Overview - Maple Help

Overview of the Distribution Object

Description

 • 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.

List of Distribution object Methods

 • 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.

Examples

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

We first build vector fields associated with 3-d cylinder (2-dim x-y rotation, and z translation and uniform scaling)

 > $\mathrm{X1}≔\mathrm{VectorField}\left(-y\mathrm{D}\left[x\right]+x\mathrm{D}\left[y\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${\mathrm{X1}}{≔}{-}{y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{x}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (1)
 > $\mathrm{X2}≔\mathrm{VectorField}\left(\mathrm{D}\left[z\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${\mathrm{X2}}{≔}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (2)
 > $\mathrm{X3}≔\mathrm{VectorField}\left(z\mathrm{D}\left[z\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${\mathrm{X3}}{≔}{z}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (3)
 > $\mathrm{\Sigma }≔\mathrm{Distribution}\left(\mathrm{X1},\mathrm{X2},\mathrm{X3}\right)$
 ${\mathrm{\Sigma }}{≔}\left\{{-}\frac{{y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}}{{x}}{+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right\}$ (4)
 > $\mathrm{\Omega }≔\mathrm{Distribution}\left(\mathrm{VectorField}\left(\mathrm{D}\left[x\right],\mathrm{space}=\left[x,y,z\right]\right)\right)$
 ${\mathrm{\Omega }}{≔}\left\{\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right\}$ (5)

We can request the dimension of this distribution

 > $\mathrm{Dimension}\left(\mathrm{\Sigma }\right)$
 ${2}$ (6)
 > $\mathrm{IsInvolutive}\left(\mathrm{\Sigma }\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{IsIntegrable}\left(\mathrm{\Sigma }\right)$
 ${\mathrm{true}}$ (8)

We can check if x-translation is subspace of Sigma

 > $\mathrm{IsSubspace}\left(\mathrm{\Omega },\mathrm{\Sigma }\right)$
 ${\mathrm{false}}$ (9)

Sum of these two distributions covers all (x,y,z) space.

 > $\mathrm{VectorSpaceSum}\left(\mathrm{\Sigma },\mathrm{\Omega }\right)$
 $\left\{\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right\}$ (10)

These two distribution don't intersect

 > $\mathrm{Intersection}\left(\mathrm{\Sigma },\mathrm{\Omega }\right)$
 ${\varnothing }$ (11)

The invariant of Sigma

 > $\mathrm{Integrals}\left(\mathrm{\Sigma }\right)$
 $\left[{{x}}^{{2}}{+}{{y}}^{{2}}\right]$ (12)

Finding other distributions..

 > $\mathrm{CauchyDistribution}\left(\mathrm{\Sigma }\right)$
 $\left\{{-}\frac{{y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}}{{x}}{+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right\}$ (13)
 > $\mathrm{DerivedDistribution}\left(\mathrm{\Sigma }\right)$
 $\left\{{-}\frac{{y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}}{{x}}{+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right\}$ (14)
 > $\mathrm{type}\left(\mathrm{\Sigma },'\mathrm{Distribution}'\right)$
 ${\mathrm{true}}$ (15)