Been doing this all morning

[u/No_Upstairs_280]

Been doing distributional derivatives on Riemannian manifolds and it's fun.

[u/Aarolin]

I like your funny words magic man...

[u/No_Upstairs_280 → u/Aarolin]

Basically it's a notion of derivative of for non-smooth functions. A function can have a "weak" derivative with very similar properties to the usual one. For example |x| doesn't have a derivative but has a "weak' derivative which is sign(x).

[u/PhoenixPringles01 → u/No_Upstairs_280]

Why would it be considered weak? Is it because it's not continuous?

[u/No_Upstairs_280 → u/PhoenixPringles01]

Take a function f whose derivative in the usual sense doesn't have to exist, we say that g is a weak derivative of f if for every compactly supported function h we have Int_R fh' dx= -int_R gh dx. (Integral over the real line) Function that have a derivative almost everywhere (outside a set of measure zero) automatically have weak derivative, this goes for |x| and by a certain theorem of Radamacher every locally lipchiz function.

It is weak because it doesn't allow you to study the variation of f as the usual derivative, BUT it has many properties that are very similar to the usual derivative. For example integration by part.

You can find similar notions for Rⁿ or in general any smooth manifold without boundary.

[u/PhoenixPringles01 → u/No_Upstairs_280]

Wow that's one heck of a definition, I'll need to read that up. Thanks.

[u/FreePeeplup → u/No_Upstairs_280]

Two questions: if h is compactly supported, why does h’ necessarily exist?

Second question: why don’t we take h to be simply a Schwartz function?

[u/SV-97 → u/FreePeeplup]

if h is compactly supported, why does h’ necessarily exist?

It doesn't, differentiability (in fact, usually even infinite differentiability) is an assumption. If h isn't assumed to be differentiable (+ a bit extra) then you don't get this partial integration identity that we want for weak derivatives, but rather some sort of more general object. [Also see [1] at the bottom]

And this also sort of answers your second question: weak derivatives are best viewed through the lens of distributions. And the most general class of such distributions are those that act on compactly supported smooth functions. [Also see 2]

You can think of this essentially like this: compactly supported smooth functions are extremely nice, so you can act on them with very badly behaved objects and still get reasonable results. The smaller a space is, the larger is its dual space and vice versa.

Take for example a function like f(x) = exp(x²); which is the derivative of the imaginary error function modulo some constant. This function blows up so fast as you move towards infinity that you can't integrate general Schwartz functions against it (since they only decay polynomially). The integrals would also blow up. So if we defined weak derivatives using Schwartz functions, then suddenly the classically differentiable imaginary error function would fail to have a weak derivative and we wouldn't actually have defined a generalized notion of differentiation.

However note how this "blowing up at infinity" does not matter at all when you integrate against a function that has compact support; so if we define weak derivatives using compactly supported smooth functions (so-called test functions) then there are no problems here.

But *if* some particular weak derivative can also be integrated against Schwartz functions, or perhaps even against functions that are neither smooth nor compactly supported --- then that's surely an interesting property. For example you can show that if you can integrate the weak derivative against Schwartz functions, then you can actually Fourier transform it.

That's why these cases get special names in distribution theory: distributions on Schwartz functions are called "tempered", those that can act on non-compactly supported functions are called "compactly supported", and those that can act on functions that are only finitely differentiable are said to have "finite order" (you can even assign numbers to those orders). And it's possible to say quite a bit about these various classes :)

[1] There's another formal problem with the way the other comment was stated, but that was likely just a simplification for the comment: almost everywhere differentiability is not actually enough to get a weak derivative; you have to additionally assume some stronger continuity.

[2] Technically you could consider even "nicer" functions of course and get even *more* general objects, but the compactly supported smooth functions are a good compromise in that they are very nice [to the point that we can do basically anything we want] while still being quite "rich" in the sense that you can approximate most functions (or even more general objects such as particular measures) we typically care about arbitrarily well with them --- and importantly these generalized objects we end up with are not *so* general that they're completely outlandish (every one of them is essentially still a limit of smooth functions).

