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<math>\operatorname{D}g</math> tells you what you would have to do, from where you are in the universe <math>[u, v],\!</math> if you want to bring about a change in the value of <math>g,\!</math> that is, if you want to get to a place where the value of <math>g\!</math> is different from what it is where you are.  In the present case, where the ruling proposition <math>g\!</math> is <math>\texttt{((u, v))},</math> the term <math>\texttt{uv} \cdot \texttt{(du, dv)}</math> of <math>\operatorname{D}g</math> tells you this:  If <math>u\!</math> and <math>v\!</math> are both true where you are, then you would have to change one or the other but not both of <math>u\!</math> and <math>v\!</math> in order to reach a place where the value of <math>g\!</math> is different from what it is where you are.

<math>\operatorname{D}g</math> tells you what you would have to do, from where you are in the universe <math>[u, v],\!</math> if you want to bring about a change in the value of <math>g,\!</math> that is, if you want to get to a place where the value of <math>g\!</math> is different from what it is where you are.  In the present case, where the ruling proposition <math>g\!</math> is <math>\texttt{((u, v))},</math> the term <math>\texttt{uv} \cdot \texttt{(du, dv)}</math> of <math>\operatorname{D}g</math> tells you this:  If <math>u\!</math> and <math>v\!</math> are both true where you are, then you would have to change one or the other but not both <math>u\!</math> and <math>v\!</math> in order to reach a place where the value of <math>g\!</math> is different from what it is where you are.

Figure&nbsp;2.4 approximates <math>\operatorname{D}g</math> by the linear form <math>\operatorname{d}g</math> that expands over <math>[u, v]\!</math> as follows:

Figure 2.4 approximates Dg in the proxy of the linear proposition

 

dg = uv (du, dv) + u(v)(du, dv) + (u)v (du, dv) + (u)(v) (du, dv).

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Noting the caste of the constant factor (du, dv) distributed over

the expansion of a tautology, dg may be digested as dg = (du, dv).

& = & \texttt{uv} \cdot \texttt{(du, dv)} & + & \texttt{u(v)} \cdot \texttt{(du, dv)} & + & \texttt{(u)v} \cdot \texttt{(du, dv)} & + & \texttt{(u)(v)} \cdot \texttt{(du, dv)}

& = & \texttt{(du, dv)}

 

Figure&nbsp;2.5 shows what remains of the difference map <math>\operatorname{D}g</math> when the first order linear contribution <math>\operatorname{d}g</math> is removed, namely:

and this is nothing but nothing at all, leaving rg = 0.

& = & \texttt{0}

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