The Distribution object provides a general toolkit for dealing with distributions. A user can query properties of a Distribution object, compute associated quantities, and combine distributions in various ways.  Some existing Maple utility functions such as indets, type, has,...etc are overloaded for use with the Distribution object.

A distribution in the differential-geometric sense is a specification of a subspace of tangent space at each point of a manifold M.

A Distribution object can be constructed via the Distribution constructor. To construct a Distribution object, see LieAlgebrasOfVectorFields[Distribution].

The Distribution object is an exported item in the LieAlgebrasOfVectorFields package. To construct and access a Distribution object, the LieAlgebrasOfVectorFields package must be loaded (i.e. with(LieAlgebrasOfVectorFields);). For more information, see Overview of the LieAlgebrasOfVectorFields package.

Once a Distribution object S has been constructed, each method in the Distribution object S can be accessed by either the short form command(S, otherArguments) or the long form S:-command(S, otherArguments).

A Distribution object is displayed (via its ModulePrint method) as a set of VectorField objects which form a basis for it at each point.

The following is a list of available commands in a Distribution object.

The following Maple builtins functions are extended so that they work for a Distribution object: type, has, hastype, indets, convert. See Distribution Object Overloaded Builtins for more detail.

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

$\mathrm{X1}≔\mathrm{VectorField}\left(-y\mathrm{D}\left[x\right]+x\mathrm{D}\left[y\right]\,\mathrm{space}=\left[x\,y\,z\right]\right)$

$\mathrm{X1}≔-y\left(\frac{\ⅆ}{\ⅆx}\right)+x\left(\frac{\ⅆ}{\ⅆy}\right)$

$\mathrm{X2}≔\mathrm{VectorField}\left(\mathrm{D}\left[z\right]\,\mathrm{space}=\left[x\,y\,z\right]\right)$

$\mathrm{X2}≔\frac{\ⅆ}{\ⅆz}$

$\mathrm{X3}≔\mathrm{VectorField}\left(z\mathrm{D}\left[z\right]\,\mathrm{space}=\left[x\,y\,z\right]\right)$

$\mathrm{X3}≔z\left(\frac{\ⅆ}{\ⅆz}\right)$

$\mathrm{\Sigma }≔\mathrm{Distribution}\left(\mathrm{X1}\,\mathrm{X2}\,\mathrm{X3}\right)$

$\mathrm{\Sigma }≔\left\{-\frac{y\left(\frac{\ⅆ}{\ⅆx}\right)}{x}+\frac{\ⅆ}{\ⅆy}\,\frac{\ⅆ}{\ⅆz}\right\}$

$\mathrm{\Omega }≔\mathrm{Distribution}\left(\mathrm{VectorField}\left(\mathrm{D}\left[x\right]\,\mathrm{space}=\left[x\,y\,z\right]\right)\right)$

$\mathrm{\Omega }≔\left\{\frac{\ⅆ}{\ⅆx}\right\}$

$\mathrm{Dimension}\left(\mathrm{\Sigma }\right)$

$2$

$\mathrm{IsInvolutive}\left(\mathrm{\Sigma }\right)$

$\mathrm{true}$

$\mathrm{IsIntegrable}\left(\mathrm{\Sigma }\right)$

$\mathrm{true}$

$\mathrm{IsSubspace}\left(\mathrm{\Omega }\,\mathrm{\Sigma }\right)$

$\mathrm{false}$

$\mathrm{VectorSpaceSum}\left(\mathrm{\Sigma }\,\mathrm{\Omega }\right)$

$\left\{\frac{\ⅆ}{\ⅆx}\,\frac{\ⅆ}{\ⅆy}\,\frac{\ⅆ}{\ⅆz}\right\}$

$\mathrm{Intersection}\left(\mathrm{\Sigma }\,\mathrm{\Omega }\right)$

$\varnothing$

$\mathrm{Integrals}\left(\mathrm{\Sigma }\right)$

$\left[x^2+y^2\right]$

$\mathrm{CauchyDistribution}\left(\mathrm{\Sigma }\right)$

$\left\{-\frac{y\left(\frac{\ⅆ}{\ⅆx}\right)}{x}+\frac{\ⅆ}{\ⅆy}\,\frac{\ⅆ}{\ⅆz}\right\}$

$\mathrm{DerivedDistribution}\left(\mathrm{\Sigma }\right)$

$\left\{-\frac{y\left(\frac{\ⅆ}{\ⅆx}\right)}{x}+\frac{\ⅆ}{\ⅆy}\,\frac{\ⅆ}{\ⅆz}\right\}$

$\mathrm{type}\left(\mathrm{\Sigma }\,'\mathrm{Distribution}'\right)$

$\mathrm{true}$