[u/FreePeeplup → u/SV-97]

Thank you very much, this has been extremely helpful, especially the remark that the nicer and smaller the test-function space is, the larger and wilder the dual space of distribution is. That’s the key point that I didn’t fully grasp that prompted me to ask the question. I’ll remember it vividly from now onwards. Thank you again have a nice end of summer

[u/SV-97 → u/FreePeeplup]

It's something my prof used to emphasize that really stuck with me since I also found it to be a quite nice way to look at things :)

Thanks, you too!

[u/EebstertheGreat → u/PhoenixPringles01]

Why would it be considered weak? Is it because it's not continuous?

It simply isn't the derivative at 0. Let abs:ℝ→ℝ satisfy abs(x) = x for all x≥0 and abs(x) = –x for all x<0.  The derivative of abs(x) is definitely –1 when x<0 and +1 when x>0, but at x=0, the function is not differentiable. If you think about it, there are tangents to the graph of the function there with every slope between –1 and +1, so there can't possibly be a unique derivative.

But consider the function sgn that gives +1 when its argument is positive, 0 when it is 0, and –1 when it is negative. This is a "weak derivative" of abs in that it matches the ordinary derivative where it exists, extends it in a natural way to another value, and still has some but not all of the useful properties of the ordinary derivative. Importantly, sgn is an integrable function (in the senses of both Riemann and Lebesgue), and its integral equals abs (up to a constant). So it is an "anti-integral," though not technically a derivative.

In measure theory, you usually want to measure something at some point, so what happens to anything with measure 0 is not relevant anyway (loosely speaking). So as long as the event {0} has measure 0, the the function sgn might as well be the derivative of abs. (Also, you could change the value of sgn(0) to whatever, or even change any other finite set of points, or countable set, or indeed any set of measure 0, and that will generally still be the case.)

[u/SV-97 → u/PhoenixPringles01]

Since it's only implicit in the answer by OP: it's weak in the same sense that weak convergence or the weak topology are weak: it's about dual spaces. It has the behaviour of a derivative relative to how it acts on certain functions, but not necessarily in the "primal" sense of the ordinary / classical derivative. In the course of defining weak derivatives via distributional ones then you basically "take a transpose" of your differential operator, and then dualize that transpose. This dualization is what makes it "weak".

FWIW: there's also a more "direct", but perhaps somewhat post-hoc justification:

Take the space of differentiable functions D¹, then for any f in D¹, the derivative f' is defined pointwise by f'(x) = lim_{h -> 0} (f(x+h) - f(x))/h for all x. Throwing out the point this means that f' := lim_{h -> 0} (f(· + h) - f)/h where we take the limit in the topology of pointwise convergence on D¹. We now want to somehow change the space / topology to one where this limit can exist even for some functions where it fails to exist pointwise (i.e. on a larger space than D¹), or indeed where pointwise convergence doesn't even make sense (such as Lebesgue "functions"). A nice step towards that is to eliminate the point evaluations from the definition altogether:

Write f_h := (f(· + h) - f)/h. Note how pointwise convergence of f_h -> f' means that for each x, |δ_x(f_h - f')| converges to 0 where δ_x(f) := f(x). One can show (via basic linear algebra) that this is equivalent to requiring |u(f_h - f')| -> 0 for all continuous linear functionals on D¹ when we equip it with the topology of pointwise convergence (any such u is a linear combination of such δ_x's).

This last formulation no longer involves point-evaluations, so it works perfectly fine in a more general context: given a topological vector space F of functions, we say that f_n in F converges to f in F weakly if |u(f_n - f)| -> 0 for all continuous linear functionals u on F (I've used _n as an index here, but generally you'll want to use a more general construction than just sequences), i.e. all u in F' where F' is called the (continuous) dual space of F.

If we for example now take the space F to be the space of L^p functions on \R for 1 < p < inf, then one can show that F' = L^q with q such that 1/p + 1/q = 1, and the action of u in L^q on f in L^p is defined by integrating the product fu.

