A slope for a curve that is no function

The slope of the mirror image differentiated a function nobody could write down — the inverse — by noticing that its graph is the original’s graph read the other way round. The curve was the same curve; only the choice of which coordinate is the input had changed, and the slope turned upside down with it.

That observation is stronger than it looks, because it does not need either choice to work everywhere. Most curves worth drawing are neither the graph of a function of $x$ nor the graph of a function of $y$. The folium of Descartes, the set of points where

$$x^3 + y^3 = 3xy,$$

runs in from the upper left, loops round through the first quadrant and leaves to the lower right, crossing itself at the origin on the way. A vertical line near $x = 1$ meets it three times, so there is no single $y$ for that $x$, and solving the cubic for $y$ produces three branches glued together at points the formula cannot see.

Nevertheless the curve plainly has a tangent at almost every point, and the tangent has a slope. The question is how to compute a slope for a curve that is not the graph of anything.

The secants in the figure answer it the way the slope of a single point always has: take a second point on the curve, measure the slope of the line through both, and let the second point approach. Nothing about that requires a formula for the curve. It requires only that the curve can be walked along, and the far ends of the secants here were found by walking — a short step along the current direction, then a correction back onto the curve — not by solving for $y$.

Differentiating the equation instead of the function #

The secants find the slope; the equation predicts it. Suppose that near the marked point the curve is the graph of some function $y(x)$, whatever it is. Then $x^3 + y(x)^3 = 3x\,y(x)$ holds for every $x$ near the point, and both sides can be differentiated with respect to $x$, using the chain rule on $y^3$ and the product rule on $xy$:

$$3x^2 + 3y^2\, y' = 3y + 3x\, y'.$$

That is linear in the unknown slope $y'$, and solving it gives

$$y' = \frac{y - x^2}{y^2 - x}.$$

At the marked point, $x = 1.001$ and $y = 0.348$, the numerator is $-0.654$ and the denominator $-0.880$, and the slope is $0.744$ — the number the secants are closing on.

The general statement is worth writing once in the form that does not depend on the curve. Write the curve as $F(x, y) = 0$. Differentiating along it gives $F_x + F_y\, y' = 0$, where $F_x$ and $F_y$ are the rates at which $F$ changes as $x$ alone or $y$ alone moves, so

$$y' = -\frac{F_x}{F_y}.$$

That is the entire technique. The equation is differentiated as it stands, and the slope comes out as a ratio of two rates that the equation supplies directly. No branch is chosen and no formula is solved.

The assumption that was smuggled in #

The derivation began with suppose the curve is the graph of some function near the point, and that is precisely what is in doubt for a curve like this one. It is the same gap the slope of the mirror image found in its chain-rule derivation: differentiating an identity presupposes that the thing being differentiated exists and has a derivative.

The implicit function theorem closes the gap, and its hypothesis is the one the formula already displays. If $F$ has continuous rates of change and $F_y \neq 0$ at a point of the curve, then in some window round that point the curve is the graph of exactly one function of $x$, that function is differentiable, and its slope is $-F_x/F_y$. The denominator that the formula divides by is the condition under which the formula is true.

The circle shows the theorem at work where everything can be checked. Here $F = x^2 + y^2 - 1$, so $F_x = 2x$ and $F_y = 2y$, and the slope is $-x/y$. At $(0.6, 0.8)$ that is $-0.75$. The upper half of the circle is the graph of $\sqrt{1 - x^2}$, whose derivative is $-x/\sqrt{1 - x^2}$, and at $x = 0.6$ the square root is $0.8$: the same $-0.75$. On the lower half the square root carries a minus sign, and so does $y$, so the one formula $-x/y$ covers both halves without being told which one it is on. The implicit formula is branch-blind in the useful way: it answers for whichever branch the point happens to be on.

It fails at exactly two points, $(1, 0)$ and $(-1, 0)$, where $y = 0$. There the tangent is vertical, and no window round either point contains a single function of $x$: to the left there are two values of $y$ for each $x$, to the right none.

Where the formula divides by nought #

The points where $F_y = 0$ are the whole story of where a curve fails to be a graph over $x$, and there are two kinds of them.

Where $F_y = 0$ but $F_x \neq 0$ the tangent is vertical. On the folium that happens once, at the right-hand edge of the loop: setting $F_y = 3y^2 - 3x$ to nought gives $x = y^2$, and substituting into the equation leaves $y^6 = 2y^3$, so $y = 2^{1/3}$ and $x = 2^{2/3}$, the point $(1.587, 1.260)$. The figure checks it the honest way, by counting: just to the left of that point a short vertical line meets the curve twice, just to the right not at all. The curve is still perfectly smooth there — it is the graph of a function of $y$, since $F_x \neq 0$ lets the theorem run with the roles of the variables exchanged — and only the choice of $x$ as the input has failed.

Where $F_x$ and $F_y$ both vanish, the formula reads nought over nought, and the curve may do anything. On the folium this is the origin, where both rates are $3x^2 - 3y$ and $3y^2 - 3x$, both zero. The curve passes through the origin twice, once along each axis, and there are two tangents rather than none. Such a point is a singular point, and no choice of input variable repairs it: near the origin the folium is not the graph of a function of $x$, nor of $y$, nor of anything, because two separate branches meet there.

