9.4 Higher Differentials

Let's learn to take the differential of a polymetric. Using polymetrics, we can iterate the differential to our hearts content!

Laws.

The differential law for addition states

\[ \begin{align} \hat{\diff} (j + k) &= \hat{\diff} j + \hat{\diff} k \end{align} \]

and the differential law for multiplication states

\[ \begin{align} \hat{\diff} (j \mult k) &= k \mult \hat{\diff} j + j \mult \hat{\diff} k \end{align} \]

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

Both of the standard metrics, \( \hat{\standard}_{x} \) and \( \hat{\standard}_{y} \), have zero differential.

\[ \begin{align} \hat{\diff} \bigl(\hat{\standard}_{x}\bigr) &= 0 & &\andSpaced & \hat{\diff} \bigl(\hat{\standard}_{y}\bigr) &= 0 \end{align} \]

As such, they look a bit like constants when taking the differential.

Laws.

The standard metrics pull out of the differential

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

where \( j : \PolyMetric_{(x, y)}^{} \) is any polymetric.

Let's see how we take the differential of a metric.

Example.

Consider the metric

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

Let's calculate the differential.

\[ \begin{align} \hat{\diff} m &= \hat{\diff} \bigl(2 x y \mult \hat{\standard}_{x} + x^3 \mult \hat{\standard}_{y}\bigr) \\ &= \hat{\diff} \bigl(2 x y \mult \hat{\standard}_{x}\bigr) + \hat{\diff} \bigl(x^3 \mult \hat{\standard}_{y}\bigr) \\ &= \hat{\diff} (2 x y) \mult \hat{\standard}_{x} + \hat{\diff} \bigl(x^3\bigr) \mult \hat{\standard}_{y} \\ &= 2 \mult \hat{\diff} (x y) \mult \hat{\standard}_{x} + 3 x^2 \mult \hat{\diff} x \mult \hat{\standard}_{y} \\ &= 2 \mult \bigl( y \mult \hat{\diff} x + x \mult \hat{\diff} y \bigr) \mult \hat{\standard}_{x} + 3 x^2 \mult \hat{\standard}_{x} \mult \hat{\standard}_{y} \\ &= 2 y \mult \hat{\standard}_{x} \mult \hat{\standard}_{x} + 2 x \mult \hat{\standard}_{y} \mult \hat{\standard}_{x} + 3 x^2 \mult \hat{\standard}_{x} \mult \hat{\standard}_{y} \\ &= 2 y \mult \hat{\standard}_{x} \mult \hat{\standard}_{x} + \bigl(2 x + 3 x^2\bigr) \mult \hat{\standard}_{x} \mult \hat{\standard}_{y} \end{align} \]

The result is a polymetric with degree two.

Taking the differential of a polymetric produces a polymetric with degree one larger. So starting with a function,

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

we find

\[ \begin{align} \hat{\diff} z &: \Metric_{(x, y)} \\ \hat{\diff} \hat{\diff} z &: \PolyMetric_{(x, y)}^{2} \\ \hat{\diff} \hat{\diff} \hat{\diff} z &: \PolyMetric_{(x, y)}^{3} \end{align} \]

and so on.

Example.

Consider the function

\[ \begin{align} z &\depends (x, y) \\ z &= x y^3 + 1. \end{align} \]

Let's take the differential twice. The first differential is a metric

\[ \begin{align} \hat{\diff} z &= \hat{\diff} \bigl(x y^3 + 1\bigr) \\ &= \hat{\diff} \bigl(x y^3\bigr) \\ &= y^3 \mult \hat{\diff} x + x \mult \hat{\diff} \bigl(y^3\bigr) \\ &= y^3 \mult \hat{\standard}_{x} + x \mult 3 y^2 \mult \hat{\diff} y \\ &= y^3 \mult \hat{\standard}_{x} + 3 x y^2 \mult \hat{\standard}_{y} \end{align} \]

and the second differential is a polymetric with degree two.

\[ \begin{align} \hat{\diff} \hat{\diff} z &= \hat{\diff} \bigl(y^3 \mult \hat{\standard}_{x} + 3 x y^2 \mult \hat{\standard}_{y}\bigr) \\ &= \hat{\diff} \bigl(y^3 \mult \hat{\standard}_{x}\bigr) + \hat{\diff} \bigl(3 x y^2 \mult \hat{\standard}_{y}\bigr) \\ &= \hat{\diff} \bigl(y^3\bigr) \mult \hat{\standard}_{x} + \hat{\diff} \bigl(3 x y^2\bigr) \mult \hat{\standard}_{y} \\ &= 3 y^2 \mult \hat{\diff} y \mult \hat{\standard}_{x} + 3 \mult \hat{\diff} (x y^2) \mult \hat{\standard}_{y} \\ &= 3 y^2 \mult \hat{\standard}_{y} \mult \hat{\standard}_{x} + 3 \mult \bigl( y^2 \mult \hat{\diff} x + x \mult \hat{\diff} \bigl(y^2\bigr) \bigr) \mult \hat{\standard}_{y} \\ &= 3 y^2 \mult \hat{\standard}_{y} \mult \hat{\standard}_{x} + 3 \mult \bigl( y^2 \mult \hat{\diff} x + x \mult 2 y \mult \hat{\diff} y \bigr) \mult \hat{\standard}_{y} \\ &= 3 y^2 \mult \hat{\standard}_{y} \mult \hat{\standard}_{x} + 3 y^2 \mult \hat{\standard}_{x} \mult \hat{\standard}_{y} + 6 x y \mult \hat{\standard}_{y} \mult \hat{\standard}_{y} \\ &= 6 y^2 \mult \hat{\standard}_{x} \mult \hat{\standard}_{y} + 6 x y \mult \hat{\standard}_{y} \mult \hat{\standard}_{y} \end{align} \]