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DifferentialGeometry

 ApplyTransformation
 evaluate a transformation at a point

 Calling Sequence ApplyTransformation(Phi, pt)

Parameters

 Phi - transformation from a manifold M to another manifold N pt - a list of coordinates or a list of equations defining a point in the domain manifold M

Description

 • ApplyTransformation(Phi, pt) returns the coordinates of the point Phi(pt) in N.
 • The second argument is of the form [a1, a2, ...] or [x1 = a1, x2 = a2, ...], where x1, x2, ... are the coordinates on M.
 • This command is part of the DifferentialGeometry package, and so can be used in the form ApplyTransformation(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-ApplyTransformation.

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$

Example 1.

Define coordinate systems M and N.

 > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right):$$\mathrm{DGsetup}\left(\left[u,v\right],N\right):$

Define a transformation Phi from M to N.

 > $\mathrm{Φ}≔\mathrm{Transformation}\left(M,N,\left[u={x}^{2}+{y}^{2}+{z}^{2},v=xy+yz+xz\right]\right)$
 ${\mathrm{Φ}}{≔}\left[{u}{=}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}{,}{v}{=}{x}{}{y}{+}{x}{}{z}{+}{y}{}{z}\right]$ (1)

Apply the transformation Phi to the points p1, p2.

 > $\mathrm{p1}≔\left[1,2,3\right]$
 ${\mathrm{p1}}{≔}\left[{1}{,}{2}{,}{3}\right]$ (2)
 > $\mathrm{ApplyTransformation}\left(\mathrm{Φ},\mathrm{p1}\right)$
 $\left[{14}{,}{11}\right]$ (3)
 > $\mathrm{p2}≔\left[x=5,y=0,z=1\right]$
 ${\mathrm{p2}}{≔}\left[{x}{=}{5}{,}{y}{=}{0}{,}{z}{=}{1}\right]$ (4)
 > $\mathrm{ApplyTransformation}\left(\mathrm{Φ},\mathrm{p2}\right)$
 $\left[{u}{=}{26}{,}{v}{=}{5}\right]$ (5)
 N >