Changes

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

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

Eventually we may want to extend our basic alphabet even further, to allow for discussion of higher order differential expressions.  For those who want to run ahead, and would like to play through, I submit the following gamut of notation (Table 9).

It eventually becomes necessary to extend the initial alphabet even further, to allow for the discussion of higher order differential expressions.  Table&nbsp;9 provides a suggestion of how these further extensions can be carried out.

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

{| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:left; width:96%"

|+ '''Table 9.  Higher Order Differential Features'''

|+ '''Table 9.  Higher Order Differential Features'''

| width=50% |

|

<p><math>\begin{array}{lllll}

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

\operatorname{d}^0 \mathcal{A}

& = & \{a_1, \ldots, a_n\}

| d⁰<font face="lucida calligraphy">A</font>

& = & \mathcal{A} \\

| =

\operatorname{d}^1 \mathcal{A}

| {''a''₁,

& = & \{\operatorname{d}a_1, \ldots, \operatorname{d}a_n\}

| &hellip;,

& = & \operatorname{d}\mathcal{A} \\

| ''a''_''n''}

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

<p><math>\begin{array}{lll}

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

& = & \{\operatorname{d}^k a_1, \ldots, \operatorname{d}^k a_n\} \\

| d¹<font face="lucida calligraphy">A</font>

\operatorname{d}^* \mathcal{A}

| =

& = & \{\operatorname{d}^0 \mathcal{A}, \ldots, \operatorname{d}^k \mathcal{A}, \ldots \} \\

| {d''a''₁,

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

| &hellip;,

|

| d''a''_''n''}

<p><math>\begin{array}{lll}

\operatorname{E}^0 \mathcal{A}

| &nbsp;

& = & \operatorname{d}^0 \mathcal{A} \\

| &nbsp;

\operatorname{E}^1 \mathcal{A}

| d^''k''<font face="lucida calligraphy">A</font>

& = & \operatorname{d}^0 \mathcal{A}\ \cup\ \operatorname{d}^1 \mathcal{A} \\

| =

\operatorname{E}^k \mathcal{A}

| {d^''k''''a''_''1'',

& = & \operatorname{d}^0 \mathcal{A}\ \cup\ \ldots\ \cup\ \operatorname{d}^k \mathcal{A} \\

| &hellip;,

\operatorname{E}^\infty \mathcal{A}

| d^''k''''a''_''n''}

& = & \bigcup\ \operatorname{d}^* \mathcal{A} \\

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

| d^*<font face="lucida calligraphy">A</font>

|}<br>

| =

| {d⁰<font face="lucida calligraphy">A</font>,

| &hellip;,

| d^''k''<font face="lucida calligraphy">A</font>,

| &hellip;}

|}

{| cellpadding="4"

| E⁰<font face="lucida calligraphy">A</font>

| =

| d⁰<font face="lucida calligraphy">A</font>

| E¹<font face="lucida calligraphy">A</font>

| =

| d⁰<font face="lucida calligraphy">A</font> &cup; d¹<font face="lucida calligraphy">A</font>

| E^''k''<font face="lucida calligraphy">A</font>

| =

| d⁰<font face="lucida calligraphy">A</font> &cup; &hellip; &cup; d^''k''<font face="lucida calligraphy">A</font>

| E^&infin;<font face="lucida calligraphy">A</font>

| =

| &cup; d^*<font face="lucida calligraphy">A</font>

|}

</font><br>

===Intentional Propositions===

===Intentional Propositions===