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{{DISPLAYTITLE:Differential Propositional Calculus}}
{{DISPLAYTITLE:Differential Propositional Calculus}}
A '''differential propositional calculus''' is a [[propositional calculus]] extended by a set of terms for describing aspects of change and difference, for example, processes that take place in a [[universe of discourse]] or transformations that map a source universe into a target universe.
A '''differential propositional calculus''' is a [[propositional calculus]] extended by a set of terms for describing aspects of change and difference, for example, processes that take place in a [[universe of discourse]] or transformations that map a source universe into a target universe.
{| align="center" cellpadding="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
|
|
<p>[[Image:Differential_Propositional_Calculus_2.jpg|500px]]</p>
<p>[[Differential_Propositional_Calculus_2.jpg|500px]]</p>
<p>'''Figure 2. Same Names, Different Habitations'''</p>
<p>'''Figure 2. Same Names, Different Habitations'''</p>
|}
|}
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| <math>((x),(y),(z))\!</math>
| <math>((x),(y),(z))~\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
|
|
<math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))</math><br>
<math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))</math><br>
<math>(\mathbb{B}^n\ +\!\to \mathbb{B})</math><br>
<math>(\mathbb{B}^n\ +\!\to \mathbb{B})\!</math><br>
<math>[\mathbb{B}^n]</math>
<math>[\mathbb{B}^n]</math>
|}
|}
</ul>
</ul>
In each case the rank <math>k\!</math> ranges from <math>0\!</math> to <math>n\!</math> and counts the number of positive appearances of the coordinate propositions <math>a_1, \ldots, a_n\!</math> in the resulting expression. For example, for <math>n = 3,\!</math> the linear proposition of rank <math>0\!</math> is <math>0,\!</math> the positive proposition of rank <math>0\!</math> is <math>1,\!</math> and the singular proposition of rank <math>0\!</math> is <math>\texttt{(} a_1 \texttt{)} \texttt{(} a_2 \texttt{)} \texttt{(} a_3 \texttt{)}.\!</math>
In each case the rank <math>k\!</math> ranges from <math>0\!</math> to <math>n\!</math> and counts the number of positive appearances of the coordinate propositions <math>a_1, \ldots, a_n\!</math> in the resulting expression. For example, for <math>n = 3,~\!</math> the linear proposition of rank <math>0\!</math> is <math>0,\!</math> the positive proposition of rank <math>0\!</math> is <math>1,\!</math> and the singular proposition of rank <math>0\!</math> is <math>\texttt{(} a_1 \texttt{)} \texttt{(} a_2 \texttt{)} \texttt{(} a_3 \texttt{)}.\!</math>
The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}\!</math> are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.
The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}\!</math> are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.
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:
<ul>
<li>
<p>The initial alphabet, <math>\mathfrak{A} = \{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \},\!</math> is extended by a ''first order differential alphabet'', <math>\operatorname{d}\mathfrak{A} = \{ {}^{\backprime\backprime} \operatorname{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \operatorname{d}a_n {}^{\prime\prime} \},\!</math> resulting in a ''first order extended alphabet'', <math>\operatorname{E}\mathfrak{A},</math> defined as follows:</p>
{| align="center" cellspacing="8" width="90%"
|
<math>\operatorname{E}\mathfrak{A} ~=~ \mathfrak{A} ~\cup~ \operatorname{d}\mathfrak{A} ~=~ \{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime}, {}^{\backprime\backprime} \operatorname{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \operatorname{d}a_n {}^{\prime\prime} \}.\!</math>
|}
</li>
<li>
<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>
{| align="center" cellspacing="8" width="90%"
|
<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>
|}
</li>
<li>
<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>
{| align="center" cellspacing="8" width="90%"
|
<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>
|}
</li>
<li>
<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>
{| align="center" cellspacing="8" width="90%"
|
<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:</p>
{| align="center" cellspacing="8" width="90%"
|
<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>
|}
</li>
</ul>
A proposition in a differential extension of a universe of discourse is called a ''differential proposition'' and forms the analogue of a system of differential equations in ordinary calculus. With these constructions, the first order extended universe <math>\operatorname{E}A^\circ</math> and the first order differential proposition <math>f : \operatorname{E}A \to \mathbb{B},</math> we have arrived, in concept at least, at the foothills of [[differential logic]].
