What is the Implicit Function Theorem good for?

Question

What are some applications of the Implicit Function Theorem in calculus? The only applications I can think of are:

1. the result that the solution space of a non-degenerate system of equations naturally has the structure of a smooth manifold;

2. the Inverse Function Theorem.

I am looking for applications that would be interesting to an advanced US math undergraduate or 1-st year graduate student who is seeing the Implicit Function Theorem for the first or second time. The context is I will be explaining this result as part of a review of manifold theory.

Answer 1

The infinite-dimensional implicit function theorem is used, among other things, to demonstrate the existence of solutions of nonlinear partial differential equations and parameterize the space of solutions. For equations of standard type (elliptic, parabolic, hyperbolic), the standard version on Banach spaces usually suffices, but you have to be clever about which Banach space to use. There is a generalization of the implicit function theorem, due to Nash who used it to demonstrate the existence of isometric embeddings of Riemannian manifolds in Euclidean space, that works for even more general types of PDE's. Moser stated and proved a simpler version of the theorem. There is a beautiful survey article by Richard Hamilton (who originally used the Nash-Moser implicit function theorem to prove the local-in-time existence of solutions to the Ricci flow) on the Nash-Moser implicit function theorem.

Answer 2

One of the major applications of Implicit Function Theorem is the lesson it teaches:

                      Locally, Manifold Theory = Linear Algebra. 

That is, locally, we can perform our calculus as if it is linear algebra. Solving simultaneous equations, discussing about linear independence of coordinates, basis set and mapping from one manifold to another can be viewed as linear transformations. Discuss the invertibility of functions as if they are linear transformations. In fact by Darboux's theorem, in Symplectic manifold theory the linear algebra aspects are more prominent.

Answer 3

I like to begin an introduction to manifolds by proving that the following conditions on a subset of $\mathbb R^n$ are equivalent:

1. Locally it can be mapped to an open subset of $\mathbb R^p\times 0$ by a diffeomorphism between open subsets of $\mathbb R^n$.

2. Locally it appears as the graph of a smooth function expressing $n-p$ of the coordinates as functions of the other $p$.

3. Locally it is $f^{-1}(0)$ for a smooth map to $\mathbb R^{n-p}$ whose derivative has maximal rank (i.e. rank $n-p$).

4. Locally it is the image of a smooth map from $\mathbb R^p$ of maximal rank (i.e. rank $p$).

2 implies 1 easily. 1 implies 3 and 4 easily. 4 implies 2 by the inverse function theorem in dimension $p$. 3 implies 2 (this implication might be called the implicit function theorem) by the inverse function theorem in dimension $n$.

I like to think that any one of these four would serve as a good definition of "smooth $p$-dimensional manifold in $\mathbb R^n$" for those who have not studied abstract manifolds, but that you haven't really begun to get into the subject until you see that they are the same.

Answer 4

A fundamental application is

Theorem Let $t\mapsto A(t)$ a $C^1$-family of $n\times n$ matrices. If at $t=0$, $A(0)$ has a simple eigenvalue $\lambda_0$, then for $t$ small enough, $A(t)$ admits a simple eigenvalue $\lambda(t)$ that is a $C^1$-function of $t$ and is such that $\lambda(0)=\lambda_0$. In addition, its derivative at $t=0$ is given by $\dot\lambda(0)=x^*\dot A(0)x$ where $x$ is the normalized ($\|x\|=1$) eigenvector associated with $\lambda_0$.

Of course, there are infinite-dimensional versions of this statement. Both finite and infinite-dimensional versions are used daily by hundreds of mathematicians.

Answer 5

If you go a little bit further than the inverse and implicit function theorems, you can get a fairly practical theorem. Kantorovich's theorem gives you fairly strong sufficient conditions for a system of smooth equations to have a solution. Moreover it tells you how quickly Newton's method converges in that situation. For example, this theorem is used by Harriet Moser to prove that SnapPea does give approximations to actual solutions to the hyperbolic gluing equations. The applications of course are pretty broad, this is one on the fairly pure end of the spectrum. Kantorovich was an economist although I do not understand the economics problems he was interested in.

