The command TangentPlane(rc, f, R) returns the tangent plane of the hypersurface defined by f at every point of F defined by rc.

The result is a list of pairs [g,ts] where ts is a zero-dimensional regular chain the zero set of which is contained in that g, and g a polynomial the zero set of which defines the tangent plane of f at ts.

It is assumed that rc is a zero-dimensional regular chain.

It is assumed that the hypersurface defined by f is non-singular at every point defined by rc.

This command is part of the RegularChains[AlgebraicGeometryTools] package, so it can be used in the form TangentPlane(..) only after executing the command with(RegularChains[AlgebraicGeometryTools]).  However, it can always be accessed through the long form of the command by using RegularChains[AlgebraicGeometryTools][TangentPlane](..).

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$$\mathrm{with}\left(\mathrm{ChainTools}\right)\:$$\mathrm{with}\left(\mathrm{AlgebraicGeometryTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[x\,y\,z\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{rc}≔\mathrm{Empty}\left(R\right)$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{rc}≔\mathrm{Chain}\left(\left[z-1\,y\,x\right]\,\mathrm{rc}\,R\right)$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{Equations}\left(\mathrm{rc}\,R\right)$

$\left[x\,y\,z-1\right]$

$f≔xz+zy+yx$

$f≔yx+xz+zy$

$\mathrm{tp}≔\mathrm{TangentPlane}\left(\mathrm{rc}\,f\,R\right)$

$\mathrm{tp}≔\left[\left[\mathrm{\_x}+\mathrm{\_y}\,\mathrm{rc}\right]\right]$

The RegularChains[AlgebraicGeometryTools][TangentPlane] command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.