A proposition in a differential extension of a universe of discourse is called a ''differential proposition'' and forms the analogue of a system of differential equations in ordinary calculus. With these constructions, the first order extended universe <math>\operatorname{E}A^\circ</math> and the first order differential proposition <math>f : \operatorname{E}A \to \mathbb{B},</math> we have arrived, in concept at least, at the foothills of [[differential logic]].
| <math>\mathbb{D}^n \to \mathbb{B}</math>
| <math>\mathbb{D}^n \to \mathbb{B}</math>
|-
|-
| <math>\operatorname{d}A^\circ</math>
| <math>\operatorname{d}A^\circ\!</math>
|
|
<math>[\operatorname{d}\mathcal{A}]</math><br>
<math>[\operatorname{d}\mathcal{A}]\!</math><br>
<math>(\operatorname{d}A, \operatorname{d}A^\uparrow)</math><br>
<math>(\operatorname{d}A, \operatorname{d}A^\uparrow)\!</math><br>
<math>(\operatorname{d}A\ +\!\to \mathbb{B})</math><br>
<math>(\operatorname{d}A\ +\!\to \mathbb{B})\!</math><br>
<math>(\operatorname{d}A, (\operatorname{d}A \to \mathbb{B}))</math><br>
<math>(\operatorname{d}A, (\operatorname{d}A \to \mathbb{B}))\!</math><br>
<math>[\operatorname{d}a_1, \ldots, \operatorname{d}a_n]</math>
<math>[\operatorname{d}a_1, \ldots, \operatorname{d}a_n]\!</math>
|
|
Tangent universe<br>
Tangent universe<br>
| height="36px" | <p><math>f_{0000}\!</math></p>
| height="36px" | <p><math>f_{0000}\!</math></p>
|-
|-
| height="36px" | <p><math>f_{0001}\!</math></p>
| height="36px" | <p><math>f_{0001}~\!</math></p>
|-
|-
| height="36px" | <p><math>f_{0010}\!</math></p>
| height="36px" | <p><math>f_{0010}\!</math></p>
| height="36px" | <p><math>(x)(y)\!</math></p>
| height="36px" | <p><math>(x)(y)\!</math></p>
|-
|-
| height="36px" | <p><math>(x)\ y\!</math></p>
| height="36px" | <p><math>(x)~y\!</math></p>
|-
|-
| height="36px" | <p><math>(x)\!</math></p>
| height="36px" | <p><math>(x)\!</math></p>
|-
|-
| height="36px" | <p><math>x\ (y)\!</math></p>
| height="36px" | <p><math>x~(y)\!</math></p>
|-
|-
| height="36px" | <p><math>(y)\!</math></p>
| height="36px" | <p><math>(y)\!</math></p>
|-
|-
| height="36px" | <p><math>(x,\ y)\!</math></p>
| height="36px" | <p><math>(x,~y)\!</math></p>
|-
|-
| height="36px" | <p><math>(x\ y)\!</math></p>
| height="36px" | <p><math>(x~y)\!</math></p>
|}
|}
|
|
| height="36px" | <p><math>\operatorname{false}</math></p>
| height="36px" | <p><math>\operatorname{false}</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{neither}\ x\ \operatorname{nor}\ y</math></p>
| height="36px" | <p><math>\operatorname{neither}~ x ~\operatorname{nor}~ y</math></p>
|-
|-
| height="36px" | <p><math>y\ \operatorname{without}\ x</math></p>
| height="36px" | <p><math>y ~\operatorname{without}~ x</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{not}\ x</math></p>
| height="36px" | <p><math>\operatorname{not}~ x</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{without}\ y</math></p>
| height="36px" | <p><math>x ~\operatorname{without}~ y</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{not}\ y</math></p>
| height="36px" | <p><math>\operatorname{not}~ y</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{not~equal~to}\ y</math></p>
| height="36px" | <p><math>x ~\operatorname{not~equal~to}~ y</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{not~both}\ x\ \operatorname{and}\ y</math></p>
