Changes

The structure of these differential fields can be described this way.  With each point of <math>X\!</math> there is associated an object of the following type:  a proposition about changes in <math>X,\!</math> that is, a proposition <math>g : \operatorname{d}X \to \mathbb{B}.</math>  In this frame of reference, if <math>X^\circ</math> is the universe that is generated by the set of coordinate propositions <math>\{ p, q \},\!</math> then <math>\operatorname{d}X^\circ</math> is the differential universe that is generated by the set of differential propositions <math>\{ \operatorname{d}p, \operatorname{d}q \}.</math>  These differential propositions may be interpreted as indicating <math>{}^{\backprime\backprime} \text{change in}\, p \, {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} \text{change in}\, q \, {}^{\prime\prime},</math> respectively.

The structure of these differential fields can be described this way.  With each point of <math>X\!</math> there is associated an object of the following type:  a proposition about changes in <math>X,\!</math> that is, a proposition <math>g : \operatorname{d}X \to \mathbb{B}.</math>  In this frame of reference, if <math>X^\circ</math> is the universe that is generated by the set of coordinate propositions <math>\{ p, q \},\!</math> then <math>\operatorname{d}X^\circ</math> is the differential universe that is generated by the set of differential propositions <math>\{ \operatorname{d}p, \operatorname{d}q \}.</math>  These differential propositions may be interpreted as indicating <math>{}^{\backprime\backprime} \text{change in}\, p \, {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} \text{change in}\, q \, {}^{\prime\prime},</math> respectively.

A differential operator <math>\operatorname{W},</math> of the first order class that we have been considering, takes a proposition <math>f : X \to \mathbb{B}</math> and gives back a differential proposition <math>\operatorname{W}f : \operatorname{E}X \to \mathbb{B}.</math>  In the field view, we see the proposition <math>f : X \to \mathbb{B}</math> as a scalar field and we see the differential proposition <math>\operatorname{W}f : \operatorname{E}X \to \mathbb{B}</math> as a vector field, specifically, a field of propositions about contemplated changes in <math>X.\!</math>

A differential operator W, of the first order sort that we have

been considering, takes a proposition f : X -> B and gives back

a differential proposition Wf: EX -> B.

 

In the field view, we see the proposition f : X -> B as a scalar field

and we see the differential proposition Wf: EX -> B as a vector field,

specifically, a field of propositions about contemplated changes in X.

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The field of changes produced by <math>\operatorname{E}</math> on <math>pq\!</math> is shown in Figure&nbsp;22-b.

The field of changes produced by <math>\operatorname{E}</math> on <math>pq\!</math> is shown in Figure&nbsp;22-b.