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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")
pair of component equations:
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{ccccc}
\operatorname{E}G_1 & = & G_1 (u + du, v + dv)
\\ \\
\operatorname{E}G_2 & = & G_2 (u + du, v + dv)
\end{array}</math>
|}
<pre>
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
