The MapToBasis(V, c) calling sequence converts the Vector or vector field V to the coordinate system c.

If V is a Vector-valued procedure, it is interpreted as a vector field.

If c is not specified, the current default coordinate system is used as the new coordinate system.

If c is specified and V represents a vector field, the new coordinate system 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.

If a coordinate system attribute is not specified on V, V is interpreted as a Vector or vector field in the current default coordinate system.  If the default coordinate system is not compatible with the new coordinate system, an error is raised.

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

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

$\mathrm{cartesian}$

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

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

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

$v≔⟨r\,\mathrm{\theta }⟩$

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

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

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

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

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

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

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

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

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

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

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

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

$\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$