Really like math. What do I learn after Differential&Integral Calculus?

I think that either differential equations or linear algebra are logical next steps. As an applied guy, I feel that differential equations are more fun: lots of applications in physics, economics, etc.

Linear Algebra

Discrete Math

Number Theory

Differential Equations

If you really want to get into it:

Groups, Rings, Fields

Topology

Fractals

Analysis

Stochastic Processes

let'sget mathematical

You're now better at english

In my first job outta school we used linear algebra for aircraft radar target tracking algorithms. Instantaneous matrix transposition to find and launch a missile or two to kill your targets is awesome.

Rhodium Nights wrote:

Linear Algebra

Discrete Math

Number Theory

Differential Equations

If you really want to get into it:

Groups, Rings, Fields

Topology

Fractals

Analysis

Stochastic Processes

Take your pick from this list, but spend time on solving generic problems outside of the books. Competition problems are a good source of interesting problems that will challenge you, make you think outside the box, discover some proofs yourself, and open doors to topics you may not know existed

Lot's of good suggestions. There are several feasible routes through the flow chart, but linear algebra is a good place to start. Lots of decent topics suggested for the longer term, the only omissions for this level are complex variable and a methods course (which will blast through calc/lin alg/diff eq/multi variable calc/complex variable/calc of variations all with an applied focus). The latter can be a nice survey course before or alongside doing a deep dive in the topics separately.

Once you have done all of that you can do some good stuff (e.g. Analytic proof of Prime Number Theorem, quite a lot of theoretical physics becomes accessible).

Rhodium Nights wrote:

Linear Algebra

Discrete Math

Number Theory

Differential Equations

If you really want to get into it:

Groups, Rings, Fields

Topology

Fractals

Analysis

Stochastic Processes

^--- solid response... My mind was instantly jumping to stochastics and game theory/probability. You could also look around for PhD programs and get some great ideas.

I'll echo everyone else so far and say that once you get comfortable with calculus, Linear Algebra and Differential Equations are a great next step and both highly applicable to many professional environments. Along the same vein, you will likely want to become comfortable with some basic programming and computational logic, take a dip in Python (potentially through Jupyter notebooks), Numerical Analysis, Mathematical Modeling.

Now, as you say you are fond of math, you might want to really dig into some other topics too. Traditionally, alongside finishing up a calculus sequence, a student might begin work in some of the more theoretical areas of math. You have "proofs" and Real Analysis in your flow, but working with proofs is less of a solitary topic and more a foundational piece of the toolkit. As an undergrad, I took a course in that area during my freshman year. Comfort with mathematical logic will open up many, many other avenues (including Real Analysis). I would recommend studying Set Theory and proofs before jumping into Linear Algebra, to get more out of further study. "Book of Proof" by Richard Hammack, available in print and online:

https://www.people.vcu.edu/~rhammack/BookOfProof/

This is a phenomenal book, especially for self-study. Very enjoyable too, and opening this window will likely grow your fondness and interest for math.

Rhodium Nights above provided some great topics for higher study. Abstract Algebra can be especially eye opening, as can Topology and Number Theory. If at some point you would like to take a break from progressing through topics (after a foundation with proofs and logic), you might like "Proofs from the Book" or "Geometry and the Imagination." I would also recommend, as would many others, "Gödel, Escher, Bach: an Eternal Golden Braid."

Welcome to the club.

Discrete Math was the hardest class I ever took. My dad says Solid state physics is harder but I wouldn't know. Some people get stuck with stochastic process. Probably a distribution of hardest math classes :)

lets get mathematical wrote:

My question is what comes after.

My engineering math courses went in this order:

(high school)

Algebra and single variable calculus

(college)

Single variable calculus

multivariable calculus

first order diff eq

linear diff eq

laplace transforms

boundary value problems

partial diff equations

series solutions of diff eq

stats

...then there were the courses applying all that math

control systems

computing

communication systems (and fourier transforms, which are very similar to laplace in terms of the mechanics)

physics

electromagnetics

fluid dynamics

chemistry - and I'm talking more like electrons moving through materials and how all that works

thermodynamics

(switching gears some) economics

... it's really a deep field with a ton of branches you can follow

lets get mathematical wrote:

Hi guys, I know there are quite a few posters here who have a strong math background. I hope to have one as well one day. I recently got a good grasp of basic calculus. My question is what comes after. Is there a natural progression one follows? Like for example: calc 1-->calc 2---->linear algebra--->multivariable calc---->proofs--->real analysis, or do you go into differential equations after linear algebra? I'm fond of math and have a strong interest in it, but I eventually want to go to graduate school for economics. Appreciate any help

Vector calculus, baby. The foundation of electrical engineering.

Complex Analysis - cool stuff.

Fizix is Phun wrote:

In my first job outta school we used linear algebra for aircraft radar target tracking algorithms. Instantaneous matrix transposition to find and launch a missile or two to kill your targets is awesome.

That sounds fascinating. Too bad when linear is taught the professors are so far up their own theoretical a$$es that all they do is work through theoretical proofs. When you actually get to use linear to do cool things, it's easy to appreciate. The problem is that the gatekeepers teaching it only care about their lives' work (proofs), which I guess actually makes sense. But it's unfortunate that there isn't a practical way to relay this sort of info in schooling, which might open the doors more to the typical student.

Ph.D. in number theory here. Math is awesome and I'm glad you're pursuing it. It's a huge field, and any area that you explore can be fascinating.

Some advice: learn linear algebra really well as it comes up all the time.

Want to study differential equations? The set of solutions to a homogeneous linear ordinary differential equation is a vector space, and a certain matrix (the Wronskian) gives you important information about that space.

Multivariable calculus? That's a generalization of the calculus you already know to vector spaces. To find the critical points of a multivariable function, you need to do some linear algebra with a matrix called the Jacobian.

Group theory? A lot of groups can represented by matrices, and some important results in group theory have been proven by using linear algebra.

Topology and differential geometry? That's the study of spaces that look just real or complex vector spaces if you zoom in close enough.

Number theory? A lot of that is the study of vector spaces over the rational numbers (or over finite fields).

Statistics? Linear regression with more than one dependent variable involves vectors and matrices.

There's more, but you get the idea. Good luck with your studies!

I did graduate level work in Economics and Econometrics and only needed a strong grasp of Calculus and Probability/Statistics. I took Linear Algebra which helps conceptually with Econometric theory but never actually used it in practice.

I thought math was racist?

Non-Euclidean geometry, triangles do weird things on the surface of a sphere.

Should have just told him to dive right into representation theory lol

lets get mathematical wrote:

Hi guys, I know there are quite a few posters here who have a strong math background. I hope to have one as well one day. I recently got a good grasp of basic calculus. My question is what comes after. Is there a natural progression one follows? Like for example: calc 1-->calc 2---->linear algebra--->multivariable calc---->proofs--->real analysis, or do you go into differential equations after linear algebra? I'm fond of math and have a strong interest in it, but I eventually want to go to graduate school for economics. Appreciate any help

Not too much I can add at this point, just a couple things:

1. Most of the math classes you take (especially after linear algebra and multivariable calculus) will be proof-based.

2. Undergraduate economics majors are very weak in math, even at top universities; if you're good at math I think it will really help. I took an upper level econ course with zero economics background (I sat next to my girlfriend and had to ask her what supply and demand were), and still got one of the highest grades in the class because the econ students struggled with math.

3. Having said that, I'm not sure how much upper level math will really help you in economics. I think statistics might be more valuable in the long run. Upper level math tends to have fewer applications -- I don't know how group theory would be very useful to an economist and I think real analysis would be a waste of time (unless you enjoy it!).