TITLE: What is the 'implicit function theorem'?
QUESTION [53 upvotes]: Please give me an intuitive explanation of 'implicit function theorem'. I read some bits and pieces of information from some textbook, but they look too confusing, especially I do not understand why they use Jacobian matrix to illustrate this theorem.

REPLY [10 votes]: There are basically two interpretations of (part of) the implicit function theorem (IMFT). 
One is that it tells you under what conditions we have solutions to some equation of the form $f=0$, which has been mentioned in Zhen Lin's answer. 
More precisely, 

let $E$ be an open set of ${\Bbb R}^{n+m}={\Bbb R}^n\times{\Bbb R}^m$ and  $f:E\to {\Bbb R}^n$ be a $C^1$ mapping such that $f(a,b)=0$, where $(a,b)\in {\Bbb R}^n\times {\Bbb R}^m$. Put $A=f'(a,b)$ and assume that $A_x$ is invertible, where $A_x$ denotes the restriction of the linear map $A$ on ${\Bbb R}^n\times\{0\}$. $\tag{*}$

Then we can "solve" the equation $f(x,y)=0$ for each of those $y$ near $b$, which means in an open neighborhood $W\subset{\Bbb R}^m$ of $b$, we have an implicitly defined function $g:W\to {\Bbb R}^n$ such that $f(g(y),y)=0$ for every $y\in W$. In particular, $f(g(b),b)=f(a,b)=0$. The word "solve" might be kind of confusing. After all, what does it mean by solving an equation $f=0$? Basically it means one can write some variables in terms of others. On the other hand, the IMFT only tells you the existence of $g$ but not what $g$ looks like. 
Another way to look at the IMFT is that it tells you when a set defined by
$$
S=\{z\in{\Bbb R}^d|f(z)=0\}
$$
is locally a graph of some function. (Here "being locally a graph" means one can find an open set $U\subset {\Bbb R}^d$ such that $U\cap S$ is a graph.) Under the assumption $(*)$, we would have the following conclusion

there exsit open sets $U\subset{\Bbb R}^{n+m}$ and $W\subset {\Bbb R}^m$, with $(a,b)\in U$ and $b\in W$ such that
  $$
U\cap S=G:=\{(g(y),y)\mid y\in W\},
$$
  where $g$ is a function from $W$ to ${\Bbb R}^n$.

Note that we call $G$ the graph of the function $g$. 
Using the notations in the second interpretation, the first one basically tells you
$$
G\subset U\cap S. 
$$