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.

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

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

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

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

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

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

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

$\mathrm{\omega }≔\mathrm{OneForm}\left(xd\left[x\right]+yd\left[y\right]+zd\left[z\right]\,\mathrm{space}=\left[x\,y\,z\right]\right)$

$\mathrm{\omega }≔x\mathrm{dx}+y\mathrm{dy}+z\mathrm{dz}$

$\mathrm{\omega }\left(R\left[x\right]\right)$

$0$

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

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

$\mathrm{\omega }\left(R\right)$

$table\left(\left[y=0\,x=0\,z=0\right]\right)$