# Which high-degree derivatives play an essential role?

Q. Which high-degree derivatives play an essential role in applications, or in theorems?

Of course the 1st derivative of distance w.r.t. time (velocity), the 2nd derivative (acceleration), and the 3rd derivative (jolt or jerk) certainly play several roles in applications. And the torsion of a curve in $$\mathbb{R}^3$$ can be expressed using 3rd derivatives.

Beyond this, I'm out of my depth of experience. I know of the biharmonic equation $$\nabla ^4 \phi=0$$. There is a literature on the solvability of quintics, but it seems this work is neither aimed at applications nor essential to further theoretic developments. (I am happy to have my ignorance corrected here.)

Q. What are examples of applications that depend on 4th-derivatives (snap/jounce) or higher? Are there substantive theorems that require existence of $$\partial^4$$ or higher as assumptions, but do not require (or are not known to require) smoothness—derivatives of all orders?

• I recall a professor from my undergraduate days who was working on modelling certain biological systems. He mentioned one model that used a 7th order differential equation. I don't recall any details now. Gerhard "Something About Blood Circulatory System?" Paseman, 2019.06.28. Jun 29 '19 at 0:18
• @GerhardPaseman: 7th-order! Impressive. Jun 29 '19 at 0:20
• @GerhardPaseman thing is, a 7th order equation in biology is likely to have been obtained from a first order equation for a vector with 7 coordinates, so I would take this with a grain of salt.. Jun 29 '19 at 11:23
• A similar question was asked many years ago at math.stackexchange.com/questions/71626/… to which I refer you for some more answers. Jun 29 '19 at 12:41
• @JosephO'Rourke, but that's really done only for convenience, As far as I can tell, you rarely need more than $C^3$ assumptions (which often can be reduced to $C^2$ if you're willing to be even more careful) to do differential geometry. Jun 29 '19 at 14:31

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 then the boundary of $$A+B$$ is only guaranteed to be '$$6\frac{2}{3}$$ times differentiable,' by which I mean $$6$$ times differentiable with $$6$$th derivative Hölder continuous with exponent $$\frac{2}{3}$$. This is known to be sharp.

• This is, to me, the most impressive occurrence of a derivative of high order so far.
– Dirk
Jun 30 '19 at 19:13
• Alternative link is mscand.dk/article/view/12183. Is it known what happens in higher dimensions? Is $6\frac{2}{3}$ the second element of a nice, or of an ugly sequence? Jul 1 '19 at 13:45
• Judging from the discussion at the beginning of this (arxiv.org/abs/1607.02753) more recent paper it seems like more is known about the smooth case, rather than the real-analytic case. The smooth case isn't as exciting. In 2D the Minkowski sum of two smooth convex sets can fail to be $C^{4+\varepsilon}$ for any $\varepsilon>0$ and in 3D or more it can fail to be $C^2$. As discussed in the original paper the issue in 2D is the presence of infinitely flat points. In the absence of infinitely flat points you get $6\frac{2}{3}$ again. Jul 1 '19 at 15:04
• I am also curious about the higher dimensional case. $20/3$ looks like a strange number, but maybe there is some structure in the optimal constant in $\mathbb{R}^n$ Jul 2 '19 at 7:40
• My mind was blown further when I clicked on the paper and saw that its abstract is written in Esperanto! Jun 22 at 2:42

Moser’s theorem in (1962) famously required estimates on the first 333 derivatives.

