8.6 Warp
The \( y x \)-warp and \( x y \)-warp of a function \( z \depends (x, y) \) are the mixed second partials:
\[
\begin{align}
\operatorname{\overset{\mathit{yx}}{\warp}} z &: \Number
&
&
&
\operatorname{\overset{\mathit{xy}}{\warp}} z &: \Number
\\
\operatorname{\overset{\mathit{yx}}{\warp}} z &= \overset{y}{\diff} \overset{x}{\diff} z
&
&\andSpaced
&
\operatorname{\overset{\mathit{xy}}{\warp}} z &= \overset{x}{\diff} \overset{y}{\diff} z.
\end{align}
\]
The first thing to know about warp is that we are free to take the partials in either order.
Theorem.
For any function \( z \depends (x, y) \), the \( y x \)-warp and the \( x y \)-warp are equal.
\[
\begin{align}
\overset{y}{\diff} \overset{x}{\diff} z &= \overset{x}{\diff} \overset{y}{\diff} z
\end{align}
\]
We'll usually just write \( \warp z \) for the warp as long as there are only two input variables.
Example.
Let's compute the warp of the function
\[
\begin{align}
z &\depends (x, y)
\\
z &= x y^3 - 5 x^2
\end{align}
\]
in both possible orders.
\[
\begin{align}
\overset{x}{\diff} z &= \overset{x}{\diff} \bigl(x y^3 - 5 x^2\bigr) = y^3 - 10 x
\\
\overset{y}{\diff} \overset{x}{\diff} z &= \overset{y}{\diff} \bigl(y^3 - 10 x\bigr) = 3 y^2
\end{align}
\]
Here we took the \( y \)-partial of the \( x \)-partial of \( z \). Now let's take the \( x \)-partial of the \( y \)-partial of \( z \).
\[
\begin{align}
\overset{y}{\diff} z &= \overset{y}{\diff} \bigl(x y^3 - 5 x^2\bigr) = 3 x y^2
\\
\overset{x}{\diff} \overset{y}{\diff} z &= \overset{x}{\diff} \bigl(3 x y^2\bigr) = 3 y^2
\end{align}
\]
We found the same result.
\[
\begin{align}
\warp z &= 3 y^2
\end{align}
\]
Let's take a look at the warp for an important example.
Example.
Recall the warp saddle,
\[
\begin{align}
z &\depends (x, y)
\\
z &= x y.
\end{align}
\]
Let's compute the warp! We'll start by taking an \( x \)-partial.
\[
\begin{align}
\overset{x}{\diff} z &= \overset{x}{\diff} (x y) = y \mult \overset{x}{\diff} x = y
\end{align}
\]
And now we'll take the \( y \)-partial of the \( x \)-partial.
\[
\begin{align}
\overset{y}{\diff} \overset{x}{\diff} z &= \overset{y}{\diff} y = 1
\end{align}
\]
We've found our result.
\[
\begin{align}
\warp z &= 1
\end{align}
\]
The warp saddle has a constant warp of one! Just as the parabola, \( z = x^2 \), is our ideal example of bend, the warp saddle is our ideal example of warp.
You may have noticed that we did not provide any explanation for our theorem about warp. Unfortunately, a full explanation requires advanced topics: integrals and limits. But by looking at the warp saddle, we see pretty good evidence for the theorem. A surface's warp tells us how quickly the vertical slices twist.
The \( (z \depends x) \)-slices and the \( (z \depends y) \)-slices of the warp saddle twist at the same rate in the same counterclockwise direction.