If you're interested, this perspective on the inverse and implicit function theorems is in "full glory" in Hubbard's multi-variable calculus text.

2nd answer: The proof of Sard's theorem is a delicate dance with the Implicit Function Theorem, Taylor's Theorem and some basic argument with Lebesgue measure zero.

Answer 6

The Implicit Function Theorem appears in a typical undergraduate course on mathematical economics, and I find that application gives me more inspiration than applications in mathematics. Here is a simple example. Suppose a company's profit $\Pi$ is a function of the number $x$ of widgets they produce (which they can choose) and a vector of other variables $\vec{y}$ over which they have no control (the price at which they can sell the widgets, cost of raw materials, etc.). Then by Calc 1, the profit-optimizing quantity $x^\ast$ will satisfy the equation $\frac{\partial}{\partial x} \Pi \vert_{x = x^\ast} = 0$.

The Implicit Function Theorem says that ${x^\ast}$ is a function of $\vec{y}$. This is just the unsurprising statement that the profit-maximizing production quantity is a function of the cost of raw materials, etc. But the IFT does better, in that in principle you can evaluate the derivatives $\partial x^* / \partial y_i$. If the company acts to maximize its profit, then these derivatives predict how the company will change its production in response to changes in external factors such as the cost of raw materials.

(You might be worried about the hypothesis in the IFT that $\frac{\partial^2}{\partial x^2} \Pi$ is nonzero. But in this example it is trivially satisfied because economists usually assume that marginal cost is increasing and that marginal revenue is decreasing, or anyway increasing more slowly than marginal cost.)

Answer 7

See the article: Nord, Gail; David Jabon, and John Nord; The Global Positioning System and the Implicit Function Theorem, SIAM Review, vol. 40:3 (1998), 692--696; for a nice application of the IFT to estimating the errors involved in GPS measurements.

Answer 8

One elementary application to ODEs: a differential equation $$u'(t)=f(t, u(t ))$$ with a continuous RHS of the form

$$f(t,x):=-\frac{\partial_t g(t,x)}{\partial_x g(t,x)},$$ for some $C^1$ function $g$ with non-vanishing $\partial_x g,$ has a one-parameter family of solutions whose graphs are the level sets of $g$. This is a largely used solution method, but it's the IFT that ensures that these solutions are indeed well-defined functions, implicitly defined by $g(t,u(t))=c.$ Incidentally, note that this is a case where there holds uniqueness for the Cauchy problem, even if the Lipschitz assumptions may fail to hold.

Answer 9

I think of the implicit function theorem as a basic result that underlies many results in transversality and Morse theory. These two ideas are the major tools in the study of manifolds. You cannot understand manifolds without them.

So if we look at the inverse image of a non-degenerate critical value in a manifold it will have finitely many singular points. Outside a neighborhood of these, the pre-image appears as a submanifold. Local coordinates are obtained via the implicit function theorem. Look at any book on manifold theory and you will see it used.

Answer 10

By changing coordinates you can make a simple function appear complicated. Have you ever asked yourself if the opposite is true: given a "complicated" function, can I make it look simpler in a neighborhood of a point by changing the coordinates near that point?

The implicit function theorem states that if that point is not a critical point, then you can find coordinates near that point such that, in these coordinates the function is a linear function.

Answer 11

In studying how the fixed points of a one parameter family of real functions $f_\mu(x)=f(x,\mu)$ as the parameter $\mu$ changes, one is lead to study the solutions of the equation $$f(x,\mu)=x.$$ The implicit function theorem is the right tool for this task, and in general, for the study of bifurcation in dynamical systems, both continuous and discrete.

Answer 12

I was just sitting down to brainstorm some applications, since we're going to finish the proof next week and I want to show some payoff. Here is what I came up with, not counting ideas already mentioned in other answers (many of which I like):

• The orthogonal group is a manifold, of dimension $\binom{n}{2}$. It's not too hard to compute the derivative of the map $X \mapsto X X^T$ from $n \times n$ matrices to $n \times n$ symmetric matrices and show that it is rank $\binom{n+1}{2}$ when $X$ is invertible. So the preimage of $\mathrm{Id}$ is a manifold of dimension $n^2 - \binom{n+1}{2}$.

