/sci/ - Science & Math

>>7281432
Riemann ftw

Lebesgue for working on things besides Euclidean space

Stochastic integral

>>7281432
>no dx

>>7282036
>implying

>>7282217
>implying integrals aren't the summation of dx times the function for its value from a to b

topkek get out of here until you learn the only real analysis worth learning

>>7282036
<div class="math"> \int _{ \{ \lambda a + (1-\lambda)b | \lambda \in [0,1] \} } \mathrm{d} f</div>

>>7282219
>summation
>dx
>analysis

>>7282219
>dx

that just gave me cancer

>>7282424
>>7282418
>what are infinitesimals/hyperreals

>>7282439
physicist cancer

>>7282451
standard analysis considered harmful

>>7282220
;_;
Never4get the biology integrals

What is the whole deal with needing/not needing a dx in an integral? I've heard it said that it's technically not necessary, but I thought it represented what you're integrating with respect to.

>>7282464
It is necessary if there is ambiguity, if you are working with multiple variables. For example in classical mechs "The Integral of Force" could be work or impulse, you need to specify dx or dt to indicate what you are integrating over

>>7282464
>I've heard it said that it's technically not necessary, but I thought it represented what you're integrating with respect to.
That's basically correct.
<div class="math">\int_S \mathrm{d}\omega = \int_{ \partial S }\omega</div>
(see e.g. Baby Rudin)

Obviously Feynman Path Integrals

<span class="math">
\int {Dx}
[/spoiler]

>>7282475
mathematicians love them

Barnett Triple Integrals

>>7282219
>implying integrals aren't the summation of the 1-form fdx evaluated on a parallelogram anchored at gamma(u) and spanned by Dgamma(u)

>>7282464
What it means and what it's representing depends largely on your definitions, level of formality, and how you're constructing the reals.

In calculus prior to the 1900's mathematics was informal and people intuitively talked about infinitesimal real numbers even though they knew the concept was flawed. In these times the dx represented an infinitesimal portion of x.

Then in the early 1900s we got formal logic and the epsilon-delta approach to analysis. At this point infinitesimals were cast out and the dx was either removed or used to only refer to notation.

Then in the 1960's Abraham Robinson for the clever idea to fix the problems with infinitesimals by instead making it so that the infinitesimals are non-real numbers, hyper-reals to be precise (of which the reals are a proper subset). In this case the d is an infinitesimally small hyper-real and the math becomes way easier and more intuitive.

At some point people also worked out differential forms and the dx are used to refer to those.

There have also recently been a couple constructive approaches to analysis where infinitesimals are introduced in different ways with different properties.

>>7282220
>prince integral
holy shit

The daniell integral gets me all excited. Probably because I learned measure theory from a book that spent waaay too long building geometric intuition.

>>7282029
this

>>7282220
>prophase metaphase anaphase telophase

my fucking sides

>>7282439
So you're saying that to understand the integral, you have to understand non-standard analysis?
You're an idiot who doesn't know what he's talking about

>>7282220
>Prince

>ctrl+f
>no Kurzweil-Henstock/Perron-Denjoy integral
>no Ward, Radon, Stietljes or Darboux integral

/sci i'm disappoint

>>7281432
all integrals are riemann integrable.

[get the fuck out of here engineer.]

>>7285894
That's not what he said. Are you literally retarded? He was just justifying the use of dx in the notation

>>7285998
Get the fuck out of here engineer. (can I interest you in the riemann-stieltjes integral? It's basically a generalized contour integral for real functions)

>>7286011
>generalized contour integral for real functions
If you mean real functions defined on the real line.

<span class="math">\int \int \int \int \int \int[/spoiler]

You'll normaly see six fold integrals in quantum physics and other areas in theoretical physics; depends on how your unis teach it. Minskowski Squares could also be used to solve nth dimensional integrals.

Just kidding :^)

>>7286470
And when you get to string theory you get 26 dimensional integrals.

>>7286474
five tripe integrals are needed to solve the flux of plasma membranes surging through a cats intenstinal tracks (with is normally used in mathematical veterinary biology)

Higher dimensional integrals tends to solve multitudes of problems.

>>7286483
I doubt those faggots do it by hand. Biologists(and all related) and the mathematicians who enable them are scum.

>>7286483
Yeah but in bosonic string theory its not a meme. You actually work in 26 dimensions.