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I watched an ICTP lecture on elementary real analysis & the lecturer went to great pains to emphasize the importance of the intermediate value theorem because it is what generalizes to higher dimensions via connectedness & how Bolzano-Weierstrass generalizes to metric spaces & how just a few results following from these are what really matters. Regardless of how true that is (I think he's biased because he is a functional analyst (analysist?)) I found that bit of intuition & motivation extremely clarifying & I'd read a lot of links, wiki's etc... just looking for that kind of motivation but it was nowhere to be found beforehand.

Similarly, I've read a lot about the implicit/inverse function theorems but I just don't understand the reasons for putting so much focus on them, but that's because I don't really know what they are saying & that's partly because I don't understand what you need to know to really appreciate these theorems.

I guess what I'm asking for is an insightful & human explanation of what these theories are, what you need to know to lead up to them, why you need to know that certain material & why these things are so powerful (for instance, I believe you can use these to prove the Lagrange Multipliers theorem, though I don't know why it has anything to do with it, I also know that knowing one means you can prove the other & that it doesn't matter which direction you approach from, but again I don't appreciate why that is).

I'm more interested in the surrounding theory, i.e. what you need to know, why you need to know that & why it is so important, than what the theorem says so I'd much rather prefer to have the motivation so that I could prove this myself.

(Yes I've read the wiki page, the threads on this site, the many articles apparently trying to motivate it, I just feel I haven't read anything that directly quells my aforementioned concerns so I think that justifies the thread).

Thanks for your time.

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4 Answers

The inverse/implicit function theorem tell you when you can (locally) solve a system of equations. This is incredibly important whenever you want to study a nonlinear problem -- e.g. differential geometry, PDE, etc. In the following I will call both of them "implicit function theorem", since they are the same thing, really.

Let me give an example: suppose you are in the plane, and want to solve for $x^2 + y^2 = 1$. You found a solution by inspection ($x = 1, y = 0$, say). Let's let $F(x,y) = x^2 + y^2 - 1$. This is $F: \mathbb{R}^2 \to \mathbb{R}$.

You have a solution $F(1,0) = 0$ and want to understand $F(x,y) = 0$. The implicit function theorem tells you to consider the derivative $dF(1,0) = (2, 0) \colon \mathbb{R}^2 \to \mathbb{R}$. This linear map is surjective, so the implicit function theorem tells us that there is a small neighbourhood of $(1,0)$ so that if we restrict our attention to this neighbourhood, there is a solution space parameterized by the kernel of the linear map, i.e. by the vertical axis. Furthermore, this describes all the solutions in this neighbourhood, and the solution space is tangent to this line at $(1,0)$.

In this example, we have used the implicit function theorem to tell us a lot about the circle. We used it to say that the circle is 1-dimensional (at least near $(1,0)$), that the circle has a vertical tangent at $(1,0)$ and that near $(1,0)$, we don't have any accumulating branches getting closer and closer. (i.e. we have ruled out something pathological like what happens if you take the union of the lines $x = 1/n$ and $x=0$.)

This is silly when we know what we are working with so specifically. However, you can use the implicit function theorem to understand much more complicated situations ... e.g. consider $n \times n$ matrices. Look at the matrices in here with determinant $1$. What does that look like? Using the implicit function theorem you can describe at least what it is tangent to at each point, and also that it is a "nice" space the way the circle is a nice space (I am trying to describe what is known as being a submanifold.)

Let me give a fancier example. Consider a differential equation $(x',y') = (f(x,y), g(x,y))$. Let's say that there are two rest points $p, q$ and you can find a trajectory connecting them by some method. You may ask yourself if this is the only trajectory connecting the two points. By using the implicit function theorem in infinite dimensions (on a Banach space), you have a tool to approach this problem. Similar methods also sometimes work in studying certain PDE.

Often, you can't find a solution by inspection, but you know that the nonlinear problem you want to solve is of the form $F(x) = 0$, where $x$ lives in some suitable space. Still, you can get some mileage from the implicit function theorem in this case. It's much harder, so I will only give details if you are interested. I suspect what I have written has already given enough to think about.

Surprisingly, the only thing you need to prove this key result is the understanding of the Banach Contraction Mapping Principle (and a good understanding of the fact that the first derivative at a point $p$ is the best linear map that approximates your nonlinear map near $p$).

