Changes

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Each of the operators E, D : X -> EX takes us from considering propositions p : X -> B, here viewed as "scalar fields" over X, to considering the corresponding "differential fields" over X, analogous to what are usually called "vector fields" over X.

Each of the operators <math>\operatorname{E}, \operatorname{D} : X \to \operatorname{E}X\!</math> takes us from considering propositions <math>p : X \to \mathbb{B},\!</math> here viewed as ''scalar fields'' over <math>X,\!</math> to considering the corresponding ''differential fields'' over <math>X,\!</math> analogous to what are usually called ''vector fields'' <math>X.\!</math>

The structure of these differential fields can be described this way.  To each point of X there is attached an object of the following type, a proposition about changes in X, that is, a proposition g : dX -> B. In this setting, if X is the universe that is generated by the set of coordinate propositions {u, v}, then dX is the differential universe that is generated by the set of differential propositions {du, dv}.  These differential propositions may be interpreted as indicating "change in u" and "change in v", respectively.

The structure of these differential fields can be described this way.  To each point of <math>X\!</math> there is attached 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 setting, if <math>X\!</math> is the universe that is generated by the set of coordinate propositions <math>\{ u, v \},\!</math> then <math>\operatorname{d}X\!</math> is the differential universe that is generated by the set of differential propositions <math>\{ \operatorname{d}u, \operatorname{d}v \}.\!</math> These differential propositions may be interpreted as indicating "change in <math>u\!</math>" and "change in <math>v\!</math>", respectively.

A differential operator F, of the first order sort that we have been considering, takes a proposition p : X -> B and gives back a differential proposition Fp : EX -> B.

A differential operator <math>\operatorname{F},\!</math> of the first order sort that we have been considering, takes a proposition <math>p : X \to \mathbb{B}\!</math> and gives back a differential proposition <math>\operatorname{F}p : \operatorname{E}X \to \mathbb{B}.\!</math>

In the field view, we see the proposition p : X -> B as a scalar field and we see the differential proposition Fp : EX -> B as a vector field, specifically, a field of propositions about contemplated changes in X.

In the field view, we see the proposition <math>p : X \to \mathbb{B}\!</math> as a scalar field and we see the differential proposition <math>\operatorname{F}p : \operatorname{E}X \to \mathbb{B}\!</math> as a vector field, specifically, a field of propositions about contemplated changes in <math>X.\!</math>

The field of changes produced by E on uv is shown in Figure 2.

The field of changes produced by <math>\operatorname{E}\!</math> on <math>uv\!</math> is shown in Figure&nbsp;2.

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