| height="36px" | <p><math>\operatorname{not~both}~ x ~\operatorname{and}~ y</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>x\ y\!</math></p>
| height="36px" | <p><math>x~y\!</math></p>
|-
|-
| height="36px" | <p><math>((x,\ y))\!</math></p>
| height="36px" | <p><math>((x,~y))\!</math></p>
|-
|-
| height="36px" | <p><math>y\!</math></p>
| height="36px" | <p><math>y\!</math></p>
|-
|-
| height="36px" | <p><math>(x\ (y))\!</math></p>
| height="36px" | <p><math>(x~(y))\!</math></p>
|-
|-
| height="36px" | <p><math>x\!</math></p>
| height="36px" | <p><math>x\!</math></p>
|-
|-
| height="36px" | <p><math>((x)\ y)\!</math></p>
| height="36px" | <p><math>((x)~y)\!</math></p>
|-
|-
| height="36px" | <p><math>((x)(y))\!</math></p>
| height="36px" | <p><math>((x)(y))\!</math></p>
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>x\ \operatorname{and}\ y</math></p>
| height="36px" | <p><math>x ~\operatorname{and}~ y</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{equal~to}\ y</math></p>
| height="36px" | <p><math>x ~\operatorname{equal~to}~ y</math></p>
|-
|-
| height="36px" | <p><math>y\!</math></p>
| height="36px" | <p><math>y\!</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{not}\ x\ \operatorname{without}\ y</math></p>
| height="36px" | <p><math>\operatorname{not}~ x ~\operatorname{without}~ y</math></p>
|-
|-
| height="36px" | <p><math>x\!</math></p>
| height="36px" | <p><math>x\!</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{not}\ y\ \operatorname{without}\ x</math></p>
| height="36px" | <p><math>\operatorname{not}~ y ~\operatorname{without}~ x</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{or}\ y</math></p>
| height="36px" | <p><math>x ~\operatorname{or}~ y</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{true}</math></p>
| height="36px" | <p><math>\operatorname{true}</math></p>
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>f_{0001}\!</math></p>
| height="36px" | <p><math>f_{0001}~\!</math></p>
|-
|-
| height="36px" | <p><math>f_{0010}\!</math></p>
| height="36px" | <p><math>f_{0010}\!</math></p>
| height="36px" | <p><math>(x)(y)\!</math></p>
| height="36px" | <p><math>(x)(y)\!</math></p>
|-
|-
| height="36px" | <p><math>(x)\ y\!</math></p>
| height="36px" | <p><math>(x)~y\!</math></p>
|-
|-
| height="36px" | <p><math>x\ (y)\!</math></p>
| height="36px" | <p><math>x~(y)\!</math></p>
|-
|-
| height="36px" | <p><math>x\ y\!</math></p>
| height="36px" | <p><math>x~y\!</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>\operatorname{neither}\ x\ \operatorname{nor}\ y</math></p>
| height="36px" | <p><math>\operatorname{neither}~ x ~\operatorname{nor}~ y</math></p>
|-
|-
| height="36px" | <p><math>y\ \operatorname{without}\ x</math></p>
| height="36px" | <p><math>y ~\operatorname{without}~ x</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{without}\ y</math></p>
| height="36px" | <p><math>x ~\operatorname{without}~ y</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{and}\ y</math></p>
| height="36px" | <p><math>x ~\operatorname{and}~ y</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>\operatorname{not}\ x</math></p>
| height="36px" | <p><math>\operatorname{not}~ x</math></p>
|-
|-
| height="36px" | <p><math>x\!</math></p>
| height="36px" | <p><math>x\!</math></p>
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>(x,\ y)\!</math></p>
