Classification — elliptic, parabolic, hyperbolic (discriminant test): Foundations

Before you can trust that one number, you need to be fluent in every symbol it is built from. This page assumes nothing. We build each piece, anchor it to a picture, and say exactly why the topic needs it.

0. What is a function of two variables?

Everything here lives on a flat sheet. Pick a point on that sheet using two coordinates: a horizontal position $x$ and a vertical position $y$ . At every such point sits a single number $u$ .

Figure s01 — the landscape. A 3-D surface sits above the flat $x y$ -floor: each floor-point $(x, y)$ is lifted to the height $u (x, y)$ . This is the picture to hold in your head for the rest of the page — every later symbol is a statement about the shape of this surface.

Figure — Classification — elliptic, parabolic, hyperbolic (discriminant test)

Why the topic needs this: the whole classification is about how the shape of $u$ is allowed to bend. So we first need the object whose shape we discuss.

1. The derivative — steepness in one direction

If $u$ were a function of just one variable, the derivative $\frac{d u}{d x}$ answers: "how fast does the height change as I step right?" It is the slope of the curve.

We use the derivative here because a PDE is a sentence about steepness and curvature — and you cannot read that sentence without knowing what "rate of change" means. See Method of Characteristics for where these slopes become curves.

2. Partial derivatives — steepness when there are two directions

Now $u$ depends on both $x$ and $y$ . Standing on the landscape, you can step east (increase $x$ ) or north (increase $y$ ), and the ground rises differently each way. A partial derivative freezes one direction and measures steepness in the other.

Figure s02 — two directions on the landscape. Blue contour lines show level curves of $u$ (equal height). From the yellow point, a red arrow points east — its slope is $u_{x}$ ; a green arrow points north — its slope is $u_{y}$ . The picture makes clear these are two different steepnesses at the same point.

Why the topic needs this: the parent PDE $A u_{xx} + B u_{x y} + \dots$ is made of these subscripts. If you cannot read $u_{x}$ you cannot read the equation.

3. Second partial derivatives — curvature

Take a slope, then ask "how fast is the slope itself changing?" That is a second derivative. On a landscape it measures curvature — whether the ground bends up like a bowl or down like a dome, and how sharply.

Figure s03 — the three flavours of curvature. Three side-by-side slices: a blue bowl ( $u_{xx} > 0$ , curving up), a red dome ( $u_{y y} < 0$ , curving down), and a yellow/green pair showing the twist ( $u_{x y}$ ) — the east-slope line shifts upward as you move north, so the two directions are coupled.

Definition The principal part (and why it is called that)

The principal part of a second-order PDE is just its highest-order piece: $A u_{xx} + B u_{x y} + C u_{y y}$ . "Principal" = "leading/most important" — it is the part that decides the type.

The deeper picture — it is a quadratic form. If you replace each derivative by a symbol for direction — $u_{xx} \to α^{2}$ , $u_{x y} \to α β$ , $u_{y y} \to β^{2}$ (where $(α, β)$ is a direction in the plane) — the principal part becomes $A α^{2} + B α β + C β^{2}$ . This is the principal symbol: a purely geometric object attached to the equation, shaped exactly like a conic $A α^{2} + B α β + C β^{2} = const$ . Whether that conic is an ellipse, parabola, or hyperbola is decided by the very same $B^{2} - 4 A C$ — which is why the three PDE types borrow those names. Everything of lower order ( $u_{x}, u_{y}, u$ , constants) sits outside this form and is ignored by the classification.

4. Coefficients $A, B, C, D, E, F, G$ — the "dials" of the equation

The parent equation is $A u_{xx} + B u_{x y} + C u_{y y} + D u_{x} + E u_{y} + F u + G = 0.$

Each capital letter is a number (or function of $x, y$ ) multiplying one derivative term — a dial that says "how much of this kind of bending counts."

Why the topic needs this: the discriminant is $B^{2} - 4 A C$ . You must be able to look at any PDE and correctly pull out these three numbers — including realising that a missing second-order term means its dial is zero (which is exactly what drives the parabolic case, $Δ = 0$ ).

5. The quadratic formula — where $B^{2} - 4 A C$ is born

A quadratic is an equation of the shape $A m^{2} + B m + C = 0$ , where $m$ is unknown. When $A \neq = 0$ , its solutions come from the quadratic formula: $m = \frac{- B \pm B ^{2} - 4 A C}{2 A} .$

The part under the square root, $B^{2} - 4 A C$ , is called the discriminant.

Figure s04 — sign of the discriminant vs number of real roots. Three parabolas $A m^{2} + B m + C$ : the red one crosses the horizontal axis twice ( $Δ > 0$ , hyperbolic), the yellow one just touches it ( $Δ = 0$ , parabolic), the blue one floats above and never touches ( $Δ < 0$ , elliptic). Where a parabola meets the axis is a real root $m = d y / d x$ — a real characteristic slope.

