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Name a better trio.

I'll wait.
>>
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I am you but stronger
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>he thinks these are three different operations
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>>9412252

>trivial babytier physishit applications

Here come's a real trio actually worth something, try not to shit your pants.

>Banach–Steinhaus theorem
>Banach–Schauder theorem
>Hahn–Banach theorem
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>>9412252
d, d and d.
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>>9412298
You forgot the most important part
[math]\star[/math]
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>>9412302
No. Fuck off with your Hodge star operator.
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>>9412305
Don't you want to construct an inverse to your exterior derivative, anon?
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>>9412292
>look at me im so smart!
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>>9412323

Yeah, brainlets can't into baby tier functional analysis. Knowing undergrad math will certainly make me look smarter than people posting HS math!
>>
Z
F
C
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>>9412252
The Analytic, Arithmetic, and Topological applications of Elliptic Curves.
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>>9412292
>try not to shit your pants
>elementary functional analysis
cringe
>>
>>9412352
that's a duo tho
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>>9412318
>inverse
You mean dual?
>>
>>9412274
BORN TO SR
THE WORLD IS A R4
LORENTZ TRANSFORM THEM ALL 1905
I am 4-vector man
410,757,864,530 FRAMES OF REFERENCE
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>>9412379
well, "inverse" in big quotes.
The codifferential.
>>
sine, cosine, tangent
send my prize money in the mail
>>
Why are there no good vector calc books?
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>>9412423
"Div, Grad, Curl and All That" by H.m. Schey. Short, concise, understandable, highly useful, and in general very well written.
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>>9412428
I want it rigorous god dammit.
>>
>>9412423
>>9412430
Spivak Calculus On Manifolds
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>>9412372
AC is independent from ZF
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>>9412457
ZF, AC -> two things, i.e. a duo
>>
I am OP, and i dont understand that shit at all. but they look and sound cool.

gimmie like a month and i will understand it tho
>>
>>9412252
If you're wondering as to why
>>9412277
>>9412298
are calling these operators the same is that they basically all converge when looking at differential forms, namely one has
[eqn]\matrix{ \Omega^0 & \stackrel{d}{\longrightarrow} & \Omega^1 & \stackrel{d}{\longrightarrow} & \Omega^2 & \stackrel{d}{\longrightarrow} & \Omega^3 \cr
\uparrow & & \uparrow & & \uparrow & & \uparrow \cr
\Omega^0 & \stackrel{\mathrm{grad}}{\longrightarrow} & \mathcal{X} & \stackrel{\mathrm{curl}}{\longrightarrow} & \mathcal{X} & \stackrel{\mathrm{div}}{\longrightarrow} & \Omega^0 \cr}[/eqn]
This math stacks answer goes into more details
https://mathoverflow.net/questions/10574/how-do-i-make-the-conceptual-transition-from-multivariable-calculus-to-different?noredirect=1&lq=1
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>>9412292
>Hahn–Banach theorem
Made me literally shit my pants desu
The Hahn Banach theorem is why I gave up math
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>>9412380
You don't know Graham then. He's ma boi!
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>>9412423

"Calculus, vol II" by Tom Apostol gives the best introductory approach.

Beyond that you want something on manifolds
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>>9412454

>"Calculus on Manifolds"
>Only defines "manifold" in the last chapter
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>>9412252
>Completeness
>Consistency
>Effective Axiomatization

You can't argue this.
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>>9413124
The protocol defines the rules syntax, semantics and synchronization of communication and possible error recovery methods.

#Acceptable values
# 1 = 1 = This usually means "I am already here/there/X" = This is the same as binary 1
# 2 = Even = This means "I am aware I am not one, but I believe it is achievable to get there from here" = This is the same as binary 0
# 3 = Prime = This means that you defer your preference for this field = This is the same as boolean "True"
# ? = Unknown = This means the value is undefined/abstained/withheld = This is the same as boolean "False"

#Eval Special Case Modification Code
##When EP/PP match, then PP counts as +1 towards grand total P
###Also calculates the grand total, was more performant to include it in this loop.
adjust = dict()

#Positional-Priority is the position that will be respected, to the exclusion of all others
#Evaluation-Priority is where computation of the problem space is locatable
#Position Priority > Evaluational Priority < Mass/Largest Priority

#Priority != None if Ident = None
#Ident != None if Priority = None
#Eval can always equal None

#If Only(Priority, Identity/Resolution):
# Translate_Problem_Space(The_Problem)
>>
What the difference between a derivative and a differential
>>
>>9413203
kek
>>
>>9413213
derivative is a rate, and a differential ain't
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>>9413213
You can't wash your hands in a derivative
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>>9413213
A differential is something which sets one apart.

A derivative is merely an uninspired replication.
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>>9413094
OP here again. Can someone explain what this theorem means in layman terms
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>>9413109
It doesn't even define a full manifold, it defines submanifolds of R^n.
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>>9413695
so it defines all manifolds...
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>>9413700
Manifolds are object independent of an ambient space.
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>>9413708
all manifolds are submanifolds of R^n (up to isomorphism obviously)
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>>9413715
>all n dimensional vector spaces are R^n !
Yes, but it's important ti show you can define these objects independent if that to discoverr intrinsix properties and shit.
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>>9413738
all ("intrinsic") properties are independent of isomorphism
>>
[math](X,\mathcal{F},\mu)[/math]
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>>9413906
good looks
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>>9413906
Measure space? Why us F for sigma algebra¿
>>
>>9412252
Can someone explain me what operators are? I'm a brainlett
>>
>>9413972
"operator" is a brainlet name for a function in a space that isn't super simple
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>>9413974
nigga what
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>>9413715
By means of a nontrivial theorem.
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>>9414024
how fucking enlightening, thank you very much for this very useful post that no one thought about
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>>9414029
Manifolds and submanifolds of R^n are not a priori the same thing. To teach them as such is wrong.
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>>9414033
elaborate
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>>9414041
https://en.wikipedia.org/wiki/Whitney_embedding_theorem
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>>9414045
see >>9414029 and fuck off if you don't have anything to say
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>>9414041
Nigger manifolds can be defined with topological spaces and homeomorphisms to R^n but the object itself is just some topological space with the appropriate separation axioms.
>>
>>9414049
see >>9414047

the only nontrivial claim is
>To teach them as such is wrong
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>>9414050
If you really think defining all manifolds as submanifolds of R^n, you have clearly never done any serious geometry or topology.

Or even in physics its a bad idea, because spacetime is modeled on a 4-manifold. So assuming it is embedded in a higher euclidean space implies things that are physically unrealistic.
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>>9414079
>youre dum if you don't agree
>physically unrealistic
wow, and here I was thinking that you had actual reasons
>>
>>9413952
yes, used F since forever and it just stuck
>>
>>9412252
Laura B
Valensiya
Hanna F




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