The eval(v, eqns) command is an extension of the top-level eval command which correctly evaluates free Vectors , rooted Vectors, position Vectors, and VectorFields for the VectorCalculus package.  If v is not a Vector, the arguments are passed to the top level eval command.

If v is a rooted Vector then both the root point or origin and the components, corresponding to the coefficients of the basis vectors, are evaluated.

If v is a VectorField, then the components are evaluated and a VectorField is returned. To properly evaluate a VectorField at a point use evalVF.

If v is a free Vector or a position Vector, then the components are evaluated. The type of the Vector does not change.

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

$\genfrac{}{}{0ex}{}{⟨1\,t\,t^2⟩}{\phantom{t\=1}}|\genfrac{}{}{0ex}{}{\phantom{⟨1\,t\,t^2⟩}}{t\=1}$

$\mathrm{v1}≔\mathrm{Vector}\left(⟨x\,y^2\,z^3⟩\,\mathrm{coords}\={\mathrm{cartesian}}_{x\,y\,z}\right)$

$\mathrm{eval}\left(\mathrm{v1}\,\left[x\=1\,y\=2\,z\=3\right]\right)$

$\mathrm{v2}≔\mathrm{RootedVector}\left(\mathrm{point}\=\left[u\,\mathrm{Pi}\right]\,\left[u\,v\right]\,{\mathrm{polar}}_{r\,t}\right)$

$\mathrm{v2}≔\left[\begin{array}{c}u\\ v\end{array}\right]$

$\genfrac{}{}{0ex}{}{\mathrm{v2}}{\phantom{u\=1\,v\=2}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{v2}}}{u\=1\,v\=2}$

$\left[\begin{array}{c}1\\ 2\end{array}\right]$

$\mathrm{GetRootPoint}\left(\mathrm{eval}\left(\mathrm{v2}\,\left[u\=1\,v\=2\right]\right)\right)$

$\mathrm{v3}≔\mathrm{RootedVector}\left(\mathrm{point}\=\left[s\,t\right]\,\left[1\,2\right]\,{\mathrm{parabolic}}_{u\,v}\right)$

$\mathrm{v3}≔\left[\begin{array}{c}1\\ 2\end{array}\right]$

$\genfrac{}{}{0ex}{}{\mathrm{v3}}{\phantom{s\=1\,t\=\mathrm{Pi}}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{v3}}}{s\=1\,t\=\mathrm{Pi}}$

$\left[\begin{array}{c}1\\ 2\end{array}\right]$

$\mathrm{GetRootPoint}\left(\mathrm{eval}\left(\mathrm{v3}\right)\right)$

$\mathrm{v4}≔\mathrm{RootedVector}\left(\mathrm{point}\=\left[1\,\frac{\mathrm{Pi}}{4}\,\frac{\mathrm{Pi}}{4}\right]\,\left[u\,v\,w\right]\,{\mathrm{spherical}}_{r\,p\,t}\right)$

$\mathrm{v4}≔\left[\begin{array}{c}u\\ v\\ w\end{array}\right]$

$\genfrac{}{}{0ex}{}{\mathrm{v4}}{\phantom{u\=1\,v\=-1\,w\=1}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{v4}}}{u\=1\,v\=-1\,w\=1}$

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

$\mathrm{GetRootPoint}\left(\mathrm{v4}\right)$

$\mathrm{pv1}≔\mathrm{PositionVector}\left(\left[t\,t\right]\,\mathrm{polar}\right)$

$\mathrm{pv1}≔\left[\begin{array}{c}t\cos \left(t\right)\\ t\sin \left(t\right)\end{array}\right]$

$\genfrac{}{}{0ex}{}{\mathrm{pv1}}{\phantom{t\=3}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{pv1}}}{t\=3}$

$\mathrm{pv2}≔\mathrm{PositionVector}\left(\left[t\,\frac{v}{\sqrt{1\+t^2}}\,\frac{vt}{\sqrt{1\+t^2}}\right]\,{\mathrm{cartesian}}_{x\,y\,z}\right)$

$\mathrm{pv2}≔\left[\begin{array}{c}t\\ \frac{v}{\sqrt{t^2+1}}\\ \frac{vt}{\sqrt{t^2+1}}\end{array}\right]$

$\genfrac{}{}{0ex}{}{\mathrm{pv2}}{\phantom{t\=3\,v\=4}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{pv2}}}{t\=3\,v\=4}$

$\mathrm{vf}≔\mathrm{VectorField}\left(⟨\frac{1}{r^2}\,0\,0⟩\,{\mathrm{spherical}}_{r\,p\,t}\right)$

$\genfrac{}{}{0ex}{}{\mathrm{vf}}{\phantom{r\=1}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{vf}}}{r\=1}$

$\mathrm{attributes}\left(\genfrac{}{}{0ex}{}{\mathrm{vf}}{\phantom{r\=1}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{vf}}}{r\=1}\right)$

$\mathrm{vectorfield},\mathrm{coords}={\mathrm{spherical}}_{r,p,t}$