Why the topic needs this: in the parent derivation the characteristic slope $m = \frac{d y}{d x}$ satisfies exactly $A m^{2} + B m + C = 0$ . So "how many real characteristic directions exist" is "how many real roots does this quadratic have" is "what is the sign of $B^{2} - 4 A C$ ." That chain is the whole classification.

6. Characteristics — the special curves (a first taste)

A characteristic is a curve in the $x y$ -plane along which the PDE loses its grip on the second derivative — the direction along which information (a kink, a wave front) is allowed to travel. The slopes of these curves are the roots $m$ from Section 5.

Full machinery lives in Method of Characteristics; here you only need to know these curves exist and that their count equals the number of real roots.

7. Linear vs quasilinear — what kind of equation we are allowed to classify

Intuition WHY this distinction matters for classification — with an example

The discriminant test reads $A, B, C$ and reports a type. But who supplies those numbers?

For a linear equation like Laplace ( $A = 1, B = 0, C = 1$ everywhere), the numbers are fixed by $x, y$ alone, so the type is settled before you ever solve. You classify, then solve.
For a quasilinear equation the top dials contain $u$ — for instance $u u_{xx} + u_{y y} = 0$ has $A = u$ , so $Δ = - 4 u$ . Now the type depends on the value of the unknown itself: wherever the solution is positive it is elliptic, wherever negative it is hyperbolic. You cannot know the type until you (partly) know the answer.

That is the whole reason the distinction earns its own name: for linear PDEs classification is a property of the equation; for quasilinear ones it can be a property of the solution. This mirrors the position-dependence of §4 but pushes it one level deeper. More in Linear vs Quasilinear PDEs.

Prerequisite map

The diagram below is written in Mermaid (a plain-text way to draw flowcharts). Read each box as a concept, and each arrow as "feeds into." In Obsidian it renders as a real diagram; if you only see text, read it top-to-bottom: the foundations on the left all funnel into the single number $B^{2} - 4 A C$ , which then decides the type.

Legend: F = the landscape; P, S = its slopes and curvature; C = the three top dials; Q = the quadratic step (valid only when $A \neq = 0$ ); D = the discriminant; CH = the characteristic slopes; T = the final verdict. Return to the parent: Classification — discriminant test. The type you land on dictates the data you need — see Well-posedness and Boundary Conditions.

Equipment checklist

Cover the right side and self-test:

What does the acronym PDE stand for?

Partial Differential Equation — an equation for an unknown function of several variables and its partial derivatives.

What does

u (x, y)

represent geometrically?

A height above the

x y

-plane — a landscape whose shape the PDE governs.

What does the symbol

\partial

mean, and why not plain

d

"Partial" derivative — differentiate with respect to one variable while holding the others fixed; the special

\partial

reminds you other variables are frozen.

What does the subscript in

u_{x}

tell you?

Which direction you stepped (

x

= east) while holding the other variable fixed.

What is the difference between

u_{x}

and

u_{xx}

u_{x}

is slope (first order, one subscript);

u_{xx}

is curvature (second order, two subscripts).

Why is there only one cross term

u_{x y}

and not also

u_{y x}

Because mixed partials commute (

u_{x y} = u_{y x}

) when the second partials are continuous — Clairaut/Schwarz theorem.

What is the principal part of a second-order PDE, and what geometric object does it form?

Its highest-order piece

A u_{xx} + B u_{x y} + C u_{y y}

; replacing derivatives by directions gives the quadratic form / principal symbol

A α^{2} + B α β + C β^{2}

— a conic.

Which coefficients enter the discriminant, and which are ignored?

Only

A, B, C

(dials on

u_{xx}, u_{x y}, u_{y y}

);

D, E, F, G

are ignored.

Can

A, B, C

be functions of position, and what follows?

Yes — then

Δ

varies with

(x, y)

and the type can change region to region (e.g. Tricomi).

What must be true of the principal part for the test to apply?

It must be nondegenerate — at least one of

A, B, C

nonzero (a genuine second-order PDE).

Write the discriminant.

Δ = B^{2} - 4 A C

Why does the sign of

B^{2} - 4 A C

matter?

It decides how many real roots the quadratic

A m^{2} + B m + C = 0

has: positive → 2, zero → 1, negative → 0.

What condition does the quadratic formula for

m

require, and what do you do if

A = 0

A \neq = 0

(it divides by

2 A

); if

A = 0

relabel or read the type from

Δ = B^{2} \geq 0

. If also

A = C = 0, B \neq = 0

the type is hyperbolic with characteristics along the coordinate lines.

What real-world object do the quadratic's roots become?

The slopes

d y / d x

of the characteristic curves.

Number of real characteristics for hyperbolic, parabolic, elliptic?

2, 1, 0.

Linear vs quasilinear in one line, and why does it matter?

Linear: all coefficients depend only on

x, y

(type fixed before solving). Quasilinear: top-derivative coefficients may depend on

u

and lower derivatives (type can depend on the solution).