The Implicit Function Theorem

Part 3, Chapter 3: Calculus of Several Variables

Learning objectives

State the Implicit Function Theorem for $F(\mathbf{x},\mathbf{y})=\mathbf{0}$
Identify the partial-Jacobian condition $\det(\partial F/\partial\mathbf{y})\neq 0$
Compute $dy/dx$ via the implicit formula $-F_x/F_y$
Predict when an equation defines one variable implicitly as a function of the others

Many equations in science are not given in the form " $y$ equals some expression in $x$ " but rather " $F(x,y)=0$ ". The unit circle is $x^2+y^2-1=0$ , not $y=\ldots$ . The level surfaces of a thermodynamic state function are similar. The Implicit Function Theorem (ImFT) tells you when such an equation locally defines one variable as a function of the others, even if you cannot solve the equation explicitly. The condition is, again, a Jacobian condition: the partial Jacobian with respect to the "to-be-solved-for" variables must be invertible.

The two-variable case (the easy version)

Let $F:\mathbb{R}^2\to\mathbb{R}$ be $C^1$ on an open set containing $(a,b)$ . Suppose $F(a,b)=0$ and $F_y(a,b)\neq 0$ y(a,b)neq0. Then there exists an open interval $I$ around $a$ and a $C^1$ function $g:I\to\mathbb{R}$ with $g(a)=b$ such that $F(x,g(x))=0$ for every $x\in I$ . Moreover $g'(x)=-F_x(x,g(x))/F_y(x,g(x))$ .

Read this as: "if the partial derivative with respect to $y$ is non-zero at the point, then near that point $y$ is locally a function of $x$ , even if we cannot write down the formula explicitly." The formula for $g'$ is what implicit differentiation gives you in any calculus textbook, the ImFT is just the theorem that justifies the procedure.

The general case

For $F:\mathbb{R}^{n+m}\to\mathbb{R}^m$ with variables split as $(\mathbf{x},\mathbf{y})\in\mathbb{R}^n\times\mathbb{R}^m$ : if $F(\mathbf{a},\mathbf{b})=\mathbf{0}$ and the $m\times m$ partial-Jacobian $\partial F/\partial\mathbf{y}$ is invertible at $(\mathbf{a},\mathbf{b})$ , then near $\mathbf{a}$ there is a smooth function $\mathbf{g}$ with $\mathbf{g}(\mathbf{a})=\mathbf{b}$ such that $F(\mathbf{x},\mathbf{g}(\mathbf{x}))=\mathbf{0}$ . The $m$ "dependent" variables are locally functions of the remaining $n$ .

Try graphing $y=\sqrt{1-x^2}$ in the widget above. That is one branch of the implicit equation $x^2+y^2=1$ ; locally near any point on the upper half it expresses $y$ as a function of $x$ . The ImFT condition $F_y=2y\neq 0$ y=2yneq0 holds everywhere on the open upper semicircle, so a smooth branch exists. At the points $(\pm 1, 0)$ , $F_y=0$ y=0 and you cannot solve for $y$ locally, the circle has vertical tangents there.

Where this shows up

Computer-aided design (CAD), constraint solving: CAD systems express geometric relationships (parallel lines, tangencies, equal lengths) as implicit equations $F(\mathbf{x})=\mathbf{0}$ . When the designer drags a vertex, the system uses the ImFT, specifically Newton iterations on the implicit equations, to update the dependent geometry consistently. The Jacobian becoming singular signals an over-constrained or degenerate sketch.
Continuation methods in numerical analysis: To find solutions of a parameterized equation $F(\mathbf{x},t)=\mathbf{0}$ as $t$ varies, the ImFT lets you predict $d\mathbf{x}/dt=-(\partial F/\partial\mathbf{x})^{-1}(\partial F/\partial t)$ . This is the heart of arclength continuation, used in nonlinear-buckling analysis, bifurcation tracking, and homotopy methods for polynomial systems.
Economics, comparative statics: Equilibrium conditions in models are usually implicit: $F(\mathbf{p},\boldsymbol{\alpha})=\mathbf{0}$ where $\mathbf{p}$ are prices and $\boldsymbol{\alpha}$ are parameters (taxes, supply shocks). Comparative-statics analysis asks "how does the equilibrium price change with the parameter?" The ImFT answers: $d\mathbf{p}/d\boldsymbol{\alpha}=-(\partial F/\partial\mathbf{p})^{-1}(\partial F/\partial\boldsymbol{\alpha})$ .

