## Main Question or Discussion Point

I'm in my 5th semester undergrad for bachelor of engineering in mechanical engineering, but I realized that I like mathematics much more than I enjoy my ME or any other applied courses that I have to take. I'm thinking about applying for graduate study in mathematics at top US schools (Berkeley, MIT, NYU, etc), but I'm afraid that I'm ill prepared and that my efforts will be for nothing, at least for those better schools. On paper, I have no advanced math courses completed. My coursework includes Calculus I and II, and multivariable calculus, differential equations, and probability and statistics. It gets worse because we were using baby texts such as Stewart's Calculus Concepts and Contexts for Calc I, II and multivariable, and not-so-bad Differential Equations by Snider and Saff. Additionally, there was no emphasis on theory whatsoever, just lots of blind computation and plug-and-chug. Nevertheless, I always read the textbooks on my own and considered the proofs to be essential and tried to follow the arguments given to the best of my abilities, even though I didn't quite understand why a proof by contradiction works, or any sort of indirect proof actually.

Things turned when I tried to go for a minor in EE and enrolled in a class called Math for EE, which basically was divided into two parts: First half was a survey of Discreet Mathematics (emphasis on propositional calc, predicate calc, valid rules of inference, valid proof methods, common fallacies, naive set theory, graph theory) from the book Discrete Mathematics by K. Rosen, and a second part was intro to complex analysis from a text by Brown and Churchill. I had to drop the course 2/3 in because I was overloaded with coursework, so I can't really add this to any graduate application but at least the knowledge stuck.
Right now I'm working myself through (self-study) Calculus vol. I by Apostol and I'm doing every proof excercise and I find that I have a talent for this, but these are still just baby steps. I try to read slowly, reflect on the proofs, and spend time on the excercises, but time is one thing I'm running out of because soon I have to decide if I want to enter the workforce or continue onto graduate level.

Any math major will probably blow me out of the water because I've looked at top undergrad math courses and I see courses such as abstract algebra, topology, differential geometry, and many many others that I just won't have time to get myself properly acquainted with before I graduate. Basically, I'm trying to establish whether graduate mathematics is out of my reach and a hopeless goal? I'm asking this on an internet forum because my parents just aren't able to help me this and I have never gotten a sound advice from counselors, but this board seems populated with some well qualified people.

I'm afraid I think you don't have the slightest chance graduating in mathematics with your current set-up. There's just too much you don't know.

Most of what you do in mathematics depends heavily on the very basic notions, and ideas and theories are built up in piece-by-piece manner, i.e. you can't say you'll let linear algebra slip and do some more calculus instead. You'll be out of the game in no time.

You'll have to spend more time on the subject, I guess. Why not start all over again?

Hi marcin,

I can identify with your situation, I studied ME before and love math. At the end, I was able to get a double major in math and ME, but it was not until I got into graduate school and did some extra course work in math that I completed the major.
Consider that you are only in your 5th semester, if you really enjoy math way more than ME, you can still consider changing to major in math. it is not really that late at all. But, of course, you must be sure u really like doing math than ME. You can also consider going for double major. If you want to switch or double major, then almost certainly you have to lengthen your time in school. Just as cliowa said, there is no way you can avoid taking the more theoretical classes if you want to go to any decent graduate school to study math.
So, my question is, do you really like proof and understand the abstract idea in math? From your description, you really have not much exposure to abstract math and proof yet, do you want to spend time on learning that? And also, do you like math because of its application in engineering or because of the beauty of abstract reasoning? I know people who studied math, love its application but hate going through the proofs. You can take a class on rigorous mathematics such as real analysis and see if you like it or not. Well, maybe you can pick up a copy of "Principles of Mathematical Analysis" by Walter Rudin, that's my first book for rigorous real analysis. Studying math takes a lot of time, there is no short-cut for that. At the end, it all come down to whether u like it deeply enough to think it is worth spending your time.

The only way to achieve your dreams, Marcin, is to take your passion and put it to work.

I know for a fact that you can go from calculus to someone that math P.h.D's will call "mathematician" in 2-3 years of training.

morphism