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<math>\begin{array}{lll}

<math>\begin{array}{lll}

\operatorname{E}f & = & \underline{((}~ u + du ~\underline{)(}~ v + dv ~\underline{))}

\operatorname{E}f & = & \texttt{(( u + du )( v + dv ))}

\\ \\

\\ \\

\operatorname{E}g & = & \underline{((}~ u + du ~,~ v + dv ~\underline{))}

\operatorname{E}g & = & \texttt{(( u + du ,~ v + dv ))}

\\ \\

\\ \\

\operatorname{D}f & = & \underline{((}~ u ~\underline{)(}~ v ~\underline{))}~ + ~\underline{((}~ u + du ~\underline{)(}~ v + dv ~\underline{))}

\operatorname{D}f & = & \texttt{((u)(v)) ~+~ (( u + du )( v + dv ))}

\\ \\

\\ \\

\operatorname{D}g & = & \underline{((}~ u ~,~ v ~\underline{))}~ + ~\underline{((}~ u + du ~,~ v + dv ~\underline{))}

\operatorname{D}g & = & \texttt{((u,~v)) ~+~ (( u + du ,~ v + dv ))}

\end{array}</math>

\end{array}</math>

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But these initial formulas are purely definitional, and help us little to understand either the purpose of the operators or the significance of the results.  Working symbolically, let's apply a more systematic method to the separate components of the mapping <math>F.\!</math>

But these initial formulas are purely definitional, and help us little to understand either the purpose of the operators or the significance of the results.  Working symbolically, let's apply a more systematic method to the separate components of the mapping <math>F.\!</math>

A sketch of this work is presented in the following series of Figures, where each logical proposition is expanded over the basic cells <math>uv, u \underline{(} v \underline{)}, \underline{(} u \underline{)} v, \underline{(} u \underline{)(} v \underline{)}</math> of the 2-dimensional universe of discourse <math>U^\circ = [u, v].\!</math>

A sketch of this work is presented in the following series of Figures, where each logical proposition is expanded over the basic cells <math>\texttt{uv}, \texttt{u(v)}, \texttt{(u)v}, \texttt{(u)(v)}</math> of the 2-dimensional universe of discourse <math>U^\circ = [u, v].\!</math>

<pre>

<pre>