# Most effective order for learning different branches of mathematics. [closed]

So, I'm just a current student with a lot of interest in mathematics. Usually I am on the site looking at the questions and most of them are about things I can't currently comprehend. As I would like to increase my knowledge of mathematics, I like to search stuff up, but often don't know where to start with a subject and quickly get lost in a wide array of topics and subjects.

The question is this:

What is the best or most useful order for learning different topics in mathematics, both in learning new material, as well as understanding more advanced stuff about old material? Also arranged in a way that seemingly builds off each other to complete understanding of the topics being learnt?

So far I've just covered the basic-general stuff in calculus. i.e. Derivatives, Integrals, Taylor Series, Multi-variable Calculus, Coordinate Systems and Linear Separable Differential Equations.

Note: In order to help narrow the focus, let's use the fact that I am an engineering student, so I would like to learn more about topics in the physics and mathematical applications in the hard sciences, so specifically the fields of calculus, algebra, differential equations, numerical and complex analysis and possible links to physics.

What next?

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## closed as too broad by Dan Rust, Old John, hardmath, Potato, Austin MohrDec 16 '13 at 2:26

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.If this question can be reworded to fit the rules in the help center, please edit the question.

what's next? Everything! unfortunately I think this is far too open-ended a question to be useful. – Dan Rust Dec 16 '13 at 1:09
@DanielRust And that is the reason why I'm asking this. I was looking for how stuff builds on top of one another, and I have no idea where to start... – Link Dec 16 '13 at 1:11
Have you considered looking at the maths syllabus of a good university? Course/module guides etc. tend to have summaries of material learnt in each module, as well as associated lecture material (notes/slides) and reading recommendations. As a bonus, they are set out in a linear fashion because they are taught in a particular order to students. – Dan Rust Dec 16 '13 at 1:15
@DanielRust, Hmmm, have any places in mind? The first thing that poped into my head was MIT or Caltech maybe? – Link Dec 16 '13 at 1:16
Every university will display this information in their own way. Using my own as an example, you can find the full module list for this academic year here (warning pdf). the vast majority of first year modules (and to a lesser extent second year modules) will cover the same backbone of material. Third year modules and beyond tend to vary widely from place to place depending on the specialties of the staff. – Dan Rust Dec 16 '13 at 1:23

You have several directions you can go here.

A. You will of course have to go on to more topics. The reason linear algebra is a good next choice is that it sits right on the intersection of so many different branches of mathematics, such as multivariate calculus, differential equations, numerical methods, and of course algebra and geometry. It's just impossible to move forward in more advanced math without it, because so much is dependent on both the fundamental concepts and the mathematical methods. Or as one student put it, "you cannot escape it".

B. Before suggesting other topics to proceed with, let me offer the idea of "backfilling" your present knowledge of calculus. The way calc is usually taught is what I call the "splat" method, wherein you are splattered with far too many topics, but provided no depth. Some of these are fundamental and essential; some are interesting, cute or helpful in special circumstances; and sadly some of them are just plain wrong. It is very rare that an effort is made to provide perspective by, say, categorizing the topics (let alone spending more time on what is important and skipping some of the other stuff). As a result, people tend to have a fuzzy view of the subject until something happens to improve their focus; such as they start getting creamed in advanced calculus; or they have to teach it.

What you need is some way of pulling things together so you have a clear viewpoint on the subjec as well as filling in anything important that you didn't quite understand. You might find some help with this in a book called "Second Year First Year Calculus" which has some good material. Or you might look around to see if you can teach a calc course somewhere or at least tutor.

C. Which topics you proceed with depends to a large extent on your interests, which may or may not be well defined at this point. But to offer some examples:

If you are interested in physics and mathematical applications to the hard sciences, you will want to take more calculus, differential equations, numerical analysis, complex analysis if offered, and by all means some physics, where you will get to use this material.

If you are interested in applications of mathematics in the life sciences, social sciences and business, you want to look at courses in statistics and modelling.

You can try out abstract algebra and topology and see if they resonate with you.

If you don't know the answers to this, then try several things and see what grabs you.

D. Finally, regarding looking stuff up and digressing all over the place, it is really easy to do. But it might be best to pick one or two specific topics and start with an introduction. For example, you can decide you will study Fourier Series, or line integrals. Start at the beginning, and give yourself some grounding. Find a tutorial or acquire a textbook. Work a bunch of problems.

It great that you have wide ranging interests, but probably would be helpful if you line them up one at a time. You don't have to get it all done immediately. There is life even after graduation, and plenty of time to get to whatever interests you; and/or to roam around in different areas of math (some of the best mathematicians do exactly that).

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Great reply! Do you have any recommendations for textbooks? Especially for Linear algebra, Numerical and Complex Analysis, Calculus and Abstract Algebra? – Link Dec 17 '13 at 15:57
@Link You know, the best book is one that presents things in a way you understand and that helps move you forward. What pleases me may not suit you at all; you are at a different place in your learning, and also learn in your own way. (In fact books that pleased me years ago often annoy me now.) There are lots of online samples of books. So say you pick Linear Algebra, and read chapters from several different books. If you find something that works for you order it. If it turns out to be a dud, return it and get something else. – Betty Mock Dec 17 '13 at 20:14
Linear algebra is a particular challenge bookwise -- look hard. If you find something that will do, then you can supplement it with internet searches, or questions here. For complex analysis, start with something short and simple if you can find one; you can then move up to a more comprehensive book. I should say that Apostol's "Real Analysis" pleased me years ago as well as now (for the more advanced calculus). He is perhaps a bit too general, but you can always focus on the simpler cases. And I have never found a mistake, so I have great trust in him. – Betty Mock Dec 17 '13 at 20:19
That is great advice, I think I'll do that then! I'll start by exploring Linear algebra books. (Though if you had to, what books please you now?) – Link Dec 18 '13 at 0:42
@Link I used a book by Jim DeFranza and Daniel Gagliardi -- well organized, fairly clear, lots of problems -- when I was trying to help a student through Strang's book, which I strongly recommend against (I found it impossible to understand and I passed a PhD exam in the subject). DeFranza/Gagliari wasn't a particular choice -- it just happened to be on my shelf and I don't even know why. For my own reference I use Peter Lax' "Linear Algebra and Its Applications", which is too abstract and general for a beginner, but a great choice for 2nd time around. – Betty Mock Dec 19 '13 at 17:21

Linear algebra is next. Let me know if you need some books.

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While I appreciate the reply, I was looking for a slightly more detailed answer which covered more topics. However I would love to have some book recommendations. – Link Dec 16 '13 at 1:12