Changes

This gives <math>\operatorname{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math>

This gives <math>\operatorname{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math>

Finally, the tangent universe E''A''^&nbsp;&bull; = [E<font face="lucida calligraphy">A</font>] is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features E<font face="lucida calligraphy">A</font>:

Finally, the tangent universe <math>\operatorname{E}A^\circ = [\operatorname{E}\mathcal{A}]</math> is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features <math>\operatorname{E}\mathcal{A},</math> and this fact is summed up in the following notation:

: E''A''^&nbsp;&bull; = [E<font face="lucida calligraphy">A</font>] = [''a''₁,&nbsp;&hellip;,&nbsp;''a''_''n'',&nbsp;d''a''₁,&nbsp;&hellip;,&nbsp;d''a''_''n''],

: <p><math>\begin{array}{lclcl}

& = & [\operatorname{E}\mathcal{A}]

& = & [a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n]. \\

\end{array}</math></p>

thus giving the tangent universe E''A''^&nbsp;&bull; the type

This gives the tangent universe <math>\operatorname{E}A^\circ</math> the type:

('''B'''^''n'' &times; '''D'''^''n'' +&rarr; '''B''') = ('''B'''^''n'' &times; '''D'''^''n'', ('''B'''^''n'' &times; '''D'''^''n'' &rarr; '''B''')).

 

: <p><math>\begin{array}{lcl}

(\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})). \\

\end{array}</math></p>

A proposition in the tangent universe [E<font face="lucida calligraphy">A</font>] is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.

A proposition in the tangent universe [E<font face="lucida calligraphy">A</font>] is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.