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Revision as of 13:48, 22 June 2009
, 13:48, 22 June 2009→Note 4: convert graphics
In the example <math>f(p, q) = pq,\!</math> the value of the difference proposition <math>\operatorname{D}f_x</math> at each of the four points in <math>x \in X\!</math> may be computed in graphical fashion as shown below:
In the example <math>f(p, q) = pq,\!</math> the value of the difference proposition <math>\operatorname{D}f_x</math> at each of the four points in <math>x \in X\!</math> may be computed in graphical fashion as shown below:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="20"
| [[Image:Cactus Graph Df = ((P,dP)(Q,dQ),PQ).jpg|500px]]
| Df = ((p, dp)(q, dq), pq) |
|-
|-
| align="center" |
| [[Image:Cactus Graph Df@PQ = ((dP)(dQ)).jpg|500px]]
|-
|-
| align="center" |
| [[Image:Cactus Graph Df@P(Q) = (dP)dQ.jpg|500px]]
|-
|-
| align="center" |
| [[Image:Cactus Graph Df@(P)Q = dP(dQ).jpg|500px]]
|-
|-
| align="center" |
| [[Image:Cactus Graph Df@(P)(Q) = dP dQ.jpg|500px]]
|}
|}
The easy way to visualize the values of these graphical expressions is just to notice the following equivalents:
The easy way to visualize the values of these graphical expressions is just to notice the following equivalents:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="20"
| [[Image:Cactus Graph Lobe Rule.jpg|500px]]
| o-o-o-...-o-o-o |
| \ / |
|-
|-
| align="center" |
| [[Image:Cactus Graph Spike Rule.jpg|500px]]
|}
|}
Laying out the arrows on the augmented venn diagram, one gets a picture of a ''differential vector field''.
Laying out the arrows on the augmented venn diagram, one gets a picture of a ''differential vector field''.
{| align="center" cellpadding="10"
{| align="center" cellspacing="20"
| [[Image:Venn Diagram PQ Difference Conj.jpg|500px]]
| [[Image:Venn Diagram PQ Difference Conj.jpg|500px]]
|}
|}
The Figure shows the points of the extended universe <math>\operatorname{E}X = P \times Q \times \operatorname{d}P \times \operatorname{d}Q</math> that are indicated by the difference map <math>\operatorname{D}f : \operatorname{E}X \to \mathbb{B},</math> namely, the following six points or singular propositions::
The Figure shows the points of the extended universe <math>\operatorname{E}X = P \times Q \times \operatorname{d}P \times \operatorname{d}Q</math> that are indicated by the difference map <math>\operatorname{D}f : \operatorname{E}X \to \mathbb{B},</math> namely, the following six points or singular propositions::
{| align="center" cellpadding="6"
{| align="center" cellspacing="20"
|
|
<math>\begin{array}{rcccc}
<math>\begin{array}{rcccc}