Pause and think: The Implicit Function Theorem can be derived from the Inverse Function Theorem. Given $F(x,y)=0$ , define $G(x,y)=(x,F(x,y))$ . Then $DG=\begin{pmatrix}1&0\\F_x&F_y\end{pmatrix}$ , with $\det DG=F_y$ y. If $F_y\neq 0$ yneq0, the IFT gives a local inverse $G^{-1}$ . Reading off the inverse's structure recovers the implicit $g$ . The two theorems are essentially the same theorem.

Try it

Predict first: at which points on the circle $x^2+y^2=1$ can we NOT solve locally for $y$ as a function of $x$ ? (Answer: where $F_y=2y=0$ y=2y=0, i.e. $(\pm 1, 0)$ , the points with vertical tangents.)
Apply the formula $y'=-F_x/F_y$ to find $dy/dx$ for $x^3+y^3=6xy$ at any point. (Answer: $-\frac{3x^2-6y}{3y^2-6x}=\frac{6y-3x^2}{3y^2-6x}$ .)
Predict: does $x^2+y^2+z^2=1$ define $z$ as a function of $(x,y)$ near the north pole $(0,0,1)$ ? (Yes; $F_z=2z=2\neq 0$ z=2z=2neq0. Near the equator $z=0$ this fails and the surface has a horizontal tangent plane.)
Show that the system $\{u^2+v^2=x,\ u+v=y\}$ locally defines $(u,v)$ as functions of $(x,y)$ near most points. Find where it fails.
Trap: the ImFT does NOT tell you the explicit formula for $g(x)$ , only that it exists and is smooth. Solving the original equation may be impossible in closed form. The theorem is about existence, not computation.

A trap to watch for

People often invoke the ImFT to conclude " $y$ is a function of $x$ " without checking the condition $F_y\neq 0$ yneq0. The unit circle $x^2+y^2=1$ shows the danger: globally $y$ is NOT a single-valued function of $x$ on $[-1,1]$ , it has two branches $\pm\sqrt{1-x^2}$ that meet at $(\pm 1, 0)$ . The ImFT correctly predicts that the branches break exactly where $F_y=2y=0$ y=2y=0. Always state the condition before invoking the conclusion.

What you now know

You can apply the ImFT to decide when an equation $F(\mathbf{x},\mathbf{y})=\mathbf{0}$ implicitly defines $\mathbf{y}=\mathbf{g}(\mathbf{x})$ , compute the derivative $\mathbf{g}'$ via the formula $-(\partial F/\partial\mathbf{y})^{-1}(\partial F/\partial\mathbf{x})$ , and connect the theorem to its twin, the IFT. The chapter's work on calculus in $\mathbb{R}^n$ is complete. Chapter 4 turns to the topological foundations that this entire framework rests on.

Mark section complete →

References

Garrity, T. (2002). All the Mathematics You Missed. Cambridge UP, ch. 3.
Spivak, M. (1965). Calculus on Manifolds. W. A. Benjamin, ch. 2.
Munkres, J. R. (1991). Analysis on Manifolds. Westview Press, ch. 2 (Theorem 8.1).
Rudin, W. (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill, ch. 9 (Theorem 9.28).
Apostol, T. M. (1974). Mathematical Analysis (2nd ed.). Addison-Wesley, ch. 13.