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.

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

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

$\mathrm{cartesian}$

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

$v≔⟨r\,\mathrm{\θ}⟩$

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

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

$\mathrm{SetCoordinates}\left({\mathrm{spherical}}_{r\,\mathrm{\φ}\,\mathrm{\θ}}\right)$

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

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

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

$\mathrm{MapToBasis}\left(\left(r\,\mathrm{\φ}\,\mathrm{\θ}\right)\→⟨\frac{1}{r^2}\,0\,0⟩\,{\mathrm{cartesian}}_{x\,y\,z}\right)$