Exact equations — exactness condition, finding potential function: Foundations

Before you can trust a single line of the parent note, you need to genuinely see what each symbol means. This page builds them one at a time, from a reader who has never met a partial derivative. Nothing here is assumed; each tool is earned before it is used.

1. The starting object: a function of TWO variables $F (x, y)$

Everything in this chapter orbits around a function that eats two numbers and returns one.

Plain words → picture → why.

Plain words: a machine with two dials $x, y$ and one output number.
Picture: a landscape. Above each floor point $(x, y)$ sits a hill surface at height $F$ .
Why the topic needs it: the whole solution of an exact equation is the sentence " $F (x, y) = c$ ". Without knowing what a two-input function is, that sentence is meaningless. See Total Differential & Partial Derivatives for the machinery we build on it.

2. Level curves: the contour lines $F (x, y) = c$

A hill is 3D and hard to draw. Mapmakers flatten it by drawing the set of all floor points that sit at the same height.

Plain words → picture → why.

Plain words: "all places at the same altitude."
Picture: loops drawn on the flat floor (look at the coloured rings in the figure). Move along a ring and your height stays frozen.
Why the topic needs it: the answer to an exact ODE is one of these rings. Solving the equation = identifying which ring you were told to stay on. Deep dive: Level Curves & Contour Lines.

Recall If you walk along a level curve, how does

$F$ change? Not at all — by definition every point on the curve has the same value $c$ , so the change in height is exactly zero. That "zero change" is the seed of the equation $d F = 0$ .

3. Partial derivatives: $\frac{\partial F}{\partial x}$ and $\frac{\partial F}{\partial y}$

To describe the hill we need its slopes. But a hill has two independent slopes at each point: steepness walking east, and steepness walking north.

Plain words → picture → why.

Plain words: slope toward east ( $F_{x}$ ) and slope toward north ( $F_{y}$ ).
Picture: two perpendicular ramps carved out of the hill at one point (the red ramp is the $x$ -slope, the mint ramp the $y$ -slope).
Why the topic needs it: the parent note's central boxed equations are $F_{x} = M$ and $F_{y} = N$ . The coefficients $M$ and $N$ in your ODE are secretly these two slopes. Full treatment: Total Differential & Partial Derivatives.

4. The differentials $d x$ , $d y$ , and the total differential $d F$

Plain words → picture → why.

Plain words: climbing a little east plus climbing a little north adds up to your total climb.
Picture: the same two ramps from figure s03, now used to estimate how much height you gain in a diagonal micro-step.
Why the topic needs it: the parent's definition of "exact" is exactly " $M d x + N d y$ is a total differential $d F$ ." So $M$ pairs with $F_{x}$ and $N$ with $F_{y}$ . This is where $M = F_{x}, N = F_{y}$ is born.

Recall Why does staying on a level curve force

$d F = 0$ ? On a contour the height never changes, so its change $d F$ over any step along the curve is $0$ . Hence $M d x + N d y = 0$ is the sentence " $d F = 0$ ", i.e. "don't leave the contour."

5. The coefficients $M (x, y)$ and $N (x, y)$

6. Mixed second partials & Clairaut: $F_{x y} = F_{y x}$

Take a slope, then take the slope of that slope in the other direction. That is a mixed second partial.

Plain words → picture → why.

Plain words: "how the east-slope changes as you go north" equals "how the north-slope changes as you go east."
Picture: a tiny square patch of hill; you may climb around it clockwise or anticlockwise — the corner heights force both mixed slopes to agree.
Why the topic needs it: since $M = F_{x}$ and $N = F_{y}$ , we get $M_{y} = F_{x y}$ and $N_{x} = F_{y x}$ . Clairaut says these are equal, giving the exactness test $M_{y} = N_{x}$ . This is the entire justification. See Clairaut's Theorem (Equality of Mixed Partials).

F (x, y)

returns and picture it

A single height number above the floor point

(x, y)

— a hill surface.

I can describe a level curve in one sentence

All floor points sharing one height,

F (x, y) = c

— a contour ring.

I know why the symbol

\partial

is used instead of

d

Because there are two inputs;

\partial

means "slope in one direction with the other input held fixed."

I can compute

F_{x}

and

F_{y}

for

F = x^{2} y

F_{x} = 2 x y

and

F_{y} = x^{2}

I can write the total differential

d F

d F = F_{x} d x + F_{y} d y

I know what

M

and

N

are

The coefficients of

d x

and

d y

M d x + N d y = 0

I can state Clairaut's theorem

For continuous second partials,

F_{x y} = F_{y x}

I can explain how Clairaut gives the exactness test

M = F_{x}, N = F_{y} \Rightarrow M_{y} = F_{x y} = F_{y x} = N_{x}

I know why

g (y)

(not just

+ C

) appears

Integrating in

x

treats

y

as constant, so the missing piece may depend on

y

I know why "simply connected" matters

No holes lets the slopes be glued into one single-valued potential

F

Recall One-line summary before you leave

$M$ and $N$ are the two slopes of a hidden hill $F$ ; if their cross-slopes match ( $M_{y} = N_{x}$ ) the hill really exists, and the ODE just says "stay on the contour $F = c$ ."

Foundations — Exact equations — exactness condition, finding potential function

1. The starting object: a function of TWO variables $F (x, y)$

2. Level curves: the contour lines $F (x, y) = c$

3. Partial derivatives: $\frac{\partial F}{\partial x}$ and $\frac{\partial F}{\partial y}$

4. The differentials $d x$ , $d y$ , and the total differential $d F$

5. The coefficients $M (x, y)$ and $N (x, y)$

6. Mixed second partials & Clairaut: $F_{x y} = F_{y x}$

7. "Simply-connected region" — the no-holes condition

8. The constant of integration that is secretly a function: $g (y)$

9. The symbol $=!$ and " $c$ "

Prerequisite map

Equipment checklist