Changes

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" cellspacing="20" style="text-align:center"

{| align="center" cellspacing="20"

| [[Image:Cactus Graph Df = ((P,dP)(Q,dQ),PQ).jpg|500px]]

| [[Image:Cactus Graph Df = ((P,dP)(Q,dQ),PQ).jpg|500px]]

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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" cellspacing="20" style="text-align:center; width:90%"

{| align="center" cellspacing="20"

| [[Image:Cactus Graph Lobe Rule.jpg|500px]]

| [[Image:Cactus Graph Lobe Rule.jpg|500px]]

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| [[Image:Cactus Graph Spike Rule.jpg|500px]]

|  (e_1, ..., e_k, ())  =        e_1 ... e_k    |

|}

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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" cellspacing="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" cellspacing="10"

{| align="center" cellspacing="20"

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<math>\begin{array}{rcccc}

<math>\begin{array}{rcccc}