Appendix E. Applying a Metric

Suppose

\[ \begin{align} m &: \Metric_{(x, y)} \\ m &= a \mult \hat{\standard}_{x} + b \mult \hat{\standard}_{y} \end{align} \]

is a metric, and

\[ \begin{align} r &: \Number & &\andSpaced & s &: \Number \end{align} \]

are numbers. By applying the metric \( m \) we produce a number

\[ \begin{align} m \of [r; s] &: \Number \\ m \of [r; s] &= a \mult r + b \mult s. \end{align} \]

Example.

Consider the metric

\[ \begin{align} m &: \Metric_{(x, y)} \\ m &= 2 \mult \hat{\standard}_{x} + 3 \mult \hat{\standard}_{y}. \end{align} \]

Let's apply our metric for a few different choices of values.

\[ \begin{align} m \of [2; 0] &= 2 \mult 2 + 3 \mult 0 = 4 \\ m \of [0; 2] &= 2 \mult 0 + 3 \mult 2 = 6 \\ m \of [1; 1] &= 2 \mult 1 + 3 \mult 1 = 5 \end{align} \]

In each case we just substitute the value on the left for \( \hat{\standard}_{x} \) and the value on the right for \( \hat{\standard}_{y} \).

The tangent metric calculates changes for a function's tangent plane.

Theorem.

Suppose

\[ \begin{align} \change_{x} &: \Number & &\andSpaced & \change_{y} &: \Number \end{align} \]

are numbers that represent changes in input. Then

\[ \begin{align} &\tangent z \of [\change_{x}; \change_{y}] \end{align} \]

is the change in height for the tangent plane.

Why?

When we substitute \( \change_{x} \) and \( \change_{y} \) into the tangent metric, we find the equation for the tangent plane.

\[ \begin{align} \tangent z \of [\change_{x}; \change_{y}] &= \operatorname{\overset{\mathit{x}}{\slope}} z \mult \change_{x} + \operatorname{\overset{\mathit{y}}{\slope}} z \mult \change_{y} \\ &= \operatorname{\overset{\mathit{x}}{\slope}} z \mult (x_T - x) + \operatorname{\overset{\mathit{y}}{\slope}} z \mult (y_T - y) \\ &= z_T - z \end{align} \]

Let's see how this works!

Example.

Consider the function

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

Let's find the tangent metric and localize it at the point \( p = (3, 1) \).

\[ \begin{align} \tangent z &= \hat{\diff} z \\ &= \bigl(2 x y - y^2\bigr) \mult \hat{\standard}_{x} + \bigl(x^2 - 2 x y\bigr) \mult \hat{\standard}_{y} \end{align} \]

Localizing at \( p \), we find

\[ \begin{align} \tangent z &= 5 \mult \hat{\standard}_{x} - 3 \mult \hat{\standard}_{y}. \end{align} \]

By applying the tangent metric, we can find changes in height due to changes in input.

\[ \begin{align} \tangent z \of [2; 0] &= 10 \\ \tangent z \of [0; 2] &= -6 \\ \tangent z \of [1; 1] &= 2 \end{align} \]

If we do not travel far from the center of the tangent plane,

\[ \begin{align} \change_{x} &\approx 0 & &\andSpaced & \change_{y} &\approx 0, \end{align} \]

then a change of height for the surface is approximately equal to the change of height for the tangent plane

\[ \begin{align} \change_{z} &\approx \tangent z \of [\change_{x}; \change_{y}]. \end{align} \]