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.

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

$\mathrm{eval}\left(⟨1\,t\,t^2⟩\,t=1\right)$

$\left(1\right)e_x\+\left(1\right)e_y\+\left(1\right)e_z$

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

$\mathrm{v1}≔\left(x\right)e_x\+\left(y^2\right)e_y\+\left(z^3\right)e_z$

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

$\left(1\right)e_x\+\left(4\right)e_y\+\left(27\right)e_z$

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

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

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

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

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

$\left(1\right)e_r\+\left(\mathrm{\pi }\right)e_t$

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

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

$\mathrm{eval}\left(\mathrm{v3}\,\left\{s=1\,t=\mathrm{\pi }\right\}\right)$

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

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

$\left(s\right)e_u\+\left(t\right)e_v$

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

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

$\mathrm{eval}\left(\mathrm{v4}\,\left\{u=1\,v=-1\,w=1\right\}\right)$

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

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

$\left(1\right)e_r\+\left(\frac{\mathrm{\pi }}{4}\right)e_p\+\left(\frac{\mathrm{\pi }}{4}\right)e_t$

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

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

$\mathrm{eval}\left(\mathrm{pv1}\,t=3\right)$

$\left[\begin{array}{c}3\cos \left(3\right)\\ 3\sin \left(3\right)\end{array}\right]$

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

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

$\mathrm{eval}\left(\mathrm{pv2}\,\left\{t=3\,v=4\right\}\right)$

$\left[\begin{array}{c}3\\ \frac{2\sqrt{10}}{5}\\ \frac{6\sqrt{10}}{5}\end{array}\right]$

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

$\mathrm{vf}≔\left(\frac{1}{r^2}\right){\stackrel{\_}{e}}_r\+\left(0\right){\stackrel{\_}{e}}_p\+\left(0\right){\stackrel{\_}{e}}_t$

$\mathrm{eval}\left(\mathrm{vf}\,r=1\right)$

$\left(1\right){\stackrel{\_}{e}}_r\+\left(0\right){\stackrel{\_}{e}}_p\+\left(0\right){\stackrel{\_}{e}}_t$

$\mathrm{attributes}\left(\mathrm{eval}\left(\mathrm{vf}\,r=1\right)\right)$

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