Builtins - Maple Help

indets

indeterminates in a Distribution object

has

test a Distribution object for a subexpression

hastype

test a Distribution object for presence of a specified type

type

type checking for a Distribution object

convert

type conversion for a Distribution object

 Calling Sequence indets( dist, typename ) has( dist, x) hastype(dist, t) type(dist, t) convert(dist, 'LAVF', infnames)

Parameters

 dist - a Distribution object typename - (optional) a Maple type x - expression, list or set of expressions t - a Maple type infnames - list of names of infinitesimals for use in the LAVF object

Description

 • These commands extend the functionality of the Maple builtins of the same name so that they work on Distribution objects. Their calling sequences and return values agree with the Maple builtin. See the help pages for the Maple builtins indets, has, hastype, type, convert for details.  Only information relevant to operation on a Distribution object is described on this page.
 • The indets of a Distribution object consist of: the coordinates of the space where the Distribution lives (see GetSpace), and any other indets occurring in the vector fields that span the Distribution.
 • has(dist, x) will test true if the vector fields spanning dist have x, or if the space coordinates of dist have x.
 • hastype(dist, t) will test true if the vector fields spanning dist contain the type t or if the space coordinates of dist do.  Thus hastype(dist, 'name') will always test true, because every Distribution exists on a space, whose coordinates are of type 'name'.
 • type(dist, t) will test true for the following types: 'anything', 'object'.  The types dependent(x) and freeof(x) are also understood. A Distribution object is dependent on x if any of its spanning vector fields is dependent on x, or if its space (coordinates) is dependent on x. A Distribution object is free of x if all its vector fields are free of x and its space (coordinates) is free of x. See type/dependent, type/freeof.
 • The only type conversion understood so far is conversion of a Distribution object to an LAVF object. For this, one must specify the names of the infinitesimals to be used in the LAVF.

Examples

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

Build vector fields...

 > $\mathrm{V1}≔\mathrm{VectorField}\left(\mathrm{D}\left[x\right]+\left({a}^{2}+{b}^{2}\right)\mathrm{D}\left[z\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${\mathrm{V1}}{≔}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}\left({{a}}^{{2}}{+}{{b}}^{{2}}\right){}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (1)
 > $\mathrm{V2}≔\mathrm{VectorField}\left(\mathrm{D}\left[y\right]+t\mathrm{D}\left[z\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${\mathrm{V2}}{≔}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{t}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (2)

Construct the associated distribution...

 > $\mathrm{\Sigma }≔\mathrm{Distribution}\left(\mathrm{V1},\mathrm{V2}\right)$
 ${\mathrm{\Sigma }}{≔}\left\{\frac{{t}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}}{{-}{{a}}^{{2}}{-}{{b}}^{{2}}}{+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}{-}\frac{\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}}{{-}{{a}}^{{2}}{-}{{b}}^{{2}}}{+}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right\}$ (3)
 > $\mathrm{indets}\left(\mathrm{\Sigma }\right)$
 $\left\{{a}{,}{b}{,}{t}{,}{x}{,}{y}{,}{z}\right\}$ (4)
 > $\mathrm{has}\left(\mathrm{\Sigma },\left[x,w\right]\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{has}\left(\mathrm{\Sigma },w\right)$
 ${\mathrm{false}}$ (6)
 > $\mathrm{hastype}\left(\mathrm{\Sigma },'\mathrm{name}'\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{hastype}\left(\mathrm{\Sigma },'\mathrm{trig}'\right)$
 ${\mathrm{false}}$ (8)
 > $\mathrm{type}\left(\mathrm{\Sigma },'\mathrm{object}'\right)$
 ${\mathrm{true}}$ (9)
 > $\mathrm{type}\left(\mathrm{\Sigma },\mathrm{dependent}\left(x,y,z\right)\right)$
 ${\mathrm{true}}$ (10)
 > $\mathrm{type}\left(\mathrm{\Sigma },\mathrm{freeof}\left(t\right)\right)$
 ${\mathrm{false}}$ (11)
 > $\mathrm{convert}\left(\mathrm{\Sigma },'\mathrm{LAVF}',\left[\mathrm{\xi },\mathrm{\eta },\mathrm{zeta}\right]\right)$
 $\left[{\mathrm{\xi }}{}\left({x}{,}{y}{,}{z}\right){}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\left({x}{,}{y}{,}{z}\right){}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{ζ}}{}\left({x}{,}{y}{,}{z}\right){}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{\mathrm{\xi }}{}\left({x}{,}{y}{,}{z}\right){=}\frac{{\mathrm{\eta }}{}\left({x}{,}{y}{,}{z}\right){}{t}{-}{\mathrm{ζ}}{}\left({x}{,}{y}{,}{z}\right)}{{-}{{a}}^{{2}}{-}{{b}}^{{2}}}\right]\right\}$ (12)

Compatibility

 • The indets, has, hastype, type and convert commands were introduced in Maple 2020.