MyWikiBiz, Author Your Legacy — Thursday September 26, 2024
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, 18:23, 5 March 2009
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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. |
| | | |
− | <pre>
| + | 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>
| + | <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 |