| height="36px" | <p><math>(x,~y)\!</math></p>
|-
|-
| height="36px" | <p><math>((x,\ y))\!</math></p>
| height="36px" | <p><math>((x,~y))\!</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>x\ \operatorname{not~equal~to}\ y</math></p>
| height="36px" | <p><math>x ~\operatorname{not~equal~to}~ y</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{equal~to}\ y</math></p>
| height="36px" | <p><math>x ~\operatorname{equal~to}~ y</math></p>
|}
|}
|
|
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>\operatorname{not}\ y</math></p>
| height="36px" | <p><math>\operatorname{not}~ y</math></p>
|-
|-
| height="36px" | <p><math>y\!</math></p>
| height="36px" | <p><math>y\!</math></p>
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>(x\ y)\!</math></p>
| height="36px" | <p><math>(x~y)\!</math></p>
|-
|-
| height="36px" | <p><math>(x\ (y))\!</math></p>
| height="36px" | <p><math>(x~(y))\!</math></p>
|-
|-
| height="36px" | <p><math>((x)\ y)\!</math></p>
| height="36px" | <p><math>((x)~y)\!</math></p>
|-
|-
| height="36px" | <p><math>((x)(y))\!</math></p>
| height="36px" | <p><math>((x)(y))\!</math></p>
{| align="center"
{| align="center"
|-
|-
| height="36px" | <p><math>\operatorname{not~both}\ x\ \operatorname{and}\ y</math></p>
| height="36px" | <p><math>\operatorname{not~both}~ x ~\operatorname{and}~ y</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{not}\ x\ \operatorname{without}\ y</math></p>
| height="36px" | <p><math>\operatorname{not}~ x ~\operatorname{without}~ y</math></p>
|-
|-
| height="36px" | <p><math>\operatorname{not}\ y\ \operatorname{without}\ x</math></p>
| height="36px" | <p><math>\operatorname{not}~ y ~\operatorname{without}~ x</math></p>
|-
|-
| height="36px" | <p><math>x\ \operatorname{or}\ y</math></p>
| height="36px" | <p><math>x ~\operatorname{or}~ y</math></p>
|}
|}
|
|
|
|
<math>\begin{smallmatrix}
<math>\begin{smallmatrix}
(y) & \operatorname{d}x\ (\operatorname{d}y) & + &
(y) & \operatorname{d}x ~(\operatorname{d}y) & + &
(x) & (\operatorname{d}x)\ \operatorname{d}y & + &
(x) & (\operatorname{d}x)~ \operatorname{d}y & + &
((x, y)) & \operatorname{d}x\ \operatorname{d}y \\
((x, y)) & \operatorname{d}x ~ \operatorname{d}y \\
y & \operatorname{d}x\ (\operatorname{d}y) & + &
y & \operatorname{d}x ~(\operatorname{d}y) & + &
(x) & (\operatorname{d}x)\ \operatorname{d}y & + &
(x) & (\operatorname{d}x)~ \operatorname{d}y & + &
(x, y) & \operatorname{d}x\ \operatorname{d}y \\
(x, y) & \operatorname{d}x ~ \operatorname{d}y \\
(y) & \operatorname{d}x\ (\operatorname{d}y) & + &
(y) & \operatorname{d}x ~(\operatorname{d}y) & + &
x & (\operatorname{d}x)\ \operatorname{d}y & + &
x & (\operatorname{d}x)~ \operatorname{d}y & + &
(x, y) & \operatorname{d}x\ \operatorname{d}y \\
(x, y) & \operatorname{d}x ~ \operatorname{d}y \\
y & \operatorname{d}x\ (\operatorname{d}y) & + &
y & \operatorname{d}x ~(\operatorname{d}y) & + &
x & (\operatorname{d}x)\ \operatorname{d}y & + &
x & (\operatorname{d}x)~ \operatorname{d}y & + &
((x, y)) & \operatorname{d}x\ \operatorname{d}y \\
((x, y)) & \operatorname{d}x ~ \operatorname{d}y \\
\end{smallmatrix}</math>
\end{smallmatrix}</math>
|}
|}
|
|
<math>\begin{smallmatrix}
<math>\begin{smallmatrix}
\operatorname{d}x\ (\operatorname{d}y) & + &
\operatorname{d}x~(\operatorname{d}y) & + &
\operatorname{d}x\ \operatorname{d}y \\
\operatorname{d}x~ \operatorname{d}y \\
