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.

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

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

$R_x≔-z\left(\frac{\ⅆ}{\ⅆy}\right)+y\left(\frac{\ⅆ}{\ⅆz}\right)$

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

$R_y≔z\left(\frac{\ⅆ}{\ⅆx}\right)-x\left(\frac{\ⅆ}{\ⅆz}\right)$

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

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

$\mathrm{\Sigma }≔\mathrm{Distribution}\left(R\left[x\right]\,R\left[y\right]\,R\left[z\right]\right)$

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

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

$\mathrm{true}$

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

$\mathrm{true}$

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

$\mathrm{IN}≔\left[x^2+y^2+z^2\right]$

$\mathrm{\rho }≔\mathrm{op}\left(\mathrm{IN}\right)$

$\mathrm{\rho }≔x^2+y^2+z^2$

$\mathrm{vfs}≔\mathrm{GetVectorFields}\left(\mathrm{\Sigma }\right)$

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

$\mathrm{map}\left(X\mapsto X\left(\mathrm{\rho }\right)\,\mathrm{vfs}\right)$

$\left[0\,0\right]$

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

For more information on Maple 2020 changes, see Updates in Maple 2020.