The GetPVDescription returns a list with the description of pvector in c coordinates; pvector is not changed.

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.

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

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

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

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

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

$\left[\left|p\right|\,\arctan \left(p\sin \left(p\right)\,p\cos \left(p\right)\right)\right]$

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

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

$c≔\arctan \left(u\right)$

$c≔\arctan \left(u\right)$

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

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

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

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

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