9.3 Polymetrics

In order to go further with the differential we'll need to understand how to work with polymetrics. The standard metrics are polymetrics,

\[ \begin{align} \hat{\standard}_{x} &: \PolyMetric_{(x, y)}^{} \\ \hat{\standard}_{y} &: \PolyMetric_{(x, y)}^{} \end{align} \]

numbers are polymetrics,

\[ \begin{align} 7 &: \PolyMetric_{(x, y)}^{} \\ -4 &: \PolyMetric_{(x, y)}^{} \end{align} \]

and we build all other polymetrics from these by adding and multiplying.

\[ \begin{align} 5 + 3 \mult \hat{\standard}_{x} - \hat{\standard}_{y} + \hat{\standard}_{x} \mult \hat{\standard}_{y} &: \PolyMetric_{(x, y)}^{} \end{align} \]

Law.

The multiplication of polymetrics is commutative. We can reorder multiplication

\[ \begin{align} k \mult j &= j \mult k \end{align} \]

where \( j : \PolyMetric_{(x, y)}^{} \) and \( k : \PolyMetric_{(x, y)}^{} \) are any two polymetrics.

Polymetrics generalize metrics: every metric is a polymetric, but there are many polymetrics which are not metrics.

Example.

Let's multiply the metrics

\[ \begin{align} m &: \Metric_{(x, y)} & & & n &: \Metric_{(x, y)} \\ m &= 3 \mult \hat{\standard}_{x} + 4 \mult \hat{\standard}_{y} & &\andSpaced & n &= 2 \mult \hat{\standard}_{x} - \hat{\standard}_{y}. \end{align} \]

We calculate.

\[ \begin{align} m \mult n &= \bigl(3 \mult \hat{\standard}_{x} + 4 \mult \hat{\standard}_{y}\bigr) \mult \bigl(2 \mult \hat{\standard}_{x} - \hat{\standard}_{y}\bigr) \\ &= 6 \mult \hat{\standard}_{x}^2 - 3 \mult \hat{\standard}_{x} \mult \hat{\standard}_{y} + 8 \mult \hat{\standard}_{y} \mult \hat{\standard}_{x} - 4 \mult \hat{\standard}_{y}^2 \\ &= 6 \mult \hat{\standard}_{x}^2 + 5 \mult \hat{\standard}_{x} \mult \hat{\standard}_{y} - 4 \mult \hat{\standard}_{y}^2 \end{align} \]

Multiplying two metrics does not result in a metric. Rather, the product of metrics is a polymetric!

In the previous example, we used the fact

\[ \begin{align} \hat{\standard}_{y} \mult \hat{\standard}_{x} &= \hat{\standard}_{x} \mult \hat{\standard}_{y}. \end{align} \]

This holds because our multiplication is commutative.

Degree of a Polymetric

Just like working with polynomials, we can always simplify a polymetric until it becomes a sum of terms. The degree of a term is the number of standard metrics it contains. So,

\[ \begin{align} 3 \mult \hat{\standard}_{x} &: \PolyMetric_{(x, y)}^{1} \\ 4 \mult \hat{\standard}_{y} &: \PolyMetric_{(x, y)}^{1} \end{align} \]

are degree one terms, and

\[ \begin{align} 6 \mult \hat{\standard}_{x}^2 &: \PolyMetric_{(x, y)}^{2} \\ 5 \mult \hat{\standard}_{x} \mult \hat{\standard}_{y} &: \PolyMetric_{(x, y)}^{2} \\ -4 \mult \hat{\standard}_{y}^2 &: \PolyMetric_{(x, y)}^{2} \end{align} \]

are degree two terms. Unlike polynomials, we expect a polymetric to be homogeneous to have a degree: each term should have the same degree.

degree	polymetric standard form
0	\( a \)
1	\( a \mult \hat{\standard}_{x} + b \mult \hat{\standard}_{y} \)
2	\( a \mult \hat{\standard}_{x}^2 + b \mult \hat{\standard}_{x} \mult \hat{\standard}_{y} + c \mult \hat{\standard}_{y}^2 \)

In degrees zero and one, polymetrics are familiar objects.

Important.

A degree zero polymetric is a number and a degree one polymetric is a metric.

We'll be most interested in polymetrics with degrees zero, one, or two, but you can find polymetrics in higher degrees, too.