Previous. Up. Next.

12.6 Higher Differentials

Let's learn to take differentials of polyrulers!
Law.
The differential law for addition states
\[ \begin{align} \bar{\diff} (p + q) &= \bar{\diff} p + \bar{\diff} q \end{align} \]
where \( p : \PolyRuler_{(x, y)}^{} \) and \( q : \PolyRuler_{(x, y)}^{} \) are polyrulers.
The differential law for addition is easy. Unfortunately, things get a bit trickier when we look at multiplication.
Laws.
The differential laws for multiplication state
\[ \begin{align} \bar{\diff} (n \mult p) &= \bar{\diff} n \mult p + n \mult \bar{\diff} p \\ \bar{\diff} (r \mult p) &= \bar{\diff} r \mult p - r \mult \bar{\diff} p \end{align} \]
where
\[ \begin{align} n &: \Number, & r &: \Ruler_{}, & p &: \PolyRuler_{}^{} \end{align} \]
are a number, a ruler, and a polyruler.
These multiplication laws are quite easy to mess up! It's difficult to remember when you need a negative sign and in what order everything goes. Luckily, we have an alternative pair of laws that are much easier to use in practice.
Laws.
The standard rulers pull out of the differential on the right side
\[ \begin{align} \bar{\diff} \bigl(p \mult \bar{\standard}_{x}\bigr) &= \bar{\diff} p \mult \bar{\standard}_{x} \\ \bar{\diff} \bigl(p \mult \bar{\standard}_{y}\bigr) &= \bar{\diff} p \mult \bar{\standard}_{y} \end{align} \]
where \( p : \PolyRuler_{(x, y)}^{} \) is any polyruler.
It's good to get some practice taking the differential.
Example.
Let's take the differential of the ruler
\[ \begin{align} r &: \Ruler_{(x, y)}{} \\ r &= x y^2 \mult \bar{\standard}_{x} + x^3 \mult \bar{\standard}_{y}. \end{align} \]
We compute.
\[ \begin{align} \bar{\diff} r &= \bar{\diff} \bigl(x y^2 \mult \bar{\standard}_{x} + x^3 \mult \bar{\standard}_{y}\bigr) \\ &= \bar{\diff} \bigl(x y^2 \mult \bar{\standard}_{x}\bigr) + \bar{\diff} \bigl(x^3 \mult \bar{\standard}_{y}\bigr) \\ &= \bar{\diff} \bigl(x y^2\bigr) \mult \bar{\standard}_{x} + \bar{\diff} \bigl(x^3\bigr) \mult \bar{\standard}_{y} \\ &= \bigl(y^2 \mult \bar{\diff} x + 2 x y \mult \bar{\diff} y\bigr) \mult \bar{\standard}_{x} + 3 x^2 \mult \bar{\diff} x \mult \bar{\standard}_{y} \\ &= \bigl(y^2 \mult \bar{\standard}_{x} + 2 x y \mult \bar{\standard}_{y}\bigr) \mult \bar{\standard}_{x} + 3 x^2 \mult \bar{\standard}_{x} \mult \bar{\standard}_{y} \\ &= y^2 \mult \bar{\standard}_{x} \mult \bar{\standard}_{x} + 2 x y \mult \bar{\standard}_{y} \mult \bar{\standard}_{x} + 3 x^2 \mult \bar{\standard}_{x} \mult \bar{\standard}_{y} \\ &= - 2 x y \mult \bar{\standard}_{x} \mult \bar{\standard}_{y} + 3 x^2 \mult \bar{\standard}_{x} \mult \bar{\standard}_{y} \\ &= \bigl(- 2 x y + 3 x^2\bigr) \mult \bar{\standard}_{x} \mult \bar{\standard}_{y} \end{align} \]
We've found a polyruler with degree two.
Let's now see an example where we take the differential of a differential.
Example.
The function
\[ \begin{align} z &\depends (x, y) \\ z &= x^2 y^3 \end{align} \]
has differential
\[ \begin{align} \bar{\diff} z &: \Ruler_{(x, y)} \\ \bar{\diff} z &= 2 x y^3 \mult \bar{\standard}_{x} + 3 x^2 y^2 \mult \bar{\standard}_{y}. \end{align} \]
Let's take the second differential.
\[ \begin{align} \bar{\diff} \bar{\diff} z &= \bar{\diff} \bigl(2 x y^3 \mult \bar{\standard}_{x} + 3 x^2 y^2 \mult \bar{\standard}_{y}\bigr) \\ &= \bar{\diff} \bigl(2 x y^3\bigr) \mult \bar{\standard}_{x} + \bar{\diff} \bigl(3 x^2 y^2\bigr) \mult \bar{\standard}_{y} \\ &= 2 \mult \bar{\diff} \bigl(x y^3\bigr) \mult \bar{\standard}_{x} + 3 \mult \bar{\diff} \bigl(x^2 y^2\bigr) \mult \bar{\standard}_{y} \\ &= 2 \mult \bigl(y^3 \mult \bar{\diff} x + 3 x y^2 \mult \bar{\diff} y\bigr) \mult \bar{\standard}_{x} + 3 \mult \bigl(2 x y^2 \mult \bar{\diff} x + 2 x^2 y \mult \bar{\diff} y\bigr) \mult \bar{\standard}_{y} \\ &= 2 \mult \bigl(y^3 \mult \bar{\standard}_{x} + 3 x y^2 \mult \bar{\standard}_{y}\bigr) \mult \bar{\standard}_{x} + 3 \mult \bigl(2 x y^2 \mult \bar{\standard}_{x} + 2 x^2 y \mult \bar{\standard}_{y}\bigr) \mult \bar{\standard}_{y} \\ &= 2 y^3 \mult \bar{\standard}_{x} \mult \bar{\standard}_{x} + 6 x y \mult \bar{\standard}_{y} \mult \bar{\standard}_{x} + 6 x y \mult \bar{\standard}_{x} \mult \bar{\standard}_{y} + 6 x^2 y \mult \bar{\standard}_{y} \mult \bar{\standard}_{y} \\ &= - 6 x y \mult \bar{\standard}_{x} \mult \bar{\standard}_{y} + 6 x y \mult \bar{\standard}_{x} \mult \bar{\standard}_{y} \\ &= 0 \end{align} \]
The second ruler differential of \( z \) is identically zero.
You might think it strange that we chose an example that has second differential equal to zero. But remarkably, every function \( z \depends (x, y) \) has second ruler differential equal to zero!
\[ \begin{align} \bar{\diff} \bar{\diff} z &= 0 \end{align} \]
We'll explore this fact in detail in the next section.
Previous. Up. Next.