8.5 Bend

A function \( z \depends (x, y) \) has two bends.

\[ \begin{align} \operatorname{\overset{\mathit{x}}{\bend}} z &: \Number & & & \operatorname{\overset{\mathit{y}}{\bend}} z &: \Number \\ \operatorname{\overset{\mathit{x}}{\bend}} z &= \frac{\overset{x}{\diff} \overset{x}{\diff} z}{2} & &\andSpaced & \operatorname{\overset{\mathit{y}}{\bend}} z &= \frac{\overset{y}{\diff} \overset{y}{\diff} z}{2} \end{align} \]

The \( x \)-bend is the second \( x \)-partial, halved. And the \( y \)-bend is the second \( y \)-partial, halved.

Example.

Consider the function

\[ \begin{align} z &\depends (x, y) \\ z &= x y^3 - 5 x^2. \end{align} \]

Let's compute the \( x \)-bend and the \( y \)-bend.

\[ \begin{align} \overset{x}{\diff} z &= y^3 - 10 x & \overset{y}{\diff} z &= 3 x y^2 \\ \overset{x}{\diff} \overset{x}{\diff} z &= -10 & \overset{y}{\diff} \overset{y}{\diff} z &= 6 x y \\ \operatorname{\overset{\mathit{x}}{\bend}} z &= -5 & \operatorname{\overset{\mathit{y}}{\bend}} z &= 3 x y \end{align} \]

This function has a constant \( x \)-bend of negative five, but the \( y \)-bend varies from point to point.

Like we saw with slopes, a graph's \( x \)-bend and \( y \)-bend can be found in its vertical slices.

Example.

Suppose a function \( z \depends (x, y) \) has vertical slices through the point \( p = (1, 2) \) as pictured.

We've drawn in the tangents at \( p \) as dotted lines. Let's see what we can say about slope and bend at \( p \).

\( \{ {y = 2} \} \)-slice.

In the \( \{ {y = 2} \} \)-slice, we see how \( z \) depends on \( x \). We can learn about the \( x \)-slope and the \( x \)-bend in this slice. At \( p \), we see a graph that has indirect variance and is concave down.

\[ \begin{align} \operatorname{\overset{\mathit{x}}{\slope}} z &\lt 0 & \operatorname{\overset{\mathit{x}}{\bend}} z &\lt 0 \end{align} \]

The slope is negative, and the bend is negative, also.

\( \{ {x = 1} \} \)-slice.

In the \( \{ {x = 1} \} \)-slice, we can see how \( z \) depends on \( y \).

\[ \begin{align} \operatorname{\overset{\mathit{y}}{\slope}} z &= 0 & \operatorname{\overset{\mathit{y}}{\bend}} z &\gt 0 \end{align} \]

Here we see zero slope and a positive bend.