MyWikiBiz, Author Your Legacy — Sunday October 20, 2024
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, 19:30, 7 May 2008
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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> |
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− | 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. |
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− | 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> |
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− | 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> |
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− | 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 2. |
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| <pre> | | <pre> |