it's a function

# It's a function

distributions are generalized functions (so all functions are distributions, but not vice versa)

Neat, infinitesimals are intuitive and explain everything.

"n-no goy, you need to use these nonsensical limits that say the same thing but make it fifty times more convoluted!"

Whoever came up with and supported this limit bullshit should be rounded up and sent to prison camps.

It's just a linear functional, that is, a function. Fucking amerimutts after their silly """""""""""""calculus""""""""""""" courses thinking only [math]\mathbf{R}\to\mathbf{R}[/math] are functions, and any other map must be some spooky "generalised function"

It's a function [math]\delta: \mathbb{R} \times \mathbb{R} \to \{0,1\}[/math]

[eqn]\delta_{ij} = \begin{cases}

1,& \text{if } i=j\\

0,& \text{if } i\not=j

\end{cases}[/eqn]

He means the Dirac delta, the continuous analogue of the Kronecker delta.

It is you who needs to get out.

it's a function

She's a liar. You are a literal god at that IQ stage.

My friend literally learned Calc 4 material DURING the final and ended up acing it. He didn't even know there was a final this day and entered in the room by mistake. He's in the 120 range. He ended up with a A++ in the class but got expelled anyway because he wasn't even registered at this uni.

the unit impulse response function isn't a function

are you guys fucking stupid or something?

What do you mean by that? Measure is a function on some family of sets that admits values from range [math][0,\infty][/math], or [math][-\infty,\infty[/math] if we don't require it to be positive

He means the Dirac delta, the continuous analogue of the Kronecker delta.

Why would he (assuming OP is a "he") mean that?

Define [math]\delta: A\,\longmapsto\,\mathbf 1_A\left(0\right)[/math] and replace [math]\int_I f\left(x\right)\, \delta\left(x\right)\, \mathrm dx[/math] with [math]\int_I f\,\mathrm d\delta[/math]. Same as having [math]\delta\left(0\right)\ =\ \infty[/math] but less handwavy.

Who says it's a function? That's just plain stupid.

Clearly it's a functional, for variance.

They are, but not from reals to reals

They are functions in the same sense that covectors are functions. They're functions that are defined over function spaces, that is they take a function as input and THEN get a real number as output. The interpretation that Dirac delta is a function R->R is just a handy interpretation that intuitively works but is wrong strictly speaking. Just like most of distribution theory, which is just a bunch of formalisms that legalise physicists' intuitions.

Because the Kronnecker delta is a function, from Z^2 to {0, 1}. And is not memed about.

Welcome to the real world, where context (including societal context) matters

Since my first semester of undergrad, my definition of function was correspondance between sets that's uniquely defined, so the clarification is not needed, and to be honest, the functional itself is used by physicists mostly for compact notation or to skip formalism in QM. The only relevant part are green functions.

I wonder if any of you retards ever actually went through theory of distributions. Fucking kek, I'd be even surprised if majority of mathfags prowling this site did.

You're all small-time

Yes. As I said, technically speaking it IS a function, but when people talk about functions they usually mean "regular functions", that is, functions that map a number set to another, not things like Schwartz space covectors. It's like saying "The Earth is not spherical" - while technically it's closer to an oblate spheroid and you are technically correct, people assume what you mean to say it's flat.

And likewise when you say "function", you intuitively mean from R^n or C^n to R or something, not from Schwartz space to R.

There is a reason those have a specific name for them in order to differentiate them: "functionals".

How the fuck would we not? It's literally Calculus III, so second year of physics/astronomy at my Uni

I assumed anyone who has used the Dirac delta (that is to say, anyone who has QM, because that's where it appears), would be taught basic distribution theory

t.

Physicists most of the time do have analysis class that covers distributions, but I've heard from most math students that undergrad doesn't even cover PDEs in required courses, so I'd be surprised if they even did distributions theory, since you can't really build PDEs without it.

desu I highly doubt that distributions are part of most Calc 3 courses, they end up with surface integrals most of the time, in good unis with differential forms.

For me, it was:

Calc 1: from the definition to naturals based on an empty set to single-variable integral calculus

Calc 2: multivariable diff and integral calculus, beginning of differential geometry

Calc 3: more diff geometry, complex calculus, sprinkled with distribution theory

Calc 4 (optional): partial differential equations

University of Warsaw, for reference

Engineers use the Dirac Delta a fuckton and are certainly not taught distribution theory

Where is measure theory, ODEs, calculus of variations, metric spaces, series, differential forms, integral transforms ???

measure theory

There is not

ODEs

calculus of variations

...what?

metric spaces, series, differential forms

Calc 1

integral transforms

If you mean Fourier transforms, also Calc 3 (as the final part of complex calc). Otherwise there was none

Basically Calc 1-3 was a speedrun to cover everything we need for QM during the 4th semester.

And that's it for compulsory mathematics

Yep, PDEs are optional during undergrad physics here

Right, I forgot engineers. My bad

They are, but not from reals to reals

Yes, they are, and that's enough. Maps [math]\mathbf{R}\to\mathbf{R}[/math] aren't better than maps between arbitrary sets. If you say Dirac delta isn't a function [math]\mathbf{R}\to\mathbf{R}[/math] then it's true, but when retards say it's not a function but some spooky generalised function (without defining what the hell does that even mean) that's just beyond retarded

What the fuck, at University of Warsaw third and begining of fourth semester of analysis is measure theory, differential forms and complex analysis is not introduced untill analytic functions, which you take during 5th semester (and which wasn't mandatory untill this year if I'm not mistaken)

Where is measure theory

Third semester of analysis

ODEs

There is a course called ODEs, usually taken during 4th semester

calculus of variations,

Im not sure, really

metric spaces

Topology and third semester of analysis

series

Numerical series are first semester of analysis, functional series are the second

differential forms

Fourth semester of analysis

integral transforms

Analytic functions or functional analysis, 5th or 6th semester

This seems okay, except for lacking calculus of variations

This is absolutely not okay and your school is really shit.

Differential forms in Calc 1

I seriously doubt that you were learning about generalized stokes' theorem in Calc 1 bruv

How does

[math] \int_{-\infty}^{\infty} f\delta \mathrm{d}x = f(0) [/math]

make sense with this definition though?

I'll save you the work, it doesn't. That's why Schwartz had to create a whole new theory to make it work.

Then what function is [math] 2\delta [/math]?

Actually, [math] \delta [/math] IS a function, but not on [math] \mathbb{R} [/math] but on a space of test functions.

By the way - OP gave us a trick question. [math] \delta [/math] is just a Greek letter.

I'm talking about Faculty of Physics which definitely has complex calc during the 3rd semester because I just finished it last semester. It's all a lot slower at the Faculty of Mathematics, where you don't have to speedrun through all the things you need for QM so you can take your time

Oh fuck, I confused the names because I'm Polish and the course language was Polish. Diff forms were between 2nd and 3rd semester, as part of diff geometry.

It's the best one in my 2nd world country.

No idea how good or bad it is related to the unis abroad