The window round the ordinary point is the positive half of the theorem made visible. Inside it, nine vertical lines evenly spaced each meet the curve exactly once, which is what being the graph of a function means. The theorem promises that such a window exists round every point where $F_y \neq 0$; it does not say how large it is, and a point near the vertical tangent has only a very narrow one.

Level ground, and a symmetry the formula keeps #

The numerator of the formula has a meaning too. Where $F_x = 0$ and $F_y \neq 0$ the slope is nought and the tangent is horizontal, and on the folium that condition is $3x^2 - 3y = 0$: $y = x^2$, which put into the equation leaves $x^6 = 2x^3$, so $x = 2^{1/3}$ and $y = 2^{2/3}$. The highest point of the loop is $(1.260, 1.587)$.

That is the vertical tangent’s point with its coordinates swapped, and the reason is visible in the equation: $x^3 + y^3 = 3xy$ is unchanged when $x$ and $y$ are exchanged, so the folium is its own mirror image in the diagonal. The slope formula respects that. Swapping the variables turns $(y - x^2)/(y^2 - x)$ into $(x - y^2)/(x^2 - y)$, which is its reciprocal — the reflection rule for inverses, applied to a curve that is its own reflection. The point where the loop is highest and the point where it reaches furthest right are mirror images of each other, and the formula’s numerator vanishing at one is its denominator vanishing at the other.

So the two rates divide the curve’s special points between them. Where $F_x$ alone vanishes the curve is level; where $F_y$ alone vanishes it is upright; where both vanish it crosses itself or has a cusp; and everywhere else it has a slope that is neither nought nor infinite, given by their ratio. Finding the horizontal and vertical tangents of a curve given by an equation is therefore the same kind of problem as finding the crossings — two polynomial equations in two unknowns — and none of it requires a formula for the curve.

The same classification on two more curves #

The lemniscate, $(x^2 + y^2)^2 = 2(x^2 - y^2)$, is a figure of eight on its side, and the classification finds its structure without any drawing.

Its rate in $y$ is $4y(x^2 + y^2 + 1)$, which vanishes only where $y = 0$, so the vertical tangents and the crossings all lie on the axis. Putting $y = 0$ into the equation leaves $x^4 = 2x^2$: the two ends of the eight at $x = \pm\sqrt 2$, and the centre. At the ends the rate in $x$ is $4x(x^2 - 1)$, which is $\pm 4\sqrt 2$, not nought, so the tangents there are vertical. At the centre both rates vanish and the two lobes cross. Three points found by solving two equations, and the whole shape of the curve’s failures follows from them.

The cubic $y^2 = x^3 - x$ is the instructive contrast, because it has vertical tangents and no crossing at all.

Here $F_y = 2y$, so the vertical tangents are where the curve meets the axis, at the three roots of $x^3 - x$: $-1$, $0$ and $1$. At each of them $F_x = 1 - 3x^2$ is $-2$, $1$ or $-2$, never nought, so none is singular. The curve falls into two pieces — a closed oval between $-1$ and $0$ and an unbounded branch from $1$ onward — and it is smooth everywhere, which is the property that makes it an elliptic curve rather than a curve with a knot in it. The window round the point on the steep branch had to be drawn tall rather than square: where the slope is steep the curve leaves a square window through its top and bottom, and the theorem’s window is a box whose proportions depend on the slope.

The mirror image was a special case #

The slope of the mirror image is recovered in one line. The graph of an inverse function is the curve $x = f(y)$, which is $F(x, y) = x - f(y) = 0$. Here $F_x = 1$ and $F_y = -f'(y)$, so the implicit formula gives

$$y' = -\frac{1}{-f'(y)} = \frac{1}{f'(y)},$$

the reciprocal of the original slope at the matching point, which is the reflection rule exactly.

And its exceptional point is this essay’s exceptional point. The cube root’s graph is $x = y^3$, whose rate in $y$ is $-3y^2$, zero at the origin. The implicit formula divides by nought there, and the curve has a vertical tangent — which is the reflection of the cube’s horizontal one. The hypothesis the original’s derivative is non-zero in the inverse-function rule is the hypothesis $F_y \neq 0$ in the implicit one, written for the particular equation that defines an inverse.

So the reflection was never really about reflection. It was about which variable is being solved for, and the implicit formula makes the choice explicit: divide by the rate of whichever variable is taken as the output, and the formula is valid exactly where that rate is not zero. Choosing $x$ as the output instead gives $dx/dy = -F_y/F_x$, the reciprocal, valid where $F_x \neq 0$ — and a point where one of the two is valid is a smooth point of the curve, whichever it is.

A tangent that adds points on a cubic #

The limit of secants on a cubic curve does something no other curve’s does, and it is the connection that made this family of curves a subject of its own.

