How can I teach myself advanced mathematics?

In calculus II we learned multivariable functions, and you need the Hessian matrix and its determinant to be able to tell if a scalar-valued function has a local minimum or maximum at a given point, for example. It's not heavy stuff, but it would be strange for someone who knows nothing about linear algebra.

I took community college calc 1-3 and we never learned the Hessian matrix. This is the first I've seen anyone suggest linear algebra before finishing calc. Not that there's a good reason to wait (although MIT linear algebra seemed to carry the implication that everyone knew diffeq in one lecture), but did I go to the weird school or did you? Merica here.

EU here, physics undergrad btw. These things are really intertwined for us, like we had our first diffeq course in the second semester, yet we basically solved a bunch of diffeqs as part of our mechanics course in the first one, and we also had to learn differentiation in the first 3 mechanics lectures (or at least those of us who didn't learn it in high school for one reason or another), even though we only got to it in the second half of the calc 1 course. Also matrix functions, diffeqs and Taylor-series were mentioned in our linear algebra course (it was a linear algebra+vector calculus course, because physics).

At our university it was pretty straightforward.
Cal 1: introductory integration differentiation

Cal 2: techniques in integration infinite series

Cal 3: multivariate Calculus, (some determinants and matrices but it was so elementary.)

I took my classes in this order cal 1, cal 2, cal 3 & diff eq, linear algebra, proofs.

I wish I would've taken linear algebra prior to Differential Equations in retrospect. I suffered so much but linear algebra was a breeze after that.

bookzz.org/

1. Go to library.
2. Find all volumes of Bourbaki.
3. Start studying the volumes in order.
4. Profit!

>study math
>profit

>Lay
>advanced
dude, that's like the #1 book for LA if you don't want to deal with proofs or autistic shit and want a book that focuses on applications rather than theory

Math feels so "scattered", how the fuck does it all connect? It always feel like a bunch of rules that came from fucking nowhere

You first start with building the idea of natural numbers, then slowly add more autism. Ofc this will be a waste of time to do every time you want to do any mathematics beyond simple arithmetic so most of it is just assumed. Which makes everything feel scattered because the entire foundation is pretty hidden from you.

Any recommendations for beginning Calc level books?
Going back to school after a 4 year break and want to refresh.

Thomas Calculus Early Transcendentals 12th Ed. There's a PDF online and there's also a solutions manual. I used it for all three cal classes and never had a problem with it.

Who is Thomas Calculus? He must be pretty good at math though, even his name is calculus.

Lax not Lay.
amazon.com/Linear-Algebra-Its-Applications-Lax/dp/0471751561/
math.wsu.edu/faculty/watkins/pdfiles/lax.pdf

this

SPIVAK
SPIVAK
SPIVAKKKKKKKKKK

who cares?

math is just a social construct. it doesn't really mean anything

ur a social construct

ty for this shit
saved that

boom

I recommend Hoffman and Kunze for Linear Algebra
LA so useful

What kind of math specifically OP?

>a bunch of rules that came from fucking nowhere
this is pretty much it. Since Euclid, math has tried to be done by first making very basic, self evident assumptions (about say the properties of numbers or geometric figures), deciding on definitions we feel are 'nice', and following the autism. Of course these basic assumptions (called axioms) don't really come up in every day math and we just take a lot of stuff for granted. If you want to read more about it, the axioms most commonly agreed upon today are called ZFC

>Math feels so "scattered"
I actually get the opposite feeling, though I have over midway through a math degree, and have done quite a bit of my own research into logic, foundations, set theory and proofs outside of what is expected of mid- to low-tier state university students.

I see "connectedness" everywhere, especially in the notation. Embarrassingly I got a C in ODEs because (in part) I was too focused on trying to uncover the underlying connections to other things I learned in discrete math and algebraic structures. As an example, at the very beginning when DEs were first introduced, I noticed for an nth order ODE would be expressed as F(x, y, y', ..., y^(n)) = . It made sense to me that we would want to be able to express them as functions as functions are well studied mathematical objects and we can do a lot of useful things with them. However, all that is expected of the student in these first sections is how to be able to classify a DE. As you may imagine, I would routinely waste time, looking up things like operator an spectral theory, trying to get the big picture, but ended up not devoting enough to the homework and thus didn't have the route experience to solve problems on the exams (especially important in an introductory DE's course) as the semester progressed. I thought if I could just understand what's going on behind the scenes I will be fine. This happened to me in a couple other classes as well. I didn't intend for this post to be near as long but hopefully this serves as a lesson to other anons just embarking on their journey.

I feel you, I was hurting on Wronskians and variation of parameters because I couldn't do baby matrices for whatever reason when I took diff eq. Currently teaching myself linear with a schaum's outline for summer break.

Why the fuck do you retards recommend Spivak? He's going back after a four year break, and wants to learn Calculus not kill himself.

Why do you spend time typing out similar paragraphs for every duplicate thread like this?

You realise they're going to try it for 10 minutes and immediately give up? You realise that you're wasting your time?

The fact that anyone is on this board at all means they don't value their time.

1) Get a book or online classnotes to follow, this is preference and you should around.
2) Find solutions to the problems so you can get feedback.
3) Study

Shop around*

Calc1->3->linear algebra->diff eq->proofs-> modern algebra-> topology.

Judging by the way you asked the question I'm assuming you have little experience in math so I'm assuming you're asking how to get to topology and shit like that. People omit modern algebra before topology sometimes. I have book recommendations for each subject, just respond to this if you want them.

If you want to get onto real higher math you'd better have topology, complex variables, analysis 1&2, Differential Geometry etc. (basically a degree). Then find yourself a database and start reading papers.

Not sure how to quote on this website, but can you post the books?

Proofs should already be in the first lecture though.

Calc, lin. alg., ODE are really intertwined. Not unrealistic to study all of these simultaneous.

Usually people don't learn linear algebra before vector calc, but you really should, it makes it a lot easier.

If you wanted to learn things in a "correct" order it would go:

calc 1 (differentiation and integration)
calc 2 (sequences, series, taylor polynomials, etc)
linear algebra (vector spaces, linear functions, determininant, etc)
vector calc (mulitvariable calc + stokes theorem things)
then you can learn a variety of things: complex analysis, diff eqs, algebra (groups, rings, fields, more abstract linear algebra) or even topology or analysis

>Usually people don't learn linear algebra before vector calc
what school do you go to so i can avoid it forever?

there's no need for any of those analytic prerequisites for topology or algebra anyway...

You can skip calculus and go straight to abstract algebra.

You don't need to know taylor series to do learn basic group, ring, field theory.