Changes

With these constructions, the differential extension <math>\operatorname{E}A</math> and the space of differential propositions <math>(\operatorname{E}A \to \mathbb{B}),</math> we have arrived, in main outline, at one of the major subgoals of this study.  Table&nbsp;8 summarizes the concepts that have been introduced for working with differentially extended universes of discourse.

With these constructions, the differential extension <math>\operatorname{E}A</math> and the space of differential propositions <math>(\operatorname{E}A \to \mathbb{B}),</math> we have arrived, in main outline, at one of the major subgoals of this study.  Table&nbsp;8 summarizes the concepts that have been introduced for working with differentially extended universes of discourse.

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"

|+ '''Table 8.  Notation for the Differential Extension of Propositional Calculus'''

|+ '''Table 8.  Differential Extension : Basic Notation'''

|- style="background:ghostwhite"

|- style="background:ghostwhite"

! Symbol

! Symbol

! Type

! Type

|-

|-

| d<font face="lucida calligraphy">A<font>

| <math>\operatorname{d}\mathfrak{A}</math>

| {d''a''₁, &hellip;, d''a''_''n''}

| <math>\lbrace\!</math>&nbsp;“<math>\operatorname{d}a_1</math>”&nbsp;<math>, \ldots,\!</math>&nbsp;“<math>\operatorname{d}a_n</math>”&nbsp;<math>\rbrace\!</math>

|

Alphabet of<br>

| <math>[n] = \mathbf{n}</math>

| <math>\operatorname{d}\mathcal{A}</math>

| <math>\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}</math>

| Basis of<br>

differential<br>

differential<br>

features

features

| [''n''] = '''n'''

| <math>[n] = \mathbf{n}</math>

|-

|-

| d''A''_''i''

| <math>\operatorname{d}A_i</math>

| {(d''a''_''i''), d''a''_''i''}

| <math>\{ (\operatorname{d}a_i), \operatorname{d}a_i \}</math>

|

| Differential<br>

Differential<br>

dimension <math>i\!</math>

dimension ''i''

| <math>\mathbb{D}</math>

| '''D'''

|-

|-

| d''A''

| <math>\operatorname{d}A</math>

|

| <math>\langle \operatorname{d}\mathcal{A} \rangle</math><br>

〈d<font face="lucida calligraphy">A</font>〉<br>

<math>\langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle</math><br>

〈d''a''₁, &hellip;, d''a''_''n''〉<br>

<math>\{ (\operatorname{d}a_1, \ldots, \operatorname{d}a_n) \}</math><br>

{‹d''a''₁, &hellip;, d''a''_''n''›}<br>

<math>\operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n</math><br>

d''A''₁ &times; &hellip; &times; d''A''_''n''<br>

<math>\textstyle \prod_i \operatorname{d}A_i</math>

&prod;_''i'' d''A''_''i''

| Tangent space<br>

|

Tangent space<br>

at a point:<br>

at a point:<br>

Set of changes,<br>

Set of changes,<br>

tangent vectors<br>

tangent vectors<br>

at a point

at a point

| '''D'''^''n''

| <math>\mathbb{D}^n</math>

|-

|-

| d''A''*

| <math>\operatorname{d}A^*</math>

| (hom : d''A'' &rarr; '''B''')

| <math>(\operatorname{hom} : \operatorname{d}A \to \mathbb{B})</math>

|

| Linear functions<br>

Linear functions<br>

on <math>\operatorname{d}A</math>

on d''A''

| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n</math>

| ('''D'''^''n'')* = '''D'''^''n''

|-

|-

| d''A''^

| <math>\operatorname{d}A^\uparrow</math>

| (d''A'' &rarr; '''B''')

| <math>(\operatorname{d}A \to \mathbb{B})</math>

|

| Boolean functions<br>

Boolean functions<br>

on <math>\operatorname{d}A</math>

on d''A''

| <math>\mathbb{D}^n \to \mathbb{B}</math>

| '''D'''^''n'' &rarr; '''B'''

|-

|-

| d''A''^&bull;

| <math>\operatorname{d}A^\circ</math>

|

| <math>[\operatorname{d}\mathcal{A}]</math><br>

[d<font face="lucida calligraphy">A</font>]<br>

<math>(\operatorname{d}A, \operatorname{d}A^\uparrow)</math><br>

(d''A'', d''A''^)<br>

<math>(\operatorname{d}A\ +\!\to \mathbb{B})</math><br>

(d''A'' +&rarr; '''B''')<br>

<math>(\operatorname{d}A, (\operatorname{d}A \to \mathbb{B}))</math><br>

(d''A'', (d''A'' &rarr; '''B'''))<br>

<math>[\operatorname{d}a_1, \ldots, \operatorname{d}a_n]</math>

[d''a''₁, &hellip;, d''a''_''n'']

| Tangent universe<br>

|

at a point of <math>A^\circ,</math><br>

Tangent universe<br>

at a point of ''A''^&bull;,<br>

based on the<br>

based on the<br>

tangent features<br>

tangent features<br>

{d''a''₁, &hellip;, d''a''_''n''}

<math>\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}</math>

|

| <math>(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))</math><br>

('''D'''^''n'', ('''D'''^''n'' &rarr; '''B'''))<br>

<math>(\mathbb{D}^n\ +\!\to \mathbb{B})</math><br>

('''D'''^''n'' +&rarr; '''B''')<br>

<math>[\mathbb{D}^n]</math>

['''D'''^''n'']

|}<br>

|}

</font><br>

The adjectives ''differential'' or ''tangent'' are systematically attached to every construct based on the differential alphabet d<font face="lucida calligraphy">A</font>, taken by itself.  Strictly speaking, we probably ought to call d<font face="lucida calligraphy">A</font> the set of ''cotangent'' features derived from <font face="lucida calligraphy">A</font>, but the only time this distinction really seems to matter is when we need to distinguish the tangent vectors as maps of type ('''B'''^''n''&nbsp;&rarr;&nbsp;'''B''')&nbsp;&rarr;&nbsp;'''B''' from cotangent vectors as elements of type '''D'''^''n''.  In like fashion, having defined E<font face="lucida calligraphy">A</font> = <font face="lucida calligraphy">A</font>&nbsp;&cup;&nbsp;d<font face="lucida calligraphy">A</font>, we can systematically attach the adjective ''extended'' or the substantive ''bundle'' to all of the constructs associated with this full complement of 2''n'' features.

The adjectives ''differential'' or ''tangent'' are systematically attached to every construct based on the differential alphabet d<font face="lucida calligraphy">A</font>, taken by itself.  Strictly speaking, we probably ought to call d<font face="lucida calligraphy">A</font> the set of ''cotangent'' features derived from <font face="lucida calligraphy">A</font>, but the only time this distinction really seems to matter is when we need to distinguish the tangent vectors as maps of type ('''B'''^''n''&nbsp;&rarr;&nbsp;'''B''')&nbsp;&rarr;&nbsp;'''B''' from cotangent vectors as elements of type '''D'''^''n''.  In like fashion, having defined E<font face="lucida calligraphy">A</font> = <font face="lucida calligraphy">A</font>&nbsp;&cup;&nbsp;d<font face="lucida calligraphy">A</font>, we can systematically attach the adjective ''extended'' or the substantive ''bundle'' to all of the constructs associated with this full complement of 2''n'' features.