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Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 15:44, 5 June 2008
, 15:44, 5 June 2008→Formal development: update
::: <p><math>\operatorname{E}\mathcal{A} = \mathcal{A}\ \cup\ \operatorname{d}\mathcal{A} = \{ a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}.</math></p>
::: <p><math>\operatorname{E}\mathcal{A} = \mathcal{A}\ \cup\ \operatorname{d}\mathcal{A} = \{ a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}.</math></p>
:* The initial space, <math>A = \langle a_1, \ldots, a_n \rangle,</math> is extended by a ''first order differential space'' or ''tangent space'', <math>\operatorname{d}A = \langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle,</math> at each point of <math>A,\!</math> resulting in a ''first order extended space'' or ''tangent bundle'', <math>\operatorname{E}A,</math> defined as follows:
:* The initial space, <math>A = \langle a_1, \ldots, a_n \rangle,</math> is extended by a ''first order differential space'' or ''tangent space'', <math>\operatorname{d}A = \langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle,</math> at each point of <math>A,\!</math> resulting in a ''first order extended space'' or ''tangent bundle space'', <math>\operatorname{E}A,</math> defined as follows:
::: <p><math>\operatorname{E}A = A \times \operatorname{d}A = \langle \operatorname{E}\mathcal{A} \rangle = \langle \mathcal{A} \cup \operatorname{d}\mathcal{A} \rangle = \langle a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle.</math></p>
::: <p><math>\operatorname{E}A = A \times \operatorname{d}A = \langle \operatorname{E}\mathcal{A} \rangle = \langle \mathcal{A} \cup \operatorname{d}\mathcal{A} \rangle = \langle a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle.</math></p>
:* Finally, the initial universe, <math>A^\circ = [ a_1, \ldots, a_n ],</math> is extended by a ''first order differential universe'' or ''tangent universe'', <math>\operatorname{d}A^\circ = [ \operatorname{d}a_1, \ldots, \operatorname{d}a_n ],</math> at each point of <math>A^\circ,</math> resulting in a ''first order extended universe'' or ''tangent bundle universe'', <math>\operatorname{E}A^\circ,</math> defined as follows:
::: <p><math>\operatorname{E}A^\circ = [ \operatorname{E}\mathcal{A} ] = [ \mathcal{A}\ \cup\ \operatorname{d}\mathcal{A} ] = [ a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n ].</math></p>
This gives <math>\operatorname{E}A^\circ</math> the type:
<br><br>
:: <p><math>[ \mathbb{B}^n \times \mathbb{D}^n ] = (\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B}) = (\mathbb{B}^n \times \mathbb{D}^n, \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}).</math></p>
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