The command evalVF(F, v) evaluates the vector field F at the point v.  The result is a rooted Vector with root point v.

The parameters, F and v, must have the same dimension, but do not need to be defined in the same coordinate system.

The vector field parameter, F, can also be specified as a Vector-valued procedure, in which case this procedure is evaluated at the coordinates of the Vector v.

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

$\mathrm{ev}≔\mathrm{evalVF}\left(\left(x\,y\,z\right)\mapsto \mathrm{RootedVector}\left(\mathrm{root}=\left[1\,0\,1\right]\,\left[\cos \left(x\right)\,\sin \left(y\right)\,z\right]\right)\,\mathrm{PositionVector}\left(\left[\mathrm{\pi }\,\frac{\mathrm{\pi }}{2}\,0\right]\right)\right)$

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

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

$\left[\begin{array}{cc}\mathrm{Type:}& \mathrm{Rooted\; Vector}\\ \mathrm{Components:}& \left[\mathrm{-1}\,1\,0\right]\\ \mathrm{Coordinates:}& {\mathrm{cartesian}}_{x,y,z}\\ \mathrm{Root\; Point:}& \left[\mathrm{\pi }\,\frac{\mathrm{\pi }}{2}\,0\right]\end{array}\right]$

$\mathrm{SetCoordinates}\left(\mathrm{polar}\left[r,\mathrm{\theta }\right]\right)$

${\mathrm{polar}}_{r,\mathrm{\theta }}$

$F≔\mathrm{VectorField}\left(⟨r^2\mathrm{\theta }\,-\mathrm{\theta }⟩\right)$

$\mathrm{simplify}\left(\mathrm{evalVF}\left(F\,⟨1\,\frac{\mathrm{\pi }}{2}⟩\right)\right)$

$\left[\begin{array}{c}\frac{\mathrm{\pi }}{2}\\ -\frac{\mathrm{\pi }}{2}\end{array}\right]$

$\mathrm{evalVF}\left(\left(a\,b\right)\mapsto ⟨a^2\,b^2⟩\,\mathrm{Vector}\left(\left[0\,1\right]\,\mathrm{attributes}=\left[\mathrm{coordinates}=\mathrm{cartesian}\right]\right)\right)$

$\left[\begin{array}{c}1\\ \frac{{\mathrm{\pi }}^2}{4}\end{array}\right]$