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

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

$\mathrm{pv}≔\mathrm{PositionVector}\left(\left[p\cos \left(p\right)\,p\sin \left(p\right)\right]\,\mathrm{cartesian}\left[x,y\right]\right)$

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

$M≔\mathrm{GetPVDescription}\left(\mathrm{pv}\,\mathrm{polar}\left[r,t\right]\right)$

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

$\mathrm{pv2}≔\mathrm{PositionVector}\left(M\,\mathrm{polar}\left[r,t\right]\right)$

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

$c≔\arctan \left(u\right)$

$c≔\arctan \left(u\right)$

$\mathrm{pv3}≔\mathrm{PositionVector}\left(\left[\cos \left(u\right)\cos \left(c\right)\,\sin \left(u\right)\cos \left(c\right)\,-\sin \left(c\right)\right]\,\mathrm{cartesian}\left[x,y,z\right]\right)$

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

$M≔\mathrm{GetPVDescription}\left(\mathrm{pv3}\,\mathrm{spherical}\left[r,p,t\right]\right)$

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

$\mathrm{simplify}\left(M\right)\mathbf{assuming}u::\mathrm{real}$

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