AreSameSpace - Maple Help

AreSameSpace

check if a sequence of VectorField and/or OneForm objects live on the same space.

 Calling Sequence AreSameSpace( obj1, obj2, ... )

Parameters

 obj1, obj2, ... - a sequence of VectorField or OneForm objects

Description

 • The AreSameSpace command returns true if all objects live on the same space, false otherwise.
 • The obj1, obj2... can be a mixture of VectorField or OneForm and other objects.
 • This method is associated with the VectorField and OneForm objects. For more detail, see Overview of the VectorField object, Overview of the OneForm object.

Examples

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

The vector fields X,Y live on the same space (x,y).

 > $X≔\mathrm{VectorField}\left(\mathrm{components}=\left[x,y\right],\mathrm{space}=\left[x,y\right]\right)$
 ${X}{≔}{x}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{y}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (1)

 > $Y≔\mathrm{VectorField}\left(\mathrm{D}\left[x\right],\mathrm{space}=\left[x,y\right]\right)$
 ${Y}{≔}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (2)

 > $\mathrm{AreSameSpace}\left(X,Y\right)$
 ${\mathrm{true}}$ (3)

Although the following vector field looks identical to Y, they live on different spaces.

 > $\mathrm{Y1}≔\mathrm{VectorField}\left(\mathrm{D}\left[x\right],\mathrm{space}=\left[x\right]\right)$
 ${\mathrm{Y1}}{≔}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (4)

 > $\mathrm{AreSameSpace}\left(Y,\mathrm{Y1}\right)$
 ${\mathrm{false}}$ (5)

 > $\mathrm{\omega }≔\mathrm{OneForm}\left(xd\left[x\right]+yd\left[y\right],\mathrm{space}=\left[x,y\right]\right)$
 ${\mathrm{\omega }}{≔}{x}{}{\mathrm{dx}}{+}{y}{}{\mathrm{dy}}$ (6)

This method can deal with multiple objects, mixing with VectorField and OneForm objects.

 > $\mathrm{AreSameSpace}\left(X,Y,\mathrm{\omega }\right)$
 ${\mathrm{true}}$ (7)

Compatibility

 • The AreSameSpace command was introduced in Maple 2020.