A line through two points of the cubic $y^2 = x^3 - x$ meets it in a third, because substituting a line $y = mx + b$ into the equation gives a cubic in $x$ with two known roots, and the three roots of that cubic add to $m^2$ — so the third is $m^2$ minus the two known ones, and it is forced. That third point, reflected in the axis, is defined to be the sum of the first two, and the rule turns the points of the curve into a group: rows of three planted on a cubic uses exactly this, with three points in line adding to nothing.

To add a point to itself there is no second point, and the secant has to be replaced by its limit: the tangent. The slope of that tangent is $-F_x/F_y = (3x^2 - 1)/(2y)$, the implicit formula, and it fails where $y = 0$ — at the three points where the tangent is vertical, which are exactly the points that added to themselves give nothing. The places where the slope formula divides by nought are the points of order two in the group, and there are three of them because the cubic $x^3 - x$ has three roots. That is the implicit function theorem’s exceptional set turning up as a piece of algebra, and it is the reason the secant-and-tangent construction on cubics is the doorway to modern number theory.

More variables, and the same condition #

A surface in space given by $F(x, y, z) = 0$ is handled the same way, and the condition that makes it locally a graph is the same condition one dimension up. Near a point where $F_z \neq 0$ the surface is the graph of a function $z(x, y)$, with rates $-F_x/F_z$ in one direction and $-F_y/F_z$ in the other; where $F_z = 0$ the tangent plane stands upright and the surface folds back over the floor. The three rates together form the gradient, the arrow pointing across the surface in the direction $F$ grows fastest, and it is perpendicular to the tangent plane — which is why the tangent line of a curve, and the tangent plane of a surface, can be written down from the gradient without parametrising anything.

With several equations in several unknowns the rates become a matrix, and the condition the rate in the output variable is not nought becomes the block of the matrix belonging to the output variables is invertible: its determinant is not zero. That is the implicit function theorem in full, and it is the flat map that fits closest doing for equations what it did for maps: the linear approximation is solvable for the chosen variables, so the curved system is too, near the point.

The rational points on the circle are found by the opposite use of the same secants. Every triple on one circle draws the line through $(-1, 0)$ with a rational slope and reads off where it meets the circle again; the second intersection is rational because the first one is. Here the second point was brought towards the first to find a slope. There the slope was fixed first and the second point was the prize — the same line, used in the opposite direction.

What the pictures can and cannot show #

The windows show that a function exists and not what it is. The implicit function theorem is an existence theorem: round each ordinary point there is a function whose graph is the curve, and nothing in the theorem produces a formula for it. The figure checks the function numerically — solving for $y$ near the marked point by Newton’s method at each nearby $x$ and measuring the resulting slope — which is how such a function is actually computed, and it is an approximation of the function rather than a description.

The size of the window is not shown to be the largest. Each box is small enough for the count of crossings to come out as one on every line tested, and that is all. How far the local function extends before a vertical tangent or a crossing stops it is a global question the theorem does not answer, and a point close to a vertical tangent has only a sliver.

Four curves are drawn, all of low degree. The classification of points by which rates vanish is general; the specific points — the loop’s edge, the eight’s ends, the cubic’s three roots — were found by solving equations numerically from a grid of starting points and checking each solution, and a curve with a singular point that none of the starts approaches would have it missed. Every point marked satisfies its equations to within a billionth, which says that what is shown is right and not that nothing is missing.

Still open: how many pieces a curve can have #

The cubic fell into two pieces, an oval and a branch, and the question of how many pieces a curve of a given degree can fall into has a clean answer. Axel Harnack proved in 1876 that a curve given by a polynomial equation of degree $n$ in the plane has at most $(n-1)(n-2)/2 + 1$ separate pieces, and that curves reaching the bound exist in every degree. For the cubic that is two, which is what the figure shows.

What is not settled is how the pieces can sit relative to one another. Closed ovals can nest inside one another or lie side by side, and David Hilbert put the question of which arrangements occur sixteenth on his list of problems in 1900. It has been answered completely for curves of degree up to seven, the last case by Oleg Viro’s construction methods around 1980. For degree eight and beyond the list of possible arrangements is not known. Every piece of every such curve is smooth wherever its two rates do not both vanish, and differentiating its equation gives its slope at every one of those points; which shapes the pieces can make together is still open.

Solving for whichever variable is available #

The habit worth keeping is the one that turned the mirror image into a general rule.

A curve given by an equation has a slope wherever the equation’s rate in one of the two variables is not nought, and the slope is the ratio of the two rates with a minus sign. No branch has to be chosen and no formula solved; the equation is differentiated as it stands, and the division that follows carries its own warning about where it is valid. The failures are not a nuisance to be excluded: they are the vertical tangents where the other variable takes over, and the crossings where no variable can, and between them they draw the whole outline of where the curve is and is not a graph.

That is the one-dimensional face of the flat map that fits closest. There the derivative of a map of the plane was a matrix, and inverting the map near a point needed the matrix to be invertible. Here the derivative of $F$ is a row of two numbers, and solving for $y$ needs the second of them to be non-zero. In both, the linear approximation decides what the curved object can do near a point, and the points where the approximation degenerates are the points worth locating first.