AsOperator - Maple Help

OneForm Object as Operator

 Calling Sequence omega( X)

Parameters

 omega - a OneForm object X - a VectorField object

Description

 • A OneForm object omega can act as an operator on a VectorField X, by contraction.
 • if $\mathrm{\omega }$ is a 1-form. and if $X=\sum _{i=1}^{n}{\mathrm{\xi }}_{i}$ is a vector field (both on a space with coordinates $\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)$), then their contraction is $\mathrm{\omega }\left(X\right)=\sum _{i=1}^{n}{\mathrm{\theta }}_{i}{\mathrm{\xi }}_{i}$.
 • Because it can act as an operator, a OneForm object is of type appliable. See Overview of OneForm Overloaded Builtins for more detail.
 • When a OneForm is acting as an operator, it will distribute itself over indexable types such as Vectors, Matrices, lists, and tables.
 • This method is associated with the OneForm object. For more detail, see Overview of the OneForm object.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $R\left[x\right]≔\mathrm{VectorField}\left(y\mathrm{D}\left[z\right]-z\mathrm{D}\left[y\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${{R}}_{{x}}{≔}{-}{z}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{y}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (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)
 > $R\left[z\right]≔\mathrm{VectorField}\left(x\mathrm{D}\left[y\right]-y\mathrm{D}\left[x\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${{R}}_{{z}}{≔}{-}{y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{x}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (3)

 > $\mathrm{\omega }≔\mathrm{OneForm}\left(xd\left[x\right]+yd\left[y\right]+zd\left[z\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${\mathrm{\omega }}{≔}{x}{}{\mathrm{dx}}{+}{y}{}{\mathrm{dy}}{+}{z}{}{\mathrm{dz}}$ (4)
 > $\mathrm{\omega }\left(R\left[x\right]\right)$
 ${0}$ (5)
 > $\mathrm{\omega }\left(\left[R\left[x\right],R\left[y\right],R\left[z\right]\right]\right)$
 $\left[{0}{,}{0}{,}{0}\right]$ (6)
 > $\mathrm{\omega }\left(R\right)$
 ${table}{}\left(\left[{x}{=}{0}{,}{z}{=}{0}{,}{y}{=}{0}\right]\right)$ (7)