IsInvolutive - Maple Help

IsInvolutive

check if a Distribution object is in involution

IsIntegrable

a synonym for IsInvolutive

Integrals

calculate the integrals of an involutive Distribution object

 Calling Sequence IsInvolutive( dist) IsIntegrable( dist) Integrals( dist)

Parameters

 dist - a Distribution object.

Description

 • The IsInvolutive (or IsIntegrable) method returns true if the distribution specified by dist is in involution.
 • A distribution is involutive (also known as integrable, or completely integrable) if the Lie bracket of any two vector fields lying in dist also lies in dist.
 • The Integrals method  returns a list of the functionally independent integrals of an involutive distribution, or the string "not known" if Maple was unable to find all the integrals.
 • A function $f\left({x}_{1},\dots ,{x}_{n}\right)$ is an integral of distribution dist on a space with coordinates $\left({x}_{1},\dots ,{x}_{n}\right)$ if every vector field X lying in dist satisfies X_(f(x[1], ..., x[n]))= 0.
 • Because successful integration of PDE by Maple cannot be guaranteed (see pdsolve), it is possible that Integrals is unable to return an answer.
 • These methods are associated with the Distribution object. For more detail see Overview of the Distribution object.

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)
 > $\mathrm{IsInvolutive}\left(\mathrm{\Sigma }\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{IsIntegrable}\left(\mathrm{\Sigma }\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{IN}≔\mathrm{Integrals}\left(\mathrm{\Sigma }\right)$
 ${\mathrm{IN}}{≔}\left[{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right]$ (7)
 > $\mathrm{\rho }≔\mathrm{op}\left(\mathrm{IN}\right)$
 ${\mathrm{\rho }}{≔}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}$ (8)

Since rho is an integral of distribution Sigma, it should be annihilated by every vector field lying in Sigma...

 > $\mathrm{vfs}≔\mathrm{GetVectorFields}\left(\mathrm{\Sigma }\right)$
 ${\mathrm{vfs}}{≔}\left[{-}\frac{{y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}}{{x}}{+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}{-}\frac{{z}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}}{{x}}{+}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]$ (9)
 > $\mathrm{map}\left(X↦X\left(\mathrm{\rho }\right),\mathrm{vfs}\right)$
 $\left[{0}{,}{0}\right]$ (10)

Compatibility

 • The IsInvolutive, IsIntegrable and Integrals commands were introduced in Maple 2020.