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Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 02:58, 5 June 2008
, 02:58, 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:
::: <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>
'''…'''
Table 5 summarizes the notations that are needed to describe the first order differential extensions of propositional calculi in a corresponding manner.
Table 5 summarizes the notations that are needed to describe the first order differential extensions of propositional calculi in a corresponding manner.