12.1 The Ruler Differential
Let's interpret the differential as a ruler!
Definition.
For any function \( z \depends (x, y) \), the ruler differential of \( z \) is a ruler
\[
\begin{align}
\bar{\diff} z &: \Ruler_{(x, y)}.
\end{align}
\]
The ruler differential, \( \bar{\diff} \), satisfies all abstract differential laws. We interpret the ruler differential of an input variable as a standard ruler:
\[
\begin{align}
\bar{\diff} x &= \bar{\standard}_{x}
&
&\andSpaced
&
\bar{\diff} y &= \bar{\standard}_{y}.
\end{align}
\]
We calculate the ruler differential in very much the same way that we calculate the metric differential.
Example.
Let's compute the differential of the function
\[
\begin{align}
z &\depends (x, y)
\\
z &= \exp (x y)
\end{align}
\]
as a ruler. We find
\[
\begin{align}
\bar{\diff} z &= \bar{\diff} (\exp (x y))
\\
&= \exp (x y) \mult \bar{\diff} (x y)
\\
&= \exp (x y) \mult \bigl(y \mult \bar{\diff} x + x \mult \bar{\diff} y\bigr)
\\
&= \exp (x y) \mult \bigl(y \mult \bar{\standard}_{x} + x \mult \bar{\standard}_{y}\bigr)
\\
&= y \mult \exp (x y) \mult \bar{\standard}_{x} + x \mult \exp (x y) \mult \bar{\standard}_{y}.
\end{align}
\]
This calculation should be familiar. A similar calculation will show that the metric differential of \( z \) is
\[
\begin{align}
\hat{\diff} z &= y \mult \exp (x y) \mult \hat{\standard}_{x} + x \mult \exp (x y) \mult \hat{\standard}_{y}.
\end{align}
\]
We can localize the ruler differential at any point \( p : \Point_{(x, y)} \) in the coordinate plane.
Example.
The bowl
\[
\begin{align}
z &\depends (x, y)
\\
z &= x^2 + y^2
\end{align}
\]
has differential
\[
\begin{align}
\bar{\diff} z &= 2 x \mult \bar{\standard}_{x} + 2 y \mult \bar{\standard}_{y}.
\end{align}
\]
Localizing at the point \( p = (-1, 1) \), we find the ruler
\[
\begin{align}
\bar{\diff} z &= -2 \mult \bar{\standard}_{x} + 2 \mult \bar{\standard}_{y}.
\end{align}
\]
Let's plot the ruler differential from the previous example. We see some similarity when comparing this ruler to a contour map for the bowl. Near \( p \), the markings of the differential resemble the contours of the bowl.
The differential is a best fit ruler!