Recall
- is differentiable at if such that .
- if is everywhere differentiable and is continuous.
- As a matrix, the entries in are the partial derivatives .
- Operator norm: .
Rules of differentiation
Usual rules of differentiation hold; in particular:
Theorem 24.1 (Chain rule). If is differentiable at and is differentiable at , then is differentiable at and
Proof. where (i.e. ), so
∎
The mean value theorem doesn’t hold: e.g. , has . However we have the mean value inequality:
Theorem 24.2 (Mean value inequality). differentiable at every point of the line segment
(This follows from the single-variable case by considering on .)
Higher-order derivatives
is if is , etc. The main important fact about higher partial derivatives is:
Proposition 24.3. If the exist and are continuous, then .
Proof. Enough to consider the case of . For and small, , consider
Writing this in terms of , this is , so by the mean value theorem for , such that this equals
In turn, by the mean value theorem for , such that this equals .
Doing the same calculation in the opposite order shows it also equals for some , . Since these 2nd derivatives are continuous by assumption, taking limits as gives the result. ∎
Hence: the Hessian matrix is symmetric, and should be interpreted as a symmetric bilinear form on tangent vectors. If then
(and so on, Taylor!).
The inverse function theorem
Because of the local approximation , the behavior of governs that of near . In particular:
- if is injective then is injective on a (sufficiently small) neighborhood of ;
- if is surjective then maps a neighborhood of surjectively onto a neighborhood of .
When both hold, is a local diffeomorphism, by the inverse function theorem.
Definition 24.4. A map between open subsets of is a diffeomorphism if it is a homeomorphism and both and are .
Theorem 24.5 (Inverse function theorem). Let open, , and suppose is an isomorphism (i.e. ). Then is a local diffeomorphism at , i.e. a neighborhood such that is a diffeomorphism between and .
The proof uses two main ingredients:
- The mean value inequality: .
- The contraction mapping principle: a complete metric space, a contraction ( for some ) has a unique fixed point.
Proof of the contraction mapping principle. Existence: let , set ; then , so , so is Cauchy, hence converges to some . Moreover , but , hence .
Uniqueness: if and then , so . ∎
Proof of the inverse function theorem. After a linear change of variables, we can assume , , .
Since , is continuous, so a ball such that for .
Now, given , let . (“The next guess using Newton’s method to find such that , given , using .”)
Key observation: iff , and for we have .
Assume . Since and for , the mean value inequality gives, for , , and also
So is a contracting map from to itself, hence fixed point (by , ). Thus
Now let and ; then , are open ( is continuous) and is a bijection by . Let be the inverse map.
Claim: is differentiable and , where ().
Proof of the claim. Fix , , and let as above, with . For small (so ), write , so . Then
But we’ve shown is contracting: . Hence by the triangle inequality, i.e. .
Given , such that . So for ,
Applying : for ,
Recalling , this yields , proving the claim.
The continuity of then follows from the continuity of and of itself. ∎
The implicit function theorem
Theorem 24.6 (Implicit function theorem). open, , , continuously differentiable ().
Write as , where (first variables) and (last variables).
Assume and is invertible () at . Then open , such that such that .
Moreover, defined by is differentiable, and .
Proof. This follows from the inverse function theorem by considering
This has an inverse over a neighborhood of . Near ,
So we let = the second component of . ∎
Given a differentiable , and a point at which is surjective, we can always find a subset of coordinates (, ) such that the corresponding part of is invertible we can apply the implicit function theorem to describe the zero set of by equations .
In particular, a hypersurface = a closed subset which is locally the zero set of a differentiable real-valued function with . Using the implicit function theorem, can be locally described as the graph of some differentiable . E.g. a differentiable curve in is locally a graph or .
Notes not reached in this lecture, covered at the start of Lecture 26
Iterated and Riemann integrals in several variables
a continuous function on an -cell we can define
(why ? clearer after differential forms) either
- as an iterated integral:
or in any order;
- as a Riemann integral: split into small cubes , and bound between the piecewise constant functions on and on , giving the estimate
If is continuous, hence uniformly continuous, then as , so this defines the integral uniquely.
Fubini’s theorem says: for continuous , the iterated integrals for different orders of integration are all equal.
If is only piecewise continuous, integrability still holds if the regions of where is continuous are sufficiently regular—e.g. delimited by smooth hypersurfaces. Specifically: when decomposing into small cubes , we want
as one subdivides further. Over such cubes, doesn’t as the step size , but if vol we still have .
Thus we can define integrals over regions of delimited by hypersurfaces, either by
- extending by outside of the given region, and integrating the resulting piecewise continuous function, or
- using changes of coordinates (via the implicit function theorem) to make the region of integration an -cell. This requires change of variables…