• “Incidentally, this number was later reduced by Russmann to $\ell \ge 5” — people.math.ethz.ch/~salamon/PREPRINTS/kam.pdf Jun 29 '19 at 2:41 • Impressive... does anyone have any slightest idea how come so many derivatives were necessary for the argument to proceed? Jun 29 '19 at 7:44 • This is an application of the so-called Nash-Moser iteration scheme, which is a generalization of the Newton iteration method and which Moser formulated, based on Nash's work on isometric embeddings. Many derivatives are lost in the iteration scheme used to prove the theorem. Jun 29 '19 at 14:25 There is a famous story that Richard Nixon once made use of the third time derivative to support his re-election, via a claim that the rate of increase of inflation was decreasing. http://www.ams.org/notices/199610/page2.pdf • I've heard this story before, but to the ordinary person, this statement doesn't necessarily refer to a third derivative (like how people use "rate of speed" to mean speed, and not acceleration, emphasizing that speed is a rate rather than intending to take the derivative of it). Has anyone actually looked at what inflation in the US economy was at the time Nixon made this statement to confirm that he was actually trying to confuse people into thinking that inflation was decreasing, when it was still increasing? Jun 29 '19 at 10:39 • Similarly Obama, “the arc of history is long, but it bends towards justice”:$J’<0$but$J’’>0$. It’s less optimistic than it appears. Jun 29 '19 at 15:22 • The question was "Q. What are examples of applications that depend on 4th-derivatives (snap/jounce) or higher?" Sad that we're allowing "humor" to trump quality answers. Hope this site doesn't go the way of Reddit. Jun 29 '19 at 17:51 • @RobertFurber, this story sounds apocryphal. Rossi did not provide any documentation, and I do not find any confirmation via Google search either. Jun 30 '19 at 9:38 • @MattF. : It's kind of late now, but Obama was quoting Martin Luther King Jr (or slightly paraphrasing, since King said "moral universe" instead of "history"). And King was quoting Theodore Parker. npr.org/templates/story/story.php?storyId=129609461 Jun 22 at 14:55 The error in Simpson's rule for integration is usually expressed in terms of the fourth derivative of the integrand. • Even better: arbitrarily high derivatives are used Taylor series approximations, and control on the size of the next higher derivative is needed to control the error. Jun 29 '19 at 0:30 • @StevenGubkin: That's not really better considering I don't really see people doing 51st-degree Taylor series, for example. The whole point was to get a realistic example. But in this case, I would've cited RKF45 which uses a 5th-order error estimator. Jul 1 '19 at 9:17 In "classical (Euler-Bernoulli) beam theory" the motion of a beam is modelled by the 4th-order PDE $$EI \frac{\partial^4 w}{\partial x^4} = -\mu \frac{\partial^2 w}{\partial t^2} + q.$$ • The 2-d analogue of Euler's elastica is the Willmore equation$\Delta_gH+2H(H^2−K)=0$, where$g$is the metric (of the underlying immersion),$H$is the mean curvature ($1/2$of the Laplacian for a flat metric) and$K$is the Gauss curvature, which is essentially the geometric version of the bi-Laplacian. This equation is also$4$th-order. Jul 1 '19 at 17:43 Nonlinear solitonic wave equations often feature high($$3^+$$) order of derivatives. The most famous one may be the KdV equation: $$\partial_t \phi + \partial_{xxx} \phi -6 \phi \partial_x \phi = 0$$. There is also the Boussinesq equation $$\partial_{tt} \phi - \partial_{xx} \phi -\alpha \partial_{tt} (\phi^2) - \partial_{xxxx} \phi = 0$$. • To add to this, Lax pairs (which these kinds of equations often have) can be used to easily form a hierarchy of high order PDEs with no restrictions on their order. Jun 29 '19 at 13:34 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 sign of this tensor. Unless the MTW tensor non-negative definite, the optimal map can fail to be continuous, even when the measures are smooth and satisfy a necessary convexity condition [2]. As a result of this, much of the regularity theory depends on the cost function being $$C^4$$, although it is possible to make some of the theory work when the cost function is $$C^3$$ [3]. For a more complete story, there is a survey paper of De Phillipis and Figalli which gives a good overview of the theory [4]. [1] Ma, Xi-Nan; Trudinger, Neil S.; Wang, Xu-Jia, Regularity of potential functions of the optimal transportation problem, Arch. Ration. Mech. Anal. 177, No. 2, 151-183 (2005). ZBL1072.49035. [2] Loeper, Grégoire, On the regularity of solutions of optimal transportation problems, Acta Math. 202, No. 2, 241-283 (2009). ZBL1219.49038. [3] Guillen, Nestor; Kitagawa, Jun, On the local geometry of maps with c-convex potentials, Calc. Var. Partial Differ. Equ. 52, No. 1-2, 345-387 (2015). ZBL1309.35038.. [4] De Philippis, Guido; Figalli, Alessio, The Monge-Ampère equation and its link to optimal transportation, ZBL06377770. • A tensor is named MTW and that doesn't stand for Misner, Thorne, and Wheeler. I am a bit surprised. Jun 29 '19 at 5:29 • Yeah, it's definitely a coincidence. The tensor is also called the "cost-sectional curvature," which might be easier to remember/google. Jun 29 '19 at 13:13 • @KConrad$-$while on the workweek calendar, MTW is dual to WTF . . . Jun 29 '19 at 14:01 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 laminar flame. There is a huge literature about it. 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 fundamental tool in geometric analysis. There are a number of important computations in the same spirit that sometimes require four derivatives (e.g. Simons' identity for minimal surfaces). 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 the Riemann curvature tensor, involve fourth derivatives of this function. As for arbitrarily high derivatives, one place these arise naturally is the study of Lie groups $$G$$ and their homogeneous spaces $$G/H.