Clairaut's theorem — mixed partials are equal (under conditions): Foundations

Before you can even read the sentence " $f_{x y} = f_{y x}$ when the mixed partials are continuous," you need to already own about a dozen little symbols and pictures. This page builds every single one from nothing, in an order where each rests on the one before. Nothing is assumed.

0. The starting object: $f (x, y)$

The picture to burn into your mind: not a curve, but a landscape — a sheet of hills and valleys hovering above the flat $x y$ -floor.

Figure — Clairaut's theorem — mixed partials are equal (under conditions)

Look at the figure. The flat grid at the bottom is the floor of input pairs $(x, y)$ . The coloured sheet floating above is the surface $z = f (x, y)$ . Pick any point on the floor, go straight up, and the sheet's height there is the output. This one picture is what every symbol below secretly refers to.

Related vault reading: Continuity of Multivariable Functions and Partial Derivatives describe this surface formally.

1. The symbol $\partial$ — "partial", meaning hold the others still

Why do we even need a new symbol? Because with two inputs, "the slope" is ambiguous — slope in which direction? The $\partial$ symbol answers "I picked a direction and I'm holding the rest constant," so the question becomes unambiguous.

Look at the figure. To measure $\partial f / \partial x$ we freeze $y$ at some value $b$ . That is a vertical slicing plane cutting the surface. The plane meets the surface in a single curve (magenta). Along that curve $x$ is the only thing changing — so now we have an ordinary one-variable curve, and "slope" means the plain old thing again.

2. $f_{x}$ and $\frac{\partial f}{\partial x}$ — the first partial (a slope)

The picture: it is the slope of the magenta slice-curve from figure 2 — "rise over run" of that curve at your point.

$f_{y}$ is the identical idea with the roles swapped: freeze $x$ , walk North.

Full detail lives at Partial Derivatives.

3. The step size $h$ (and $k$ ) and the symbol $lim$

The picture: a shrinking horizontal arrow of length $h$ on the floor, and the chord it produces on the slice-curve tipping over until it becomes the tangent line.

Why needed: the whole proof of Clairaut is built from a second difference using two step sizes $h$ and $k$ , then squeezing both to zero. Without a firm grip on "nudge, then shrink the nudge," the proof is unreadable.

4. The mixed partial $f_{x y}$ — a twist

Now the star of the topic. We already have $f_{x}$ : a slope-in-the-East-direction. But $f_{x}$ is itself a new function of $(x, y)$ — the East-slope changes as you wander around the surface. So we can differentiate it again.

Look at the figure. On the near edge (small $y$ ) the East-slope arrow points gently up. On the far edge (larger $y$ ) the East-slope arrow is steeper. The change between those two arrows, per unit of North-travel, is $f_{x y}$ . If the two arrows were identical, $f_{x y} = 0$ — no twist.

$f_{y x}$ is the mirror: North-slope first, then see how it changes going East. Clairaut's claim is $f_{x y} = f_{y x}$ .

More on stacking derivatives: Higher Order Partial Derivatives.

5. "Continuous" — the price of admission

Why the topic obsesses over this: the proof produces the twist at some unknown nearby point and then lets that point slide toward $(a, b)$ . Only if the twist is continuous can "twist at a nearby point" become "twist at $(a, b)$ ." Continuity is literally the hinge the whole theorem swings on. See Continuity of Multivariable Functions.

6. The Mean Value Theorem — the engine of the proof

Look at the figure. The orange secant joins the two endpoints; the violet tangent touches the curve at some interior point $c$ and is exactly parallel to the secant. MVT guarantees such a $c$ exists.

Why the topic needs it: Clairaut's proof converts each difference of the function into a derivative times a step, four times over (twice in $x$ , twice in $y$ ). Every one of those conversions is one use of MVT. Deep dive: Mean Value Theorem.

7. The second difference $Δ$ — the symmetric gadget

The picture: a small rectangle with corners $(a, b)$ , $(a + h, b)$ , $(a, b + k)$ , $(a + h, b + k)$ ; $Δ$ combines the four heights with signs $+, -, -, +$ . The crucial feature: swapping $x$ with $y$ leaves $Δ$ unchanged — it is symmetric. That symmetry is why the same number can be read as $f_{x y}$ one way and $f_{y x}$ the other. This is the seed of the entire theorem; it appears fully grown in the parent note.

Prerequisite map

Reading the map: the surface $f$ feeds partial derivatives; partials plus limits build the twist $f_{x y}$ ; MVT plus $f$ build the symmetric $Δ$ ; and the theorem stands on the twist, the symmetric gadget, and continuity all at once. Downstream, these ideas power Hessian Matrix and Exact Differential Equations.

Equipment checklist

Test yourself — cover the right side and answer aloud before revealing.

What does

\partial

warn you about?

More than one input exists; freeze all but the chosen one while differentiating.

In one plain sentence, what is

f_{x}

The steepness of the surface when you walk in the

+ x

direction with

y

held fixed.

Why can't you just set

h = 0

in the slope formula?

You'd get

0/0

;

lim_{h \to 0}

instead watches where the ratio settles as the step shrinks.

What does the mixed partial

f_{x y}

physically measure?

How the East-slope

f_{x}

changes as you walk North — the twist of the surface.

f_{x y}

, which variable is differentiated first?

x

(subscripts read left to right).

What does MVT convert, and into what?

A difference

g (a + h) - g (a)

into a derivative times a step,

h g^{'} (c)

Why does the theorem demand continuity of the mixed partials?

The proof yields the twist at an unknown nearby point; continuity lets its limit equal the twist exactly at

(a, b)

What is special about the second difference

Δ

It is symmetric — swapping

x

and

y

leaves it unchanged — so one number equals both

f_{x y}

and

f_{y x}