• Suppose I have a smooth map $(x,y) \mapsto (f,g)$ from $\mathbb{R}^2 \to \mathbb{R}^2$, so that $\mathrm{rank} (Df)$ is everywhere $1$. Then one of $f$ and $g$ is locally a function of the other. (Proof: After some initial fiddling around, we can assume that $\partial f/\partial x \neq 0$. Then $f=\mathrm{constant}$ is a curve. Differentiating $g$ along this curve gives $0$.) I remember this trick being used to great effect in Kenyon-Okounkov to show that two functions obeyed a relation without actually constructing the relation; I'll try to find some simple examples where it is very nonobvious what the relation is.

• This top voted proof of the fundamental theorem of algebra uses centrally that a submersion is an open map, which is the key lemma in all proofs of IFT that I know.

I'll come back in a few weeks and report on how these went.

Answer 13

One thing I've learned recently (moving into symplectic geometry from topology) is that people often underestimate the value of regarding manifolds locally as graphs of functions, or submanifolds of Euclidean space. The IFT helps with this, and provides actual applications, but here's possible motivation.

--My recent favorite example for intuition: How does a surface of negative curvature behave? Like the graph of a harmonic function on $\mathbb R^2$ (where the Hessian is not 0). That is to say, if you have trouble imagining a surface in $\mathbb R^3$ which has no hills and valleys but only saddles, try graphing a function like $xy$ or $e^x\cos y$.

--Related to this: use IFT to study the curvature of a surface in $\mathbb R^3$ (or $n$-manifold in $\mathbb R^{n+1}$) nearby a strict maximum in any one coordinate. I'm not sure what the nicest precise statement should be here to cover all non-generic cases, but it's fun route to differential geometry if someone hasn't had it.

--Less directly related to the IFT: What does a Morse critical point "look like''? We like to say that the Morse function can be given in some local coordinates by $x_1^2+\ldots +x_k^2-x_{k+1}^2-\ldots x_n^2$, but when we visualize this, we usually see the manifold (OK, surface) itself as the graph of that function over a piece of $\mathbb R^n$. Thus making the Morse function "height" as usual.

Answer 14

The implicit function theorem is critical in the theory of manifolds (especially that of Riemann surfaces) in showing that a subvariety of affine or projective space is actually a submanifold. I found this out bluntly after trying to read a textbook on Riemann surfaces and realizing that I didn't entirely understand it for this reason. Furthermore, the conditions of the implicit function theorem motivate the definition of a non-singular point of a variety, and in more advanced algebraic geometry, the notion of an étale map. Another important notion in algebraic geometry motivated by the implicit function theorem is that of a local complete intersection. To read an elementary account of this latter notion, see I.3 of Algebraic Curves and Riemann Surfaces by Miranda.

A good book to see how multivariable calculus and commutative algebra interact is Smooth Manifolds and Observables by Nestruev.

Answer 15

For polynomials over the $p$-adic numbers there is a version of the implicit function theorem which makes the domain of the implicit function precise. I'll state the two-variable version here, along with an application to counting congruences modulo powers of $p$. Both generalise easily to any number of variables.

Let $f(X,Y) \in \mathbf{Z}_p[X,Y]$. Suppose that $a, b \in \mathbf{Z}_p$ are such that $f(a,b) = 0$ and $\frac{\partial f}{\partial X}(a,b) \not\equiv 0$ mod $p$. Then there exists a continuous function $g : b + p\mathbf{Z}_p \rightarrow a + p\mathbf{Z}_p$ such that if $y \in b + p\mathbf{Z}_p$ then $f(x,y) = 0$ if and only if $x = g(y)$.

I learned of an application of this result from Benjamin Klopsch. Suppose that $f(X,Y) \in \mathbf{Z}[X,Y]$ is non-singular and let $N_k(f)$ be the number of residue classes $(x,y) \in (\mathbf{Z} / p^k \mathbf{Z})^2$ such that $f(x,y) \equiv 0$ mod $p^k$. Applying the result above to each solution of $f(a,b) \equiv 0$ mod $p$ gives $N_k(f) = N_1(f)p^{k-1}$ for all $k \ge 1$.