Operators - Maple Help

VectorField & OneForm Object Operator Methods

overview of VectorField and OneForm object operators

Description

 • The + and - operators provide addition and subtraction of two objects X and Y, and returns a new object Z. These objects are either all VectorField or all OneForm objects.
 • These objects can also be added using the existing commands add and sum.
 • The * operator provides scalar multiplication of an object X, and returns a new object Y. These two objects are either both VectorField or both OneForm objects.
 • The = operator tests equality of two VectorField or OneForm objects. The two objects are regarded as equal if they live on the same space and have the same components.
 • The ?[] operator provides a look-up of components of a VectorField or OneForm object. The object is indexed by its space coordinates, see examples below.
 • These operators are 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 +, -, $*$ Operators

We first construct some suitable VectorField objects: rotation, translations in (x,y) coordinates

 > $R≔\mathrm{VectorField}\left(x\mathrm{D}\left[y\right]-y\mathrm{D}\left[x\right],\mathrm{space}=\left[x,y\right]\right)$
 ${R}{≔}{-}{y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{x}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (1)
 > $\mathrm{Tx}≔\mathrm{VectorField}\left(\mathrm{D}\left[x\right],\mathrm{space}=\left[x,y\right]\right)$
 ${\mathrm{Tx}}{≔}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (2)
 > $\mathrm{Ty}≔\mathrm{VectorField}\left(\mathrm{D}\left[y\right],\mathrm{space}=\left[x,y\right]\right)$
 ${\mathrm{Ty}}{≔}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (3)
 > $\mathrm{Tx}+\mathrm{Ty}$
 $\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (4)
 > $-R$
 ${y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{-}{x}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (5)
 > $x\mathrm{Tx}+y\mathrm{Ty}$
 ${x}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{y}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (6)
 > $R+y\left[0\right]\mathrm{Tx}-x\left[0\right]\mathrm{Ty}$
 $\left({{y}}_{{0}}{-}{y}\right){}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}\left({-}{{x}}_{{0}}{+}{x}\right){}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (7)

One-form arithmetic works same as vector field arithmetic.

 > $\mathrm{\omega }\left[x\right]≔\mathrm{OneForm}\left(d\left[x\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${{\mathrm{\omega }}}_{{x}}{≔}{\mathrm{dx}}$ (8)
 > $\mathrm{\omega }\left[y\right]≔\mathrm{OneForm}\left(d\left[y\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${{\mathrm{\omega }}}_{{y}}{≔}{\mathrm{dy}}$ (9)
 > $\mathrm{\omega }\left[z\right]≔\mathrm{OneForm}\left(d\left[z\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${{\mathrm{\omega }}}_{{z}}{≔}{\mathrm{dz}}$ (10)
 > ${x}^{2}\mathrm{\omega }\left[z\right]+x\mathrm{\omega }\left[y\right]+\mathrm{\omega }\left[x\right]$
 ${{x}}^{{2}}{}{\mathrm{dz}}{+}{x}{}{\mathrm{dy}}{+}{\mathrm{dx}}$ (11)

Taking a linear combination of these one-forms using add (or sum) command.

 > $\mathrm{add}\left(a\left[i\right]\mathrm{\omega }\left[i\right],i\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left[x,y,z\right]\right)$
 ${{a}}_{{x}}{}{\mathrm{dx}}{+}{{a}}_{{y}}{}{\mathrm{dy}}{+}{{a}}_{{z}}{}{\mathrm{dz}}$ (12)

The = Operator

 > $X≔\mathrm{VectorField}\left(x\left(x-1\right)\mathrm{D}\left[x\right]+y\mathrm{D}\left[y\right],\mathrm{space}=\left[x,y\right]\right)$
 ${X}{≔}{x}{}\left({x}{-}{1}\right){}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{y}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (13)
 > $Y≔\mathrm{VectorField}\left(\left({x}^{2}-x\right)\mathrm{D}\left[x\right]+y\mathrm{D}\left[y\right],\mathrm{space}=\left[x,y\right]\right)$
 ${Y}{≔}\left({{x}}^{{2}}{-}{x}\right){}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{y}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (14)
 > $\mathrm{evalb}\left(X=Y\right)$
 ${\mathrm{false}}$ (15)
 > $\mathrm{evalb}\left(\mathrm{expand}\left(X\right)=\mathrm{expand}\left(Y\right)\right)$
 ${\mathrm{true}}$ (16)
 > $\mathrm{θ1}≔\mathrm{OneForm}\left(d\left[x\right],\mathrm{space}=\left[x\right]\right)$
 ${\mathrm{θ1}}{≔}{\mathrm{dx}}$ (17)
 > $\mathrm{θ2}≔\mathrm{OneForm}\left(d\left[x\right],\mathrm{space}=\left[x,y\right]\right)$
 ${\mathrm{θ2}}{≔}{\mathrm{dx}}$ (18)

Although θ1,θ2 look the same, they are not equal because they live on different spaces.

 > $\mathrm{evalb}\left(\mathrm{θ1}=\mathrm{θ2}\right)$
 ${\mathrm{false}}$ (19)

The ?[] Operator

 > $X≔\mathrm{VectorField}\left({x}^{2}\mathrm{D}\left[z\right]+\mathrm{D}\left[y\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${X}{≔}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{{x}}^{{2}}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (20)

 > $\left[X\left[x\right],X\left[y\right],X\left[z\right]\right]$
 $\left[{0}{,}{1}{,}{{x}}^{{2}}\right]$ (21)

Compatibility

 • The VectorField & OneForm Object Operator Methods command was introduced in Maple 2020.