「仕事ポテンシャル場」の構成要素は，以下のものであった：
\[
f : S \longrightarrow K \\
f^{'} : {\bf x} \longmapsto
\left( \frac{\partial f({\bf x})}{\partial x}, \frac{\partial f({\bf x})}{\partial y}, \frac{\partial f({\bf x})}{\partial z} \right)
\ \ \ \ ({\bf x} \in S) \\
{\bf X} : {\bf x} \longmapsto \left( {\bf X}_x({\bf x}), {\bf X}_y({\bf x}), {\bf X}_z({\bf x}) \right) \,; \ \ \ S \longrightarrow K \times K \times K \\
f^{'}({\bf x}) \cdot {\bf X}({\bf x})
=
\frac{\partial f({\bf x})}{\partial x} {\bf X}_x({\bf x}) \,+\,
\frac{\partial f({\bf x})}{\partial y} {\bf X}_y({\bf x}) \,+\,
\frac{\partial f({\bf x})}{\partial z} {\bf X}_z({\bf x}) \\
=
\left( \frac{\partial f({\bf x})}{\partial x} ,\,
\frac{\partial f({\bf x})}{\partial y} ,\,
\frac{\partial f({\bf x})}{\partial z} \right)
\left(
\begin{array}{c}
{\bf X}_x({\bf x}) \\
{\bf X}_y({\bf x}) \\
{\bf X}_z({\bf x}) \\
\end{array}
\right)
\]
このうち，ｆを択んで，これの次元拡張を行う。
即ち，f をつぎの関数に替える：
\[
Y = ( Y^1, \cdots, Y^n) : S \longrightarrow \overbrace{K \times \cdots \times K}^{n}
\]
これに伴い，上の内容でｆが係わるところは，つぎのように変更される：
\[
Y^{'} : {\bf x} \longmapsto
\left(
\begin{array}{ccc}
\frac{\partial Y^1({\bf x})}{\partial x} & \frac{\partial Y^1({\bf x})}{\partial y} & \frac{\partial Y^1({\bf x})}{\partial z} \\
& \cdots & \\
\frac{\partial Y^n({\bf x})}{\partial x} & \frac{\partial Y^1({\bf x})}{\partial y} & \frac{\partial Y^n({\bf x})}{\partial z} \\
\end{array}
\right) \\
\\
Y^{'}({\bf x}) \, {\bf X}({\bf x})
=
\left(
\begin{array}{ccc}
\frac{\partial Y^1({\bf x})}{\partial x} & \frac{\partial Y^1({\bf x})}{\partial y} & \frac{\partial Y^1({\bf x})}{\partial z} \\
& \cdots & \\
\frac{\partial Y^n({\bf x})}{\partial x} & \frac{\partial Y^1({\bf x})}{\partial y} & \frac{\partial Y^n({\bf x})}{\partial z} \\
\end{array}
\right)
\left(
\begin{array}{c}
{\bf X}_x({\bf x}) \\
{\bf X}_y({\bf x}) \\
{\bf X}_z({\bf x}) \\
\end{array}
\right)
\]
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