$$ For instance, one may ask if there is a "differential operator" acting on locally defined diffeomorphisms of $$G/H$$ whose kernel consists exactly of those arising from the left $$G$$-action. As noted by RBega2 above, for $$G=PSL(2, \mathbb{C})$$ and $$G/H=\mathbb{CP}^{1}$$ one is led to the Schwarzian derivative, a non-linear third order differential operator. This question was studied in general by Spencer in the 1960's resulting in the so-called Spencer complex. Nonetheless, both of these cases might be more high brow than the OP was looking for. • This is a good answer and I think it's worth noting the connection of your first example to optimal transport. In particular, the MTW tensor can be understood as a curvature tensor for a certain pseudo-Riemannian metric or (in some cases) as the orthogonal bisectional curvature of a particular Kahler metric. Jul 2 '19 at 12:55 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 to as the "jerk" and quantifies the smoothness of a motion. See http://courses.shadmehrlab.org/Shortcourse/minimumjerk.pdf article for a great description of jerk. In general, people will try to minimize the jerk to obtain a smooth motion for the robot. If position as a function of time is specified by $$x(t)$$, then the jerk is $$\dddot{x}(t) = \frac{d^3 x(t)}{dt^3}$$ 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 order $$3$$. But then is it not also true that a curve in $$\mathbb R^4$$ is described up to isometry by the derivatives up to order $$4$$. and a curve in $$\mathbb R^5$$ is described up to isometry by the derivatives up to order $$5$$. and so on? Similarly to Martin Fevre's answer: When routing a road for fast vehicles you need to minimize the rate of change of the road curvature so that the drivers wouldn't have to move the steering wheels in quick jerks. For example, a perfect transition from a circular roundabout to a linear road tangent to the roundabout would require an instantaneous steering wheel adjustment; that can be avoided by connecting the circle and the line with a ramp that has smoothly varying curvature. Since the curvature is essentially the 2nd derivative the rate of change of the curvature is the 3rd derivative, minimizing the rate of change of the curvature requires the 4th derivative. • I've always been fascinated by the paths in grassy areas that emerge from people crossing with their preferred optimization. About 10 years ago, I googled this idea and found a paper by some robotics researchers in which they described experiments sending people walking around and mapping their trajectories. As I recall, their paths were VERY well approximated by the paths that minimized the total variation in the curvature (just as you say). Jul 1 '19 at 22:09 What are examples of applications that depend on 4th-derivatives ... or higher? The Dirac equation is a system of four partial differential equations for four complex functions. However, in a general case, it is equivalent to one fourth-order equation for just one function (see references to my article). In geometric measure theory (GMT), many theorems require more than four derivatives. This is in part due to the use of Nash embedding theorem (which require $$C^3$$). We will give two examples. Almgren's big regularity theorem (1983, 2000) required the manifold to be $$C^5$$, but De Lellis and Spadaro (2014) managed to reduce the needed regularity to $$C^{3,\alpha}$$ for some $$\alpha>0$$ (they also simplified and shortened a lot the proof). In Almgren-Pitts theory (1976-present day), the original result of Pitts (1981) concerning the existence of one closed minimal hypersurface in closed Riemannian manifolds of dimension $$3\leq n\leq 6$$ required the manifold to be $$C^k$$ with $$k\geq \mathrm{max}\{4,n-1\}$$ (and the minimal hypersurface will also be of class $$C^k$$). • Quick question: For these theorems, is there a conjecture on what the minimal regularity is needed for these theorems? Jul 2 '19 at 14:43 • @Deane Yang. Thank you, this is an interesting question. First, as the Simon's identity is used, one definitely requires the manifold be$C^3$in Pitts's book. Likewise, the stronger curvature estimate of Schoen-Simon ((1981)(mathscinet.ams.org/mathscinet/search/…)) requires a$C^3$regularity (and Nash embedding theorem is used). So this is probably the best one can do. Furthermore, by reading more carefully some statements in Pitts's book, theorems such as the "Decomposition theorem" are stated for manifolds at least$C^5$. Jul 3 '19 at 13:42 A robotics application (similar to the one given by Martin Fevre) involves minimizing a function of the snap of a quadrotor's trajectory, where $$\mathrm{snap}(t) :=\frac{\mathrm{d}^4x}{\mathrm{d}t^4}$$; here is a reference. This is probably a stretch, but the theory of distributions is highly relevant to a large chunk of applied math and heavily relies on functions which have derivatives of arbitrarily high order. • For example, in microlocal analysis sufficient regularity of functions is often needed so that their Fourier transforms decay fast enough, which, in turn, is needed in order that certain integrals converge. Although the theory of distributions is always presented in the smooth category, I believe it's not necessary. As far as I know, derivatives of arbitrarily high order are never really needed in any application of distribution theory. Do you know an example of where they are? Jun 29 '19 at 14:12 • @Deane Yang: maybe Ruelle resonances? Given a hyperbolic map$T$, the eigenvalues of the composition operator$f \mapsto f \circ T\$ are not functions, but distributions; and in general, there are such eigendistributions of arbitrarily high order (so which have to be integrated against arbitrarily smooth functions). Jun 30 '19 at 19:53