\operatorname{d}x\ (\operatorname{d}y) & + &
\operatorname{d}x~(\operatorname{d}y) & + &
\operatorname{d}x\ \operatorname{d}y \\
\operatorname{d}x~ \operatorname{d}y \\
\end{smallmatrix}</math>
\end{smallmatrix}</math>
|}
|}
|
|
<math>\begin{smallmatrix}
<math>\begin{smallmatrix}
\operatorname{d}x\ (\operatorname{d}y) & + &
\operatorname{d}x ~(\operatorname{d}y) & + &
(\operatorname{d}x)\ \operatorname{d}y \\
(\operatorname{d}x)~ \operatorname{d}y \\
\operatorname{d}x\ (\operatorname{d}y) & + &
\operatorname{d}x ~(\operatorname{d}y) & + &
(\operatorname{d}x)\ \operatorname{d}y \\
(\operatorname{d}x)~ \operatorname{d}y \\
\end{smallmatrix}</math>
\end{smallmatrix}</math>
|}
|}
|
|
<math>\begin{smallmatrix}
<math>\begin{smallmatrix}
(\operatorname{d}x)\ \operatorname{d}y & + &
(\operatorname{d}x)~\operatorname{d}y & + &
\operatorname{d}x\ \operatorname{d}y \\
\operatorname{d}x ~\operatorname{d}y \\
(\operatorname{d}x)\ \operatorname{d}y & + &
(\operatorname{d}x)~\operatorname{d}y & + &
\operatorname{d}x\ \operatorname{d}y \\
\operatorname{d}x ~\operatorname{d}y \\
\end{smallmatrix}</math>
\end{smallmatrix}</math>
|}
|}
|
|
<math>\begin{smallmatrix}
<math>\begin{smallmatrix}
y & \operatorname{d}x\ (\operatorname{d}y) & + &
y & \operatorname{d}x ~(\operatorname{d}y) & + &
x & (\operatorname{d}x)\ \operatorname{d}y & + &
x & (\operatorname{d}x)~ \operatorname{d}y & + &
((x, y)) & \operatorname{d}x\ \operatorname{d}y \\
((x, y)) & \operatorname{d}x ~ \operatorname{d}y \\
(y) & \operatorname{d}x\ (\operatorname{d}y) & + &
(y) & \operatorname{d}x ~(\operatorname{d}y) & + &
x & (\operatorname{d}x)\ \operatorname{d}y & + &
x & (\operatorname{d}x)~ \operatorname{d}y & + &
(x, y) & \operatorname{d}x\ \operatorname{d}y \\
(x, y) & \operatorname{d}x ~ \operatorname{d}y \\
y & \operatorname{d}x\ (\operatorname{d}y) & + &
y & \operatorname{d}x ~(\operatorname{d}y) & + &
(x) & (\operatorname{d}x)\ \operatorname{d}y & + &
(x) & (\operatorname{d}x)~ \operatorname{d}y & + &
(x, y) & \operatorname{d}x\ \operatorname{d}y \\
(x, y) & \operatorname{d}x ~ \operatorname{d}y \\
(y) & \operatorname{d}x\ (\operatorname{d}y) & + &
(y) & \operatorname{d}x ~(\operatorname{d}y) & + &
(x) & (\operatorname{d}x)\ \operatorname{d}y & + &
(x) & (\operatorname{d}x)~ \operatorname{d}y & + &
((x, y)) & \operatorname{d}x\ \operatorname{d}y \\
((x, y)) & \operatorname{d}x ~ \operatorname{d}y \\
\end{smallmatrix}</math>
\end{smallmatrix}</math>
|}
|}
|
|
<p><math>\operatorname{d}f = </math></p>
<p><math>\operatorname{d}f = </math></p>
<p><math>\partial_x f \cdot \operatorname{d}x\ +\ \partial_y f \cdot \operatorname{d}y</math></p>
<p><math>\partial_x f \cdot \operatorname{d}x ~+~ \partial_y f \cdot \operatorname{d}y</math></p>
|
|
<p><math>\operatorname{d}^2 f = </math></p>
<p><math>\operatorname{d}^2 f = </math></p>
<p><math>\partial_{xy} f \cdot \operatorname{d}x\, \operatorname{d}y</math></p>
<p><math>\partial_{xy} f \cdot \operatorname{d}x \; \operatorname{d}y</math></p>
| <math>\operatorname{d}f|_{x\, y}\!</math>
| <math>\operatorname{d}f|_{ x \; y}\!</math>
| <math>\operatorname{d}f|_{x\, (y)}\!</math>
| <math>\operatorname{d}f|_{ x \;(y)}\!</math>
| <math>\operatorname{d}f|_{(x)\, y}\!</math>
| <math>\operatorname{d}f|_{(x)\; y}\!</math>
| <math>\operatorname{d}f|_{(x)(y)}\!</math>
| <math>\operatorname{d}f|_{(x)(y)}\!</math>
|-
|-