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These theorems are so fundamentally important because they say what you intuitively think should be true, is actually true. I shall explain this for the implicit function theorem; but you could give similar explanations for the inverse function theorem or for the rank theorem.

Assume, e.g., than three physical quantities like $p$, $V$, $T$ are related by a certain law $\Phi(p, V, T)=0$, where the left side is a complicated expression in the variables $p$, $V$, $T$. Counting "degrees of freedom" you have the impression that whenever values are given for two of the three variables the corresponding value of the third variable should be determined, even if you are not able to solve the equation $\Phi(p,V,T)=0$ explicitly for the third variable.

That's where the theorem on implicit functions comes in: It not only guarantees under certain hypotheses that the values of, say, $V$ and $T$ determine the value of $p$, but that $p$ is actually a differentiable function $\phi(\cdot,\cdot)$ of $V$ and $T$, and it gives formulas for the partial derivatives ${\partial \phi\over\partial V}$, ${\partial \phi\over\partial T}$ that do not require the solution of $\Phi(p, V, T)=0$ for $p$ in variable terms.

Now the implicit function theorem (as well as the other theorems mentioned) is only a local theorem. This means that you need a feasible point $(p_0,V_0,T_0)$ to start with, and the graph of the function $\phi: (V,T)\to p:=\phi(V,T)$ is only defined in a small box with center $(p_0,V_0,T_0)$. In addition there is a certain "technical condition" which I won't explain here. It excludes cases like the equation $x+y^2=0$ which does not define $y$ as a good function of $x$ in the neighborhood of $(0,0)$.

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Suppose $f:\mathbb{R}^m\to\mathbb{R}^n$ is a smooth map. Recall that, at any given point $x \in \mathbb{R}^m,$ the derivative of $f$ is a linear map $D_x f: \mathbb{R}^m \to \mathbb{R}^n$ given by

$$ D_x f(\vec{v}) = \lim_{t\to 0} \frac{f(x + t \vec{v}) - f(x)}{t}.$$

Recall that, even when $f$ is not invertible, for any given point $y \in \mathbb{R}^n,$ we may still define the preimage $f^{-1}(y)$ to be the set of all points $x \in \mathbb{R}^m$ such that $f(x) = y.$

We say that a point $y \in \mathbb{R}^n$ is a regular point when for every point $x \in f^{-1}(y),$ $D_x f$ has full rank. In particular, if $m > n,$ this implies that $D_x f$ is surjective. Whenever $D_x f$ is surjective, we say that $f$ is a submersion at $x.$

Local Submersion Theorem: If $f$ is a submersion at $x,$ then there are local coordinates $(\tilde{x}_1,\ldots, \tilde{x}_m)$ around $x$ such that $f(\tilde{x}_1,\ldots,\tilde{x}_m) = (\tilde{x}_1,\ldots,\tilde{x}_n).$

Preimage Theorem: If $y \in \mathbb{R}^n$ is a regular point, then $f^{-1}(y)$ is a smooth $(m-n)$-dimensional submanifold of $\mathbb{R}^m.$ That is, $f^{-1}(y)$ is locally homeomorphic (by a smooth invertible map) to $\mathbb{R}^{(m-n)}.$ That is, up close, $f^{-1}(y)$ looks just like $\mathbb{R}^{(m-n)}.$

The proof of the local submersion theorem depends critically upon IFT. In turn, the proof of the preimage theorem depends critically upon the local submersion theorem.

Transversality and intersection theory---as given in, for instance, Guillemin and Pollack's classic Differential Topology---depend critically upon the preimage theorem. These tools are very useful; they can be used to prove the Borsuk-Ulam theorem and the fundamental theorem of algebra, and they are essential tools for the development of Lefschetz fixed point theory and the nebula of results called the "Poincare-Hopf index theorem." The Poincare-Hopf index theorem, in turn, is used to prove the general Gauss-Bonnet theorem.

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Try the Krantz-Parks book The implicit function theorem: History, theory, and applications.

See also What is the Implicit Function Theorem good for? at MO.

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the book is excellent. –  Laura Dec 15 '12 at 6:46
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