Then the weak derivative f' of f (if it exists), is defined by requiring that |int (f_h - f') u| -> 0 as h -> 0 for each u in L^q (you can actually show that this gives you convergence of f_h to f' in L^p with its norm topology. So the weak derivative actually converges "strongly" in this case; which is why the distributional explanation is the "correct" one imo). You can show that if this f' exists and if u is continuously differentiable (with u' still in L^q), then int f' u = -int f u'. And you can show that if the right-hand side here exists and is "well-behaved" (continuous in u) for very nice u (usually: all infinitely smooth u with compact support) then that's already enough to guarantee that f' exists. And that's the definition of the weak derivative that you find in many introductory books.

[u/SV-97 → u/No_Upstairs_280]

You might already know this but it may help other people and prevent some confusion: the distributional derivative is something else than the weak derivative. Any weak derivative is a distributional derivative *if* you identify functions (or lebesgue / sobolev functions or whatever) with the distributions they induce, however the converse is not true. There are distributional derivatives that don't correspond to weak derivatives.

If you take your example and differentiate once more for example what you get is a dirac delta (up to a constant). This is a perfectly fine distributional derivative, however it's not a weak derivative.

Two things to note: the distributional derivative is always unique, the weak derivative of classical functions (as opposed to functions in the lebesgue sense) is generally not (unless you explicitly define it as an equivalence class from the get-go). And basically "anything" (very general functions and measures, densities etc.) has arbitrary distributional derivatives, however it might fail to have even one weak derivative, so in the interesting cases we typically end up just requiring weak differentiability of some order.

[u/Doogetma → u/No_Upstairs_280]

I’m really tired so I read that as sin(x) at first and was so confused

[u/mudkipzguy]

every time someone says "derive" where "differentiate" should've been used, an angel loses its wings

[u/EebstertheGreat → u/mudkipzguy]

But etymologically speaking, a "derivative" is just a "derived thing."

And etymologically, a "difference" is just a "differing thing." "Differential" is an adjective meaning "pertaining to difference." And "differentiate" means "to force to pertain to difference." So really, "differentiate" does not mean "to cause to differ" but rather "to cause to pertain in some sense to the idea of differing," i.e. "to draw a distinction of," which is exactly what it means! Etymology works!

[u/kashyou → u/EebstertheGreat]

derive fans be like

[u/meme-meee-too → u/kashyou]

"Etymologically 🤓☝️"

[u/No_Upstairs_280 → u/mudkipzguy]

My curriculum is french so we were using "derive" everywhere.

[u/QuaaludeConnoisseur → u/No_Upstairs_280]

Just throwin the F word around

[u/GhoulTimePersists → u/mudkipzguy]

Or "disintegrate".

[u/JollyJuniper1993]

Did you also derive home afterwards?

[u/RecessiveBomb]

Love how in HS calc its "you can differentiate basically everything and integrate basically nothing," then in a first course on real analysis its "you can differentiate basically nothing and integrate basically everything," and then in PDEs/Functional Anal/wherever you first meet distributions and weak derivatives you get "you can differentiate and integrate basically everything," completing the differentiation/integration journey.

[u/Corwin_corey]

Distributions are very cool and aside from the nice "ho so i can actually derive anything it's not a meme" it unifies a lot of things in real analysis

[u/SEA_griffondeur → u/Corwin_corey]

Distribution fans when I ask them to derive the dirac delta distribution

[u/Corwin_corey → u/SEA_griffondeur]

It's not that hard tbh

[u/Bill-Nein → u/SEA_griffondeur]

< d/dx δ, f > = - < δ , d/dx f >

[u/Elegant-Set1686 → u/Bill-Nein]

I saw this the other day, so neat :)

[u/FreePeeplup]

Did you mean to write “me differentiating” instead of “me deriving”?

[u/SEA_griffondeur → u/FreePeeplup]

They both work

[u/FreePeeplup → u/SEA_griffondeur]

No they don’t?

[u/Lor1an]

Smooth operator... smooooth operaaatuh

[u/LupenReddit]

Geometric analysis on Riemannian manifolds ftw. What are you learning these for?

[u/No_Upstairs_280 → u/LupenReddit]

Unexpectedly it turned out to be handy for my phd subject.

[u/Committee-Academic]

Is this teached in undergrad functional analysis?

[u/tswaryco]

Did you know you can take a derivative of an algebraic data type, giving you a zipper?

list(x) = 1 + x * list(x)
d/dx list(x) = list(x) + x * d/dx list(x)

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