorField}\left(-x\mathrm{D}\left[z\right]+z\mathrm{D}\left[x\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${{R}}_{{y}}{≔}{z}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{-}{x}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (2)
 > $R\left[z\right]≔\mathrm{VectorField}\left(-y\mathrm{D}\left[x\right]+x\mathrm{D}\left[y\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${{R}}_{{z}}{≔}{-}{y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{x}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (3)

Construct the associated distribution...

 > $\mathrm{\Sigma }≔\mathrm{Distribution}\left(R\left[x\right],R\left[y\right],R\left[z\right]\right)$
 ${\mathrm{\Sigma }}{≔}\left\{{-}\frac{{y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}}{{x}}{+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}{-}\frac{{z}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}}{{x}}{+}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right\}$ (4)

A trivial Distribution has empty basis...

 > $\mathrm{Distribution}\left("trivial",\mathrm{space}=\left[x,y,z\right]\right)$
 ${\varnothing }$ (5)

The universal Distribution is spanned by the basis vectors of tangent space at each point.

 > $\mathrm{Distribution}\left("universal",\mathrm{space}=\left[x,y,z\right]\right)$
 $\left\{\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right\}$ (6)

Compatibility

 • The LieAlgebrasOfVectorFields[Distribution] command was introduced in Maple 2020.