The implicit and inverse function theorems
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
Letand
be open sets and let
be a continuously differentiable mapping. Let
be a point for which
and for which the square matrix
is invertible. Then there are open neighborhoodsof
and
of
and a continuously differentiable mapping
such that that for all
,
if and only if
.
and
Inverse Function Theorem
Letbe an open set in
and let
be a continuously differentiable mapping with the property that the Jacobian at a fixed point
is invertible. Then there is a neighborhood
with
on which the mapping
is also invertible, i.e.,
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:
- Differential of the determinant (6/29/2007)
- Exactness of differential forms (7/28/2007)
- Differential Geometry (9/16/2004)
It’s typical in modern mathematics for otherwise straitforward concepts, theorems and definitions to be garbled beyond all recognition and left incomprehensible to all but those who already know their “true” meaning. Sometimes I think that these “correct” definitions allow misconceptions to creep in and find refuge in the minds of mathematicians. The inverse function theorem is a good example of this.
The inverse function theorem does not in fact give conditions for our ability to find an inverse function. As the text states, it places conditions on our ability to find a diffeomorphism between the domain and the range, i.e. find an inverse function which is itself differentiable.
For example, the inverse function for x^3, i.e. x^(1/3), exists everywhere, including at x=0. It’s not differentiable at x=0, but it is an inverse function. After reading the inverse function theorem, a lot of people might in fact think it wasn’t.
I think a lot of people don’t really understand the theorems they put in their books.
I agree with your sentiment– some writers do seem to drown simple ideas in layers of formality– but I don’t think this is because the authors are confused; it’s probably that they (wrongly) value conciseness and exactness over simplicity and clarity. All are desirable, but the latter should always prevail over the former.