I was once president of a PhD defense in chemistry (this is standard in our faculty; the president should not belong to the same department as the student). Most of his thesis involved simulations of behaviours of the seventh derivatives of something. It was claimed that this was testing for chaotic chemical reactions (certainly periodic reactions exist). The student however, was mathematically quite weak, so I don't really know how effective his results were.

I think third-order and fourth-order derivatives in particular are very common in both pure and applied mathematics. For example, I once worked with some equations modelling the local film thickness of a moving liquid film which contained third-order derivatives:

$$\delta q_{t} = \frac{5}{6}h - \frac{5}{2}\frac{q}{h^{2}} + \delta \Bigg( \frac{9}{7}\frac{q^{2}}{h^{2}}h_{x} - \frac{17}{7}\frac{q}{h}q_{x} \Bigg) + \frac{5}{6}hh_{xxx} + \eta \Bigg[ 4\frac{q}{h^{2}}(h_{x})^{2} - \frac{9}{2h}q_{x}h_{x} - 6 \frac{q}{h}h_{xx} + \frac{9}{2}q_{xx} \Bigg].$$

I agree with Dirk's answer that there exist also applications for arbitrarily high order derivatives. Another example is the infinite order Kosterlitz–Thouless phase transition

• Could you give a sketch of why an infinite number of derivatives are needed? Jun 29 '19 at 14:28

Analytic regularity of a $$C^\infty$$ function can be characterized by using an infinite number of derivatives. A function $$f\in C^\infty(\Omega)$$ where $$\Omega$$ is an open subset of $$\mathbb R^n$$ is real-analytic on $$\Omega$$ iff for all $$K$$ compact subsets of $$\Omega$$, there exist positive constants $$C_K, \rho_K$$, such that $$\begin{array}{ccc} \forall \alpha=(\alpha_1, \dots, \alpha_n) \in \mathbb N^n \,, & ~~ & \sup\limits_{x\in K}\left|(\partial_x^\alpha f)(x)\right| \le C_K \rho_K^{-\vert \alpha\vert} \alpha ! \,, \\[10px] \vert \alpha\vert=\sum \alpha_j \,, & & \alpha!=\prod \alpha_j! \,. \end{array}$$

• More generally, any theorem in the real analytic category necessarily involves derivatives of all orders. Jun 29 '19 at 18:54

To perhaps state the obvious, there are many places in theory (and, I'm told, practice) where analytic continuations are used. For instance, the Riemann zeta function was discovered as the analytic continuation of Euler's product formula.

Of course you can't make any sense of this without all the derivatives of a function.

• I think what OP is after is situations where you need, say, the 4th derivative, but you don't need the 5th or 6th or any other high-order derivative. Jul 2 '19 at 22:50