The MapToBasis(V, c) command converts Vectors and vector fields between different coordinate systems.

If V is a Vector valued procedure, it is interpreted as a vector field. Otherwise, a vector field is a Vector created by a call to the VectorField routine.

If c is not specified, the current coordinate system is used. If V represents a vector field, the implied coordinates must be indexed with the names of the new coordinates. Otherwise, an error is raised. If V represents a Vector, no coordinate names are required.

If a coordinate system attribute is specified on V, V is interpreted in this coordinate system. Otherwise, the object is interpreted as a Vector or vector field in the current coordinate system. If the two are not compatible, an error is raised.

If p is specified and v is a free Vector in Cartesian coordinates, the result will be a rooted Vector with a root point p. If p is a list, it will be interpreted as Cartesian coordinates.

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

$\mathrm{GetCoordinates}\left(\right)$

$\mathrm{cartesian}$

$\mathrm{v1}≔\mathrm{MapToBasis}\left(⟨1\,1⟩\,\mathrm{polar}\right)$

$\mathrm{v1}≔\left(\sqrt{2}\right)e_r\+\left(\frac{\mathrm{\pi }}{4}\right)e_{\mathrm{\θ}}$

$\mathrm{MapToBasis}\left(\mathrm{v1}\,\mathrm{cartesian}\right)$

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

$\mathrm{v2}≔⟨r\,\mathrm{\theta }⟩$

$\mathrm{v2}≔\left(r\right)e_x\+\left(\mathrm{\theta }\right)e_y$

$\mathrm{SetCoordinates}\left(\mathrm{v2}\,\mathrm{polar}\right)$

$\left(r\right)e_r\+\left(\mathrm{\theta }\right)e_{\mathrm{\θ}}$

$\mathrm{MapToBasis}\left(\mathrm{v2}\right)$

$\left(r\cos \left(\mathrm{\theta }\right)\right)e_x\+\left(r\sin \left(\mathrm{\theta }\right)\right)e_y$

$\mathrm{v3}≔\mathrm{MapToBasis}\left(⟨a\,b⟩\,\mathrm{polar}\,\left[0\,1\right]\right)$

$\mathrm{v3}≔\left[\begin{array}{c}b\\ -a\end{array}\right]$

$\mathrm{About}\left(\mathrm{v3}\right)$

$\left[\begin{array}{cc}\mathrm{Type:}& \mathrm{Rooted\; Vector}\\ \mathrm{Components:}& \left[b\,-a\right]\\ \mathrm{Coordinates:}& \mathrm{polar}\\ \mathrm{Root\; Point:}& \left[1\,\frac{\mathrm{\pi }}{2}\right]\end{array}\right]$

$\mathrm{SetCoordinates}\left(\mathrm{spherical}\left[r,\mathrm{\phi },\mathrm{\theta }\right]\right)$

${\mathrm{spherical}}_{r,\mathrm{\phi },\mathrm{\theta }}$

$\mathrm{VF}≔\mathrm{VectorField}\left(⟨r\,0\,0⟩\right)$

$\mathrm{VF}≔\left(r\right){\stackrel{\_}{e}}_r\+\left(0\right){\stackrel{\_}{e}}_{\mathrm{\φ}}\+\left(0\right){\stackrel{\_}{e}}_{\mathrm{\θ}}$

$\mathrm{MapToBasis}\left(\mathrm{VF}\,\mathrm{cartesian}\left[x,y,z\right]\right)$

$\left(x\right){\stackrel{\_}{e}}_x\+\left(y\right){\stackrel{\_}{e}}_y\+\left(z\right){\stackrel{\_}{e}}_z$

$\mathrm{MapToBasis}\left(\left(r\,\mathrm{\phi }\,\mathrm{\theta }\right)\mapsto ⟨\frac{1}{r^2}\,0\,0⟩\,\mathrm{cartesian}\left[x,y,z\right]\right)$

$\left(\frac{x}{{\left(x^2+y^2+z^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right){\stackrel{\_}{e}}_x\+\left(\frac{y}{{\left(x^2+y^2+z^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right){\stackrel{\_}{e}}_y\+\left(\frac{z}{{\left(x^2+y^2+z^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right){\stackrel{\_}{e}}_z$