somewhere near the beginning.

The implicit and inverse function theorems

Filed under: Mathematics — Alex @ 1:03 pm 9/12/2006

The implicit and inverse function theorems are two of the most important theorems in analysis, and are foundational in differential geometry, yet I’ve found that the way they are presented is more often confusing than not. Here’s my attempt to provide a simple intuitive approach to looking at the two theorems:

The gist of the implicit function theorem is that a set which satisfies a continuously differentiable implicit relation also locally satisfies explicit continuous differentiable equations.

The gist of the inverse function theorem is that a continuously differentiable mapping is locally a diffeomorphism.

Here are formal statements of the theorems (taken from “Differential Geometry: Curves, Surfaces, Manifolds” by Kuhnel):

Implicit Function Theorem
Let U_1 \subset \R^k and U_2 \subset \R^m be open sets and let F : U_1 \times U_2 \rightarrow \R^m be a continuously differentiable mapping. Let (a,b) \in U_1 \times U_2 be a point for which F(a,b)=0 and for which the square matrix
\frac{\partial F}{\partial y} = \left( \frac{\partial F_i}{\partial y_j} \right)_{i,j =1,\ldots,m}
is invertible. Then there are open neighborhoods V_1 of a and V_2 of b and a continuously differentiable mapping g : V_1 \times V_2 \rightarrow \R^m such that that for all (x,y) \in V_1 \times V_2, F(x,y)=0 if and only if y = g(x).

and

Inverse Function Theorem
Let U be an open set in \R^n and let f : U \rightarrow \R^n be a continuously differentiable mapping with the property that the Jacobian at a fixed point u_0 is invertible. Then there is a neighborhood V with u_0 \in V \subset U on which the mapping f is also invertible, i.e., f|_V : V \rightarrow f(V) is a diffeomorphism.

The crucial hypothesis of both theorems is that the derivative of the map has to be full rank; this is often condensed to ‘the map is full ranked’. The term immersion, which refers to an everywhere (within its domain) differentiable and everywhere full ranked map, is suggestive of the conclusion of the implicit function theorem: an immersion locally immerses a lower dimensional hypersurface in a larger ambient space in a recoverable manner, due to the injectivity of (an appropriate ‘partition’ of) the derivative.

Hmm… what I just wrote makes sense to me, but I doubt it will be clear to anyone else.

Possibly relevant posts:

2 Comments »

RSS feed for comments on this post. TrackBack URL

Leave a comment