108
votes

### Which high-degree derivatives play an essential role?

Given two sets $A$ and $B$ in $\mathbb{R}^n$, the Minkowski sum written $A+B$ is the set $\{a+b:a\in A,b\in B\}$.
If $A$ and $B$ are convex subsets of $\mathbb{R}^2$ with real-analytic boundaries ...

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63
votes

### Which high-degree derivatives play an essential role?

Moserâ€™s theorem in (1962) famously required estimates on the first 333 derivatives.

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42
votes

### Which high-degree derivatives play an essential role?

There is a famous story that Richard Nixon once made use of the third time derivative to support his re-election, via a claim that the rate of increase of inflation was decreasing.
http://www.ams.org/...

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39
votes

### Which high-degree derivatives play an essential role?

The error in Simpson's rule
for integration is usually expressed in terms of the fourth derivative of the integrand.

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39
votes

### Which high-degree derivatives play an essential role?

In
"classical
(Euler-Bernoulli) beam theory" the motion of a beam
is modelled by the 4th-order PDE
$$
EI \frac{\partial^4 w}{\partial x^4}
= -\mu \frac{\partial^2 w}{\partial t^2} + q.
$$

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30
votes

### History of differential forms and vector calculus

V.J. Katz in History of Topology:
Although Cartan realized in 1899 [1] that the three theorems of vector calculus
(Gauss, Green, Stokes) could be easily stated using differential
forms, it was ...

30
votes

### Which high-degree derivatives play an essential role?

Nonlinear solitonic wave equations often feature high($3^+$) order of derivatives.
The most famous one may be the KdV equation: $\partial_t \phi + \partial_{xxx} \phi -6 \phi \partial_x \phi = 0$.
...

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25
votes

### Which high-degree derivatives play an essential role?

In optimal transport, there is a quantity known as the MTW tensor [1] which depends on the fourth derivatives of the cost function. The regularity theory of transport depends in a crucial way on the ...

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22
votes

### Why is there no symplectic version of spectral geometry?

From a certain point of view the premise of the question is wrong. The study of sympletic manifolds with no additional structure is akin to differential topology rather than differential geometry. ...

20
votes

Accepted

### Why is there no symplectic version of spectral geometry?

The characteristic variety (i.e. vanishing locus of the symbol) of a symplectomorphism invariant scalar differential equation is a real projective hypersurface invariant under the group of ...

19
votes

### Method of characteristics for higher order PDEs in more than two variables

I hope to use this answer to convince you that in general the method of characteristics cannot work for higher order PDEs in more than 2 variables. Nevertheless, there are some ideas in PDEs that are, ...

18
votes

### Which high-degree derivatives play an essential role?

The Kuramoto-Sivashinsky equation
$$\partial_tu+\Delta^2u+\Delta u+\frac12|\nabla u|^2=0$$
where $\Delta$ is the Laplace operator (second order) was derived to model diffusive instabilities in a ...

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18
votes

### Analytic functions where all derivatives vanish at infinity and which are bounded

Yes.
Let $\phi$ be any smooth function with compact support on the interval $[-1,1]$.
Set $f$ to be the inverse Fourier transform of $\phi$.
Since $\phi$ is in Schwartz class, so is $f$, and all of ...

16
votes

Accepted

### Projective-invariant differential operator

There's a straightforward abstract answer that you may not like, but, because it clarifies your question and explains a uniform way to answer similar questions, I'll sketch it here.
First, consider a ...

16
votes

Accepted

### Hodge decomposition of smooth n-forms: is it an isomorphism of topological vector spaces?

Maybe an even more elementary argument than the one of Tobias:
The continuity of all involved operators is easy: simply all differential operators with smooth coefficients between sections of vector ...

15
votes

### Which high-degree derivatives play an essential role?

In Kahler geometry, and many other places, (Riemannian) metrics are often (sometimes) prescribed via a potential function, which involves two derivatives. Therefore, the fundamental invariants, like ...

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14
votes

### Which high-degree derivatives play an essential role?

Third derivatives (and higher) show up naturally in the study of affine and projective geometry. Perhaps most notably in the Schwarzian.
The Bochner formula also involves three derivatives and is a ...

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14
votes

### Which high-degree derivatives play an essential role?

As a roboticist, I always pay close attention to the 3rd derivative of position with respect to time when generating a motion for a given robot. The 3rd derivative of position is most often referred ...

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14
votes

Accepted

### The principal symbol as an element in the K-theory

It's a bit easier to see this using a slightly non-standard definition of topological K-theory. Given a locally compact Hausdorff space $X$, let $\bf{E}$ be a complex of vector bundles, i.e. a ...

13
votes

Accepted

### Relation between harmonic vector field and harmonic 1-form

The two notions are related, but they are not the same.
The condition for a unit vector field $X$ on a Riemannian manifold $(M,g)$ to be harmonic is not the same as the condition that the dual $1$-...

13
votes

### Eigenvalues of the Laplace-Beltrami operator on a compact Riemannnian manifold

A modern "simple philosophical" explanation is that this problem can be restated as an eigenvalue problem for a compact operator in an appropriate Hilbert space, whose eigenvalues are reciprocal to ...

13
votes

### Differentiability of operator norm

It need not be differentiable everywhere. Let $P$ and $Q$ be mutually orthogonal self-adjoint projections. Then the norm of $P+ t Q$ is 1 for $|t| \leq 1$ and $|t|$ for $|t| > 1$.
However, $\|A + ...

13
votes

### Hodge decomposition in elliptic complexes

Yes, there is a Hodge decomposition for elliptic complexes on compact oriented Riemannian manifolds.
$L^2$-version. Let $(M,g)$ be a compact oriented Riemannian manifold and let
$$
0 \to \Gamma(E_0) \...

12
votes

### Harmonic spinors on closed hyperbolic manifolds

I'm happy to be able to answer my own question! John Ratcliffe, Steven Tschantz and I showed that the Dirac operator on the Davis manifold (a closed hyperbolic 4-manifold constructed by Mike Davis) ...

12
votes

Accepted

### Boundary terms of formal adjoints of differential operators

Yes, there is a generalization of such a "boundary term" for an arbitrary linear differential operator (smooth coefficients assumed, of course). My favorite way to define the formal adjoint $D^*$ of a ...

12
votes

Accepted

### Semantics of derivations as derivatives

In all of these contexts, derivations are infinitesimal automorphisms, in the sense that $D$ is a derivation on $A$ (an algebra, a Lie algebra, etc.) iff $\exp(Dt)$ is an automorphism of $A \otimes k[...

12
votes

Accepted

### Self-adjointness and choosing appropriate function spaces

The question of self-adjointness is quite often all about the boundary conditions. In order to get the domains of the operator and its adjoint to match, boundary conditions need to be 'distributed' ...

12
votes

Accepted

### Surjectivity of differential operators with constant coefficients

Here is another approach. Let $R$ be a non-zero homogeneous polynomial of degree $n$. We want to show that the mapping $Q\mapsto R(\partial)Q$ is surgective from $V_{m+n}$ to $V_m$ where $V_k$ is the ...

12
votes

### Line graphs called "graph derivatives": any intuition?

If one considers a graph $G=(V,E)$ and a function $f:V\to\mathbb{R}$, it makes sense to look at the finite differences $f(v_i)-f(v_j)$ for neighboring vertices $v_i,v_j\in V$ as a sort of discrete ...

11
votes

### Which high-degree derivatives play an essential role?

The OP says
And the torsion of a curve in $\mathbb R^3$
can be expressed using 3rd derivatives.
More generally,
a curve in $\mathbb R^3$ is described up to isometry by the derivatives up to ...

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