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An initial universe of discourse, <math>A^\circ,</math> supplies the groundwork for any number of further extensions, beginning with the ''first order differential extension'', <math>\operatorname{E}A^\circ.</math>  The construction of <math>\operatorname{E}A^\circ</math> can be described in the following stages:

An initial universe of discourse, <math>A^\circ,</math> supplies the groundwork for any number of further extensions, beginning with the ''first order differential extension'', <math>\operatorname{E}A^\circ.</math>  The construction of <math>\operatorname{E}A^\circ</math> can be described in the following stages:

:* The initial alphabet, <math>\mathfrak{A} = \lbrace\!</math>&nbsp;“<math>a_1\!</math>”&nbsp;<math>, \ldots,\!</math>&nbsp;“<math>a_n\!</math>”&nbsp;<math>\rbrace,\!</math> is extended by a ''first order differential alphabet'', <math>\operatorname{d}\mathfrak{A} = \lbrace\!</math>&nbsp;“<math>\operatorname{d}a_1\!</math>”&nbsp;<math>, \ldots,\!</math>&nbsp;“<math>\operatorname{d}a_n\!</math>”&nbsp;<math>\rbrace,\!</math> resulting in a ''first order extended alphabet'', <math>\operatorname{E}\mathfrak{A},</math> defined as follows:

:* <p>The initial alphabet, <math>\mathfrak{A} = \lbrace\!</math>&nbsp;“<math>a_1\!</math>”&nbsp;<math>, \ldots,\!</math>&nbsp;“<math>a_n\!</math>”&nbsp;<math>\rbrace,\!</math> is extended by a ''first order differential alphabet'', <math>\operatorname{d}\mathfrak{A} = \lbrace\!</math>&nbsp;“<math>\operatorname{d}a_1\!</math>”&nbsp;<math>, \ldots,\!</math>&nbsp;“<math>\operatorname{d}a_n\!</math>”&nbsp;<math>\rbrace,\!</math> resulting in a ''first order extended alphabet'', <math>\operatorname{E}\mathfrak{A},</math> defined as follows:</p><blockquote><math>\operatorname{E}\mathfrak{A} = \mathfrak{A}\ \cup\ \operatorname{d}\mathfrak{A} = \lbrace\!</math>&nbsp;“<math>a_1\!</math>”&nbsp;<math>, \ldots,\!</math>&nbsp;“<math>a_n\!</math>”<math>,\!</math>&nbsp;&nbsp;“<math>\operatorname{d}a_1\!</math>”&nbsp;<math>, \ldots,\!</math>&nbsp;“<math>\operatorname{d}a_n\!</math>”&nbsp;<math>\rbrace.\!</math></blockquote>

::: <p><math>\operatorname{E}\mathfrak{A} = \mathfrak{A}\ \cup\ \operatorname{d}\mathfrak{A} = \lbrace\!</math>&nbsp;“<math>a_1\!</math>”&nbsp;<math>, \ldots,\!</math>&nbsp;“<math>a_n\!</math>”<math>,\!</math>&nbsp;&nbsp;“<math>\operatorname{d}a_1\!</math>”&nbsp;<math>, \ldots,\!</math>&nbsp;“<math>\operatorname{d}a_n\!</math>”&nbsp;<math>\rbrace.\!</math></p>

:* <p>The initial basis, <math>\mathcal{A} = \{ a_1, \ldots, a_n \},</math> is extended by a ''first order differential basis'', <math>\operatorname{d}\mathcal{A} = \{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \},</math> resulting in a ''first order extended basis'', <math>\operatorname{E}\mathcal{A},</math> defined as follows:</p><blockquote><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></blockquote>

:* The initial basis, <math>\mathcal{A} = \{ a_1, \ldots, a_n \},</math> is extended by a ''first order differential basis'', <math>\operatorname{d}\mathcal{A} = \{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \},</math> resulting in a ''first order extended basis'', <math>\operatorname{E}\mathcal{A},</math> defined as follows:

:* <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 space'', <math>\operatorname{E}A,</math> defined as follows:</p><blockquote><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></blockquote>

:* <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><blockquote><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></blockquote><p>This gives <math>\operatorname{E}A^\circ</math> the type:</p><blockquote><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></blockquote>

 

:* 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>

 

:* 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:

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:: <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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