What do you learn after calculus?

It's a subject in EE that built on Fourier/Laplace/Z-Transforms.

take some real linear algebra and maybe also multivariable calc if you hate yourself.
if you really hate yourself, take diff eq.

absolutely positively take stats if you haven't already. basic stats are obnoxious and annoy math purists, but they're essential for any kind of empiricism.

Thank you for the comprehensive list. It seems a bit overwhelming at first glance but I suppose everyone finds their niche in time.

A lot of the stuff on his list could be put together, so it's not that scary.

>logic, set theory

You will have a good enough grasp of it after going through real analysis, number theory and topology.

>matrix algebra

Just put it together with linear algebra

etc

What uni course/career involves the most/hardest calculus and algebra?

en.wikipedia.org/wiki/Math_55

>>logic, set theory
>You will have a good enough grasp of it after going through real analysis, number theory and topology.
You never learned logic and set theory properly, did you?

But yeah, that list is kinda crap.

>Topology
What do you want topology for? I understand how calculus is very useful, but topology?

Topology has a lot of applications in Biology as well as certain problems in engineering.

Topology is useful even in algebra (search for Zariski and Krull topology).
In order to study PDEs you'll need lots of functional analysis, and some function spaces are not so nice that they are normalizable, so you'll need the general theory of topological vector spaces.
Topology is also a prerequisite for studying dynamical systems.

Topological spaces appear naturally in most of mathematics because of its relation to order theory, adjointness (or Galois connections) and Moore closure.

>certain problems in engineering.
examples? im finishing my engineering degree and ive only ever heard about topology on Veeky Forums

I'd recommend going through in this order

>Lang, Basic mathematics (precalc, conics, etc)
as at least a refresher
>Stewart, calculus
as a refresher + calc 2,3 + some other basic stuff
>How to prove it, Velleman and/or how to solve it, Polya
As a basic introduction to proofs
>Strang, Linear algebra
Basic linear algebra is needed in EVERY field
>Spivak, Calculus
To get a taste of real analysis, you can skip this though
>Pinter, A book on abstract algebra
A taste of basic abstract algebra, explained in a very intuitive way, with a LOT of exercises, as well as interesting exercises (you develop some code theory on the side)
>Halmos, Naive set theory
Short book, gives you an idea on the importance of set theory. You can skip this, but I really recommend it, and it's really short.
>Apostol, Mathematical analysis or Abbot, Understanding analysis
Intro to real analysis, it is important to give you the notion of why rigour is important, and gives a lot of motivation for other fields (especially topology).
>Hoffman/Kunze, Linear algebra or Halmos, Finite Dimensional Vector Spaces. Axler, Linear Algebra done right to supplement
More rigourous linear algebra, useful for differential geometry
>Artin, Algebra
Rigorous abstract algebra
>Jones, Elementary number theory
Skippable although it is a very beautiful subject

Do you prefer analysis or algebra? You can skip what you don't like, but some are a must

>Analysis
>Munkres, Topology
A very good, self contained text, you can skip the algebraic topology section
>Complex analysis: no real classic book, but Needham, Visual complex analysis is good and supplement with Lang or Stein/Sakarchi
>Tao, Intro to measure theory
>Evans, PDEs
>Do Carmo, Differential geometry of curves and surfaces
Intro to DG
>Milnor, Topology from a Differentiable viewpoint
Very good book on Differential topology
>Lee, Introduction to smooth manifolds
Differential geometry, alternatively, Spivak's 5 volumes on differential geometry are a classic

>Algebra
>Steward/Tall, Algebraic number theory with a view towards Fermat's Last Theorem
optional, but it is a satisfying subject
>Eisenbud, Commutative Algebra with a view towards algebraic geometry
>Serre, A course on Arithmetic
>hatcher, Algebraic topology
classic, and you can get it for free
>Shafarevich, Basic Algebraic Geometry, followed by Hartshorne

Obviously this is not a complete list, since I'm missing notably geometry, probability, analytic number theory, etc, but it should be something to strive for.

topology shows up in almost every branch of mathematics, minimally as a tool

in logic - stone spaces, topological sheaves for semantics
algebra - ring spectra, lattice theory (coherent locales)
analysis - metric spaces, differential topology

"continuous variation" shows up everywhere in math, and you often need special notions of continuity which topology makes precise

en.wikipedia.org/wiki/Topology_optimization
is the only one I know.
I don't even see how it's topology (no open sets?), but I'm just getting into that

Halmos is the worst book on set theory, use The Joy of Sets by Devlin or Elements of Set Theory by Enderton if the former is too hard for you, and Jech and Hrbaceck if that's too easy.

Munkres is also not great, although I learned from it first, it has no uniform spaces, filters and nets, try Willard instead.

Ahlfors is the classic complex analysis textbook, but no one likes it. I'd recommend Conway though.

Starting PDEs with Evans is too much, but I don't know any other book on PDEs...

Lee is good, but Bredons Geometry and Topology (Topology and Geometry?) is better.

I'm still learning algebraic geometry but Hatcher looked horrible. I'm enjoying Homology Theory by Vick and A Concise Course in Algebraic Topology by Peter May (which you can download for free).

Also, after measure theory and before PDEs, add:
>Functional Analysis
Elements of the Theory of Functions and Functional Analysis - Kolmogorov and Fomin
A course in functional analysis - Conway

>I went on to pursue a liberal arts degree
wew lad

Functional Analysis and Algebraic Geometry

kind of off topic but I'm starting inear algebra in the fall. my skills are definitely weak. what should I study over the summer to prepare me to dominate linear algebra? I'm thinking if I brush up on at least basic calculus and algebra

Calculus doesn't come up much. Get a book on Linear Algebra and read ahead.

that's not NEARLY a comprehensive list, it's a list on engineering classes and then a couple of random math classes

>Analysis and Topology
What's the difference? Honest question here

Analysis is all about putting you epsilon into that cute sigma. Topology is the art of getting the bra off.

I don't know what you're getting at. From what I've seen until now, analysis is usually thought of as normed + metric spaces, which is just a special case of topological spaces so I don't understand the distinction

I wanted to emphasise this. Nearly every point set topology book contains an introduction to set theory even.

double trips

lmao tru tho

Linear algebra, differential equations and partial differential equations, linear algebra 2, stat and prob. That's the start of it.

Topology is too general to be used as an introduction to higher level mathematics in the way analysis is. What you say is true but come on. An introductory topology class has a very different tone than an introductory analysis class.

In introductory analysis you revisit ideas you already almost completely understand and see how the process of turning your ideas into reality works. In introductory topology you for the first time see many ideas that you have no idea where they come from nor why specifically they were invented.

You learn Mathematical Analysis.

en.wikipedia.org/wiki/Mathematical_analysis

Wtf? Undergrads learn this in their 1st semester, right after set theory

>Is X a meme?
You're a fucking meme and deserve to be recommended nothing.