The GetPVDescription returns a list with the description of pvector in c coordinates.

If no coordinate argument is provided, the description of pvector is given with respect to the current coordinate system as long as the dimensions match.

The position Vector is always a cartesian Vector rooted at the origin, but its description varies with the choice of coordinates. For more details about position Vectors, see VectorCalculus,Details

with(VectorCalculus):

pv := PositionVector([p*cos(p), p*sin(p)], cartesian[x,y]);

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

M :=GetPVDescription(pv,polar[r,t]);

$M≔\left[\sqrt{p^2{\cos \left(p\right)}^2+p^2{\sin \left(p\right)}^2}\,\arctan \left(p\sin \left(p\right)\,p\cos \left(p\right)\right)\right]$

pv2 := PositionVector(M,polar[r,t]);

$\mathrm{pv2}≔\left[\begin{array}{c}p\cos \left(p\right)\\ p\sin \left(p\right)\end{array}\right]$

c := arctan(u);

$c≔\arctan \left(u\right)$

pv3 := PositionVector([cos(u)*cos(c), sin(u)*cos(c), -sin(c)], cartesian[x,y,z]);

$\mathrm{pv3}≔\left[\begin{array}{c}\frac{\cos \left(u\right)}{\sqrt{u^2+1}}\\ \frac{\sin \left(u\right)}{\sqrt{u^2+1}}\\ -\frac{u}{\sqrt{u^2+1}}\end{array}\right]$

M := GetPVDescription(pv3, spherical[r,p,t]);

$M≔\left[\sqrt{\frac{{\cos \left(u\right)}^2}{u^2+1}+\frac{{\sin \left(u\right)}^2}{u^2+1}+\frac{u^2}{u^2+1}}\,\arctan \left(\sqrt{\frac{{\cos \left(u\right)}^2}{u^2+1}+\frac{{\sin \left(u\right)}^2}{u^2+1}}\,-\frac{u}{\sqrt{u^2+1}}\right)\,\arctan \left(\frac{\sin \left(u\right)}{\sqrt{u^2+1}}\,\frac{\cos \left(u\right)}{\sqrt{u^2+1}}\right)\right]$

simplify(M) assuming u::real;

$\left[1\,\arctan \left(1\,-u\right)\,\arctan \left(\sin \left(u\right)\,\cos \left(u\right)\right)\right]$