Name a better trio

Z
F
C

The Analytic, Arithmetic, and Topological applications of Elliptic Curves.

>try not to shit your pants
>elementary functional analysis
cringe

that's a duo tho

>inverse
You mean dual?

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

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?

"Div, Grad, Curl and All That" by H.m. Schey. Short, concise, understandable, highly useful, and in general very well written.

I want it rigorous god dammit.

Spivak Calculus On Manifolds

AC is independent from ZF

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

If you're wondering as to why
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
mathoverflow.net/questions/10574/how-do-i-make-the-conceptual-transition-from-multivariable-calculus-to-different?noredirect=1&lq=1

>Hahn–Banach theorem
Made me literally shit my pants desu
The Hahn Banach theorem is why I gave up math

You don't know Graham then. He's ma boi!

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

Beyond that you want something on manifolds

>"Calculus on Manifolds"
>Only defines "manifold" in the last chapter

>Completeness
>Consistency
>Effective Axiomatization

You can't argue this.

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

kek

derivative is a rate, and a differential ain't

You can't wash your hands in a derivative

A differential is something which sets one apart.

A derivative is merely an uninspired replication.

OP here again. Can someone explain what this theorem means in layman terms

It doesn't even define a full manifold, it defines submanifolds of R^n.

so it defines all manifolds...

Manifolds are object independent of an ambient space.

all manifolds are submanifolds of R^n (up to isomorphism obviously)

>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.

all ("intrinsic") properties are independent of isomorphism

[math](X,\mathcal{F},\mu)[/math]

good looks

Measure space? Why us F for sigma algebra¿

Can someone explain me what operators are? I'm a brainlett

"operator" is a brainlet name for a function in a space that isn't super simple

nigga what

By means of a nontrivial theorem.

how fucking enlightening, thank you very much for this very useful post that no one thought about

Manifolds and submanifolds of R^n are not a priori the same thing. To teach them as such is wrong.

elaborate

en.wikipedia.org/wiki/Whitney_embedding_theorem

see and fuck off if you don't have anything to say

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.

see the only nontrivial claim is
>To teach them as such is wrong

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.

>youre dum if you don't agree
>physically unrealistic
wow, and here I was thinking that you had actual reasons

yes, used F since forever and it just stuck

Laura B
Valensiya
Hanna F