The VectorSpace(c, origin) command returns a module representing the tangent vector space rooted at point origin in c coordinates.

The order of the arguments may be reversed. If no coordinate system argument is present, the current coordinate system is used.

The root point origin can be specified as a free or position Vector or as a list of coordinates. If it is a free or position Vector, the coordinate system attribute is checked and conversion done accordingly. If it is a list, the components are expected to be in c coordinates.

Once coordinates c are specified, the unit basis vectors are evaluated at the point origin. If any of the unit basis vectors evaluate to zero an error is raised.

The returned module is a valid right hand side for the space attribute on Vectors. If this attribute is present, it qualifies the Vector as a rooted Vector, and determines its root point and coordinate system.

The module itself has 3 useful exports:

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

$\mathrm{vs}≔\mathrm{VectorSpace}\left(\mathrm{spherical}\left[r,\mathrm{\phi },\mathrm{\theta }\right]\,\left[1\,\frac{\mathrm{\pi }}{2}\,\frac{\mathrm{\pi }}{2}\right]\right)$

$\mathrm{vs}≔\mathbf{module}\left(\right)\mathbf{local}\mathrm{\_origin}\,\mathrm{\_coords}\,\mathrm{\_coords\_dim}\;\mathbf{export}\mathrm{GetCoordinates}\,\mathrm{GetRootPoint}\,\mathrm{Vector}\,\mathrm{eval}\;\mathbf{end\; module}$

$\mathrm{About}\left(\mathrm{vs}\right)$

$\left[\begin{array}{cc}\mathrm{Type:}& \mathrm{Vector\; Space}\\ \mathrm{Coordinates:}& {\mathrm{spherical}}_{r,\mathrm{\phi },\mathrm{\theta }}\\ \mathrm{Root\; Point:}& \left[1\,\frac{\mathrm{\pi }}{2}\,\frac{\mathrm{\pi }}{2}\right]\end{array}\right]$

$v≔\mathrm{vs}:-\mathrm{Vector}\left(\left[0\,1\,0\right]\right)$

$v≔\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]$

$\mathrm{About}\left(v\right)$

$\left[\begin{array}{cc}\mathrm{Type:}& \mathrm{Rooted\; Vector}\\ \mathrm{Components:}& \left[0\,1\,0\right]\\ \mathrm{Coordinates:}& {\mathrm{spherical}}_{r,\mathrm{\phi },\mathrm{\theta }}\\ \mathrm{Root\; Point:}& \left[1\,\frac{\mathrm{\pi }}{2}\,\frac{\mathrm{\pi }}{2}\right]\end{array}\right]$

$\mathrm{MapToBasis}\left(v\,'\mathrm{cartesian}'\right)$

$\left[\begin{array}{c}0\\ 0\\ \mathrm{-1}\end{array}\right]$

$v≔\mathrm{VectorSpace}\left(\mathrm{polar}\left[r,\mathrm{\theta }\right]\,⟨x\,y⟩\right):-\mathrm{Vector}\left(\left[a\,b\right]\right)\:$

$\mathrm{About}\left(v\right)$

$\left[\begin{array}{cc}\mathrm{Type:}& \mathrm{Rooted\; Vector}\\ \mathrm{Components:}& \left[a\,b\right]\\ \mathrm{Coordinates:}& {\mathrm{polar}}_{r,\mathrm{\theta }}\\ \mathrm{Root\; Point:}& \left[\sqrt{x^2+y^2}\,\arctan \left(y\,x\right)\right]\end{array}\right]$

$\mathrm{About}\left(\mathrm{GetSpace}\left(v\right)\right)$

$\left[\begin{array}{cc}\mathrm{Type:}& \mathrm{Vector\; Space}\\ \mathrm{Coordinates:}& {\mathrm{polar}}_{r,\mathrm{\theta }}\\ \mathrm{Root\; Point:}& \left[\sqrt{x^2+y^2}\,\arctan \left(y\,x\right)\right]\end{array}\right]$