Changes

differential analysis of <math>G\!</math> is based on the definition of a couple of further transformations, derived by way of operators on <math>G,\!</math> that ply between the (first order) extended universes, <math>\operatorname{E}U^\circ = [u, v, du, dv]</math> and <math>\operatorname{E}X^\circ = [x, y, dx, dy],</math> of <math>G \operatorname{'s}\!</math> own source and target universes.

differential analysis of <math>G\!</math> is based on the definition of a couple of further transformations, derived by way of operators on <math>G,\!</math> that ply between the (first order) extended universes, <math>\operatorname{E}U^\circ = [u, v, du, dv]</math> and <math>\operatorname{E}X^\circ = [x, y, dx, dy],</math> of <math>G \operatorname{'s}\!</math> own source and target universes.

First, the ''enlargement map'' (or the ''secant transformation'') <math>\operatorname{E}G = (\operatorname{E}G_1, \operatorname{E}G_2) : \operatorname{E}U^\circ \to \operatorname{E}X^\circ</math> is defined by the following pair of component equations:

First, the "enlargement map" (or the "secant transformation")

EG = <EG_1, EG_2> : EU% -> EX% is defined by the following

pair of component equations:

EG_1  = G_1 <u + du, v + dv>

{| align="center" cellpadding="8" width="90%"

 

EG_2  = G_2 <u + du, v + dv>

\operatorname{E}G_1 & = & G_1 (u + du, v + dv)

\operatorname{E}G_2 & = & G_2 (u + du, v + dv)

\end{array}</math>

Second, the "difference map" (or the "chordal transformation")

Second, the "difference map" (or the "chordal transformation")

DG = <DG_1, DG_2> : EU% -> EX% is defined in a component-wise

DG = <DG_1, DG_2> : EU% -> EX% is defined in a component-wise