9.5 The Saddle Discriminant

The Hessian polymetric of a function \( z \depends (x, y) \) is the second differential, halved.

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

The Hessian describes how a surface peels away from its tangent plane by taking the bend and warp into account.

Formula.

The Hessian

\[ \begin{align} \hessian z &= a \mult \hat{\standard}_{x}^2 + b \mult \hat{\standard}_{x} \mult \hat{\standard}_{y} + c \mult \hat{\standard}_{y}^2 \end{align} \]

records the \( x \)-bend, the warp, and the \( y \)-bend.

\[ \begin{align} a &= \operatorname{\overset{\mathit{x}}{\bend}} z & b &= \warp z & c &= \operatorname{\overset{\mathit{y}}{\bend}} z \end{align} \]

Why?

Let's first find a formula for the second differential.

\[ \begin{align} \hat{\diff} \hat{\diff} z &= \hat{\diff} \Bigl( \overset{x}{\diff} z \mult \hat{\standard}_{x} + \overset{y}{\diff} z \mult \hat{\standard}_{y} \Bigr) \\ &= \hat{\diff} \Bigl( \overset{x}{\diff} z \mult \hat{\standard}_{x} \Bigr) + \hat{\diff} \Bigl( \overset{y}{\diff} z \mult \hat{\standard}_{y} \Bigr) \\ &= \hat{\diff} \overset{x}{\diff} z \mult \hat{\standard}_{x} + \hat{\diff} \overset{y}{\diff} z \mult \hat{\standard}_{y} \\ &= \Bigl( \overset{x}{\diff} \overset{x}{\diff} z \mult \hat{\standard}_{x} + \overset{y}{\diff} \overset{x}{\diff} z \mult \hat{\standard}_{y} \Bigr) \mult \hat{\standard}_{x} + \Bigl( \overset{x}{\diff} \overset{y}{\diff} z \mult \hat{\standard}_{x} + \overset{y}{\diff} \overset{y}{\diff} z \mult \hat{\standard}_{y} \Bigr) \mult \hat{\standard}_{y} \\ &= \overset{x}{\diff} \overset{x}{\diff} z \mult \hat{\standard}_{x}^2 + 2 \mult \overset{y}{\diff} \overset{x}{\diff} z \mult \hat{\standard}_{y} \mult \hat{\standard}_{x} + \overset{y}{\diff} \overset{y}{\diff} z \mult \hat{\standard}_{y}^2 \end{align} \]

We get our formula for the Hessian by dividing by two.

\[ \begin{align} \hessian z &= \frac{\overset{x}{\diff} \overset{x}{\diff} z}{2} \mult \hat{\standard}_{x}^2 + \overset{y}{\diff} \overset{x}{\diff} z \mult \hat{\standard}_{x} + \frac{\overset{y}{\diff} \overset{y}{\diff} z}{2} \mult \hat{\standard}_{y}^2 \end{align} \]

The saddle discriminant of a function \( z \depends (x, y) \) is the number

\[ \begin{align} \discriminant z &: \Number \\ \discriminant z &= (\warp z)^2 - 4 \mult \operatorname{\overset{\mathit{x}}{\bend}} z \mult \operatorname{\overset{\mathit{y}}{\bend}} z. \end{align} \]

The discriminant helps us describe the shape of the graph.

Theorem.

Suppose we localize the saddle discriminant at a point \( p : \Point_{(x, y)} \).

If the discriminant is positive, \( \discriminant z \gt 0 \), then the graph of \( z \) is saddle-shaped near \( p \).
If the discriminant is negative, \( \discriminant z \lt 0 \), then the graph of \( z \) is bowl-shaped near \( p \).

We'll say that the discriminant is inconclusive at \( p \) if it is equal to zero.

Why?

We learn how the graph peels away from its tangent plane by looking at the Hessian

\[ \begin{align} \hessian z &= a \mult \hat{\standard}_{x}^2 + b \mult \hat{\standard}_{x} \mult \hat{\standard}_{y} + c \mult \hat{\standard}_{y}^2 \end{align} \]

where

\[ \begin{align} a &= \operatorname{\overset{\mathit{x}}{\bend}} z, & b &= \warp z, & c &= \operatorname{\overset{\mathit{y}}{\bend}} z. \end{align} \]

By using a bit of algebraic dark magic—completing the square—we can rewrite the Hessian polymetric as

\[ \begin{align} \hessian z &= a \mult \biggl( \biggl(\hat{\standard}_{x} + \frac{b}{2 a} \mult \hat{\standard}_{y}\biggr)^2 - \frac{b^2 - 4 a c}{(2 a)^2} \mult {\hat{\standard}_{y}}^2 \biggr). \end{align} \]

The discriminant

\[ \begin{align} \discriminant z &= b^2 - 4 a c \end{align} \]

determines whether our polymetric factors, and thereby determines the shape of the graph. You might recognize the discriminant: it also shows up in the quadratic formula!

Let's take a look at saddle discriminants for some familiar surfaces.

shape	function	discriminant
bowl	\( z = x^2 + y^2 \)	\( \discriminant z = -4 \)
parabolic saddle	\( z = x^2 - y^2 \)	\( \discriminant z = 4 \)
warp saddle	\( z = x y \)	\( \discriminant z = 1 \)
thick parabola	\( z = x^2 \)	\( \discriminant z = 0 \)
plane	\( z - z_p = s_x \mult (x - x_p) + s_y \mult (y - y_p) \)	\( \discriminant z = 0 \)

Each of these functions has a constant saddle discriminant. This will actually be uncommon! For most functions, the saddle discriminant will vary from point to point, depending on whether the graph is saddle-shaped or bowl-shaped. If the discriminant is zero at a point, the graph may be saddle-shaped, bowl-shaped, or some other shape. In this case, we might look at a contour map to better understand the graph.