9.2 Parts

The metric differential, \( \hat{\diff} \), is closely related to the partials, \( \overset{x}{\diff} \) and \( \overset{y}{\diff} \). In fact, we may think of partials as "part differentials."

Formula.

The metric differential of a function \( z \depends (x, y) \) is the metric

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

whose parts

\[ \begin{align} a &: \Number & & & b &: \Number \\ a &= \overset{x}{\diff} z & &\andSpaced & b &= \overset{y}{\diff} z \end{align} \]

are the partials of \( z \).

Let's see what this looks like with an example.

Example.

The function

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

has differential

\[ \begin{align} \hat{\diff} z &= 3 \mult (x - y)^2 \mult \hat{\standard}_{x} - 3 \mult (x - y)^2 \mult \hat{\standard}_{y}. \end{align} \]

The partials of \( z \) are the parts of \( \hat{\diff} z \).

\[ \begin{align} \overset{x}{\diff} z &= 3 \mult (x - y)^2 \\ \overset{y}{\diff} z &= - 3 \mult (x - y)^2 \end{align} \]

The metric differential can be read as a concise description of the tangent plane: at any point, the differential records the graph's \( x \)-slope and \( y \)-slope. To emphasize a geometric point of view, we'll call the differential the tangent metric.

\[ \begin{align} \tangent z &: \Metric_{} \\ \tangent z &= \hat{\diff} z \end{align} \]

Let's rewrite our formula for the metric differential using geometric language.

Important.

The parts of the tangent metric are the \( x \)-slope and the \( y \)-slope. That is,

\[ \begin{align} \tangent z &= \operatorname{\overset{\mathit{x}}{\slope}} z \mult \hat{\standard}_{x} + \operatorname{\overset{\mathit{y}}{\slope}} z \mult \hat{\standard}_{y} \end{align} \]

where \( z \depends (x, y) \) is any function.

We can localize the tangent metric at any point \( p : \Point_{(x, y)} \) of interest.

Example.

Consider the parabolic saddle.

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

Let's compute the metric differential.

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

We've found the tangent metric.

\[ \begin{align} \tangent z &= 2 x \mult \hat{\standard}_{x} - 2 y \mult \hat{\standard}_{y} \end{align} \]

Let's see what the tangent metric looks like at the point \( (3, 1) : \Point_{(x, y)} \).

\[ \begin{align} \tangent z &= 6 \mult \hat{\standard}_{x} - 2 \mult \hat{\standard}_{y} \end{align} \]

At \( (3, 1) \), the tangent plane has a positive \( x \)-slope and a negative \( y \)-slope. Can you find a point where both the \( x \)-slope and the \( y \)-slope are positive? Or a point where the tangent plane is horizontal?

For more the tangent metric, see Appendix E. Applying a Metric.