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

4th year applied math major here. You should study Complex Analysis up to and including the basics of residual calculus. A lot of extremely difficult elementary integrals and many non-elementary integrals are most easy to solve by going through the complex plane and using residues. But before you open a textbook on the subject, make sure to study geometric series and the most common tests for convergence. Series are fundamental to the concept of residues.

You can study differential equations without knowing anything about complex analysis, but there will be certain topics difficult (or impossible) to understand without a proper background in the subject. Laplace Transforms are immensely important for solving differential equations and they operate using properties of the complex plane. I made the mistake of taking ODEs in University before Complex Analysis and I got my ass kicked for it.

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

You should learn Real Analysis as quickly as possible. Simply buy a book, Rudin is good, and work through the proofs. You don't need a course for this. This will be the first time you do actual math.

From a practical perspective, you should master the basics of Linear Algebra. This is probably the most useful area of mathematics conceptually for any applications.

Computational topology is an interesting emerging field and may encourage you to improve your coding skills as well.

Rising sophomore studying CS here. I've taken a required CS probability class, and learned some of the basics. Stochastic processes really interest me and I want to learn more about them, how would you guys suggest I self-study to understand stochastic processes?

WantToGetBack wrote:

Rising sophomore studying CS here. I've taken a required CS probability class, and learned some of the basics. Stochastic processes really interest me and I want to learn more about them, how would you guys suggest I self-study to understand stochastic processes?

I did my PhD in algebraic geometry and arithmetic number theory so I'm a bit out of my depth on stats and probability, but I'd say it depends on how advanced you want to go. In terms of machine learning and "AI" then that's really just basic Gaussian processes and setting up linear classifiers I believe. in that case, design of experiments and classes like that will be really beneficial. A class in signal processing will actually probably be more beneficial than anything else.

In terms of hard math, you'll need real analysis and linear algebra, probably also topology; this will then allow you to study measure theory. From there you'll probably need Ito calculus and almost certainly functional analysis (Hilbert spaces come to mind). Again, it depends on how in depth you want to go.

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

1 x 100m= 100m

2 x 100m= 200m

4 x 100m= 400m (1 lap around outdoor track)

2 x 400m= 800m (2 laps around outdoor track)

4 x 400m= 1600m (4 laps around outdoor track- aka Metric Mile)

Anything longer is considered "Advanced" math and requires a "Mathlete" degree.

"Go Run One"

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

I give a higher priority to good (hard, theoretical) courses in probability and statistics than many of the respondents so far, in order to apply your math skills in a wide variety of areas. I also agree with the recommendation of signal processing and linear algebra. These topics are all interrelated in interesting ways and are of great help for university and corporate research.

Real examples from my work include: linear algebra and statistics to understand shoulder biomechanics, and to analyze balance in different kinds of running shoes. Nonlinear differential equations to explain the effect of gravity or its absence on fluid pressures in the brain (NASA-funded). Neural network control of an extruder to produce uniform thickness of a thin plastic sheet. Convolution integration and Markov chain theory to understand how membrane proteins control the amount of calcium in cardiac muscle. Network models to understand how the brain generates the breathing rhythm and adjusts it in response to external stimuli. Maximum likelihood estimation of brain control of the circulation. Spectrum analysis for symptom tracking in Parkinson's disease. Quaternion operations to analyze soccer heading accelerometer data. You get the idea. The foundations are linear algebra, differential equations, probablility, statistics.

The maths (plural because I'm English) that made me struggle the most was matrix algebra. The concept in maths that confused me the most was imaginary numbers. So if you want to test your maths, go for matrices, if you want to confuse your noodle try and explain i.

I am not OP and I am no math major but EXCELLENT thread here folks. I'm not surprised about the plurality of math majors here, but the advice you have given here about additional topics to study is very useful to me.

Let me ask you guys about two topics that I'm interested in studying and the math associated with them.

Thermodynamics of solids. I am not a physics major nor that type of in-depth simulation guy, but I am very interested in thi stopic and it will come up as proof for a lot of the stuff I'm learning. What math background is necessary for this?

Process optimization: Kind of an industrial engineering topic. Is this just applied linear algebra stuff?

Finance/Econometrics: Like so many people I like to make money on the side gambling, but I like to do the more informed version of gambling (that's mostly safer these days) called options trading. I like quantitative finance. It's a bit late for me to make a career of it, but what do you suggest I study to make this a hobby?

Thanks.

Good points, but on #3 - if the person is looking to get a PhD in Econ, they like math majors (because of your point #2) and real analysis is not at all a waste of time, could be quite useful.

This.

Just look in the college catalog! Genius

mathy wrote:

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.

Not sure how that is a solid response. It's basically just a list of "all" the options. It does not provide any guidance at all.

It's as if someone were to ask what are some good states to live in for a runner and someone responded by listing all 50 states. No value in that.

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

I have a PhD in mathematics, I did most of my research on quantum mechanical systems and used a lot of functional analysis.

I think a lot of posters here are getting very ahead of themselves. OP literally just learned calculus the other day and indicated they wanted to pursue economics or an applied field. In no universe do they need to know how to formally define a limit much less know measure theory or other concepts from real analysis.

At the undergraduate level mathematics only loosely follows a progression. I feel like there are courses appropriate for underclassmen (Calc 1/Calc 2/Multivariate/Linear Algebra/ODE/Basic Proofs) and from there it really depends on the interest of the student. Engineers or other students in applied maths take courses like PDE/stochasics/probability/numerical analysis/complex variables, while students interested in pure maths will take courses like real analysis/algebra/topology. Grad school math branches off in all kinds of different directions.

OP if I had to give you advice, the most valuable thing you can do is keep your enthusiasm for learning. Seek out research papers on google scholar about whatever you're interested in, try and fail to understand them, seek out texts, just immerse yourself. In the meantime, take all of the math courses you can stomach; anything you miss in undergrad you'll eventually build a strong enough base to learn it on your own. For example, I literally never took a topology course but with all of my background I was able to learn it on my own for grad school.

For more tactical advice, pay lots of attention in Linear Algebra because it is the basis (no pun intended) of so much that you do in mathematics. Also definitely spend some time learning how to code, probably some combination on Python/MATLAB/R.

Theology & Geometry. You'll never use Algebra in real life.

No relations to the Fake Hingle McCringleberry

"Éléments de mathématique" by Nicolas Bourbaki would be a good start.

I've never met an economist who was actually good at math/stats.

They have one trick: linear regressions. Everything is a linear model to those jokers.

Proofy courses like real analysis may not seem relevant, but they will make you better at everything else you do, especially econ. It takes a few proof course to get a feel for it, but once you do, you will be able to teach your self anything, and I mean anything. In the world of econ, you will among the few, instead of merely one of the many.

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

Take an intro to proofs course as soon as possible! Ideally at the same time as linear algebra or differential equations.

Based on how much you enjoy proofs, and logic, vs. more applied calculations, you'll have a better sense of what you want to do from then on.

If it turns out you really enjoy the calculations and problem-solving in calculus, but the abstract proofs in your proofs course a bit less, you'll want to focus your upper-division course work on more applied topics. If you really enjoy proofs and abstraction, you'll want to focus more on pure math topics. You want to learn this about yourself as soon as possible.

Linear algebra is the foundation of machine learning and data science, whereas differential equations is the foundation of much of the natural sciences. Both are great subjects.

Thermo of solids- heat transfer, material science classes from ME depth

Optimization-take optimization classes, lots to learn there. Newton methods, interior point methods, stochastic annealing. etc.