Part III Rudin has a nice chapter on Fourier series, that is, representing a periodic function f(t) with a sum of e^(iwt) for overtones at frequencies w and wh

Part III Rudin has a nice chapter on Fourier series, that is, representing a periodic function f(t) with a sum of e^(iwt) for overtones at frequencies w and where each frequency w is a whole number multiple of a fundamental frequency. The linear algebra and geometry here are that the e^(iwt) are perpendicular projections of the f(t). So, the e^(iwt) are orthogonal axes. The line integral generalizes to the exterior calculus of differential forms (keep track of signs from the direction do the integrations, i.e., as in line integrals) and the fundamental theorem of calculus generalizes to Stokes theorem crucial in Maxwell's equations, fluid flow, and partial differential equations. Now we are close to differential geometry, and Rudin also does the inverse and implicit function theorems important in differential geometry and parts of nonlinear optimization, e.g., Lagrange multipliers. I warmly suggest that take a really fun, one weekend, pass through Stokes theorem parts of Tom M. Apostol, Mathematical Analysis: A Modern Approach to Advanced Calculus, Addison-Wesley, Reading, Massachusetts, 1957. Get it used, and pay whatever you have to. This way get to see the 2-3 dimensional versions and, really, mostly enough for what physics and engineering do with vector analysis and Stokes theorem. In H. Royden, Real Analysis and the first half of W. Rudin, Real and Complex Analysis, can see the Lebesgue integral, due to H. Lebesgue near 1900. In essentially all cases where the Riemann integral is defined, the Lebesgue integral is also defined and gives the same numerical value. But the Lebesgue integral has more powerful theorems and is defined in more general situations. As in a 1933 paper of A. Kolmogorov, the Lebesgue integral gives a solid foundation to probability, statistics, and stochastic processes. In W. Rudin, Functional Analysis get a treatment of distributions that cleans up what physics tries to do with the Dirac delta function and also covers spectral theory. From there, can go for a Ph.D. For that will need mostly (A) pass the qualifying exams and (B) do some original research. The standards are commonly something like "an original contribution to knowledge worthy of publication" and for publication, "new, correct, and significant". For the qualifying exams, first pick a department, pure/applied math or some math area in engineering, optimization, probability, economics, computer science, etc., get their description of their qualifying exam topics and references, study, and take and pass the exams. For the research, do some. One suggestion: For the research, pick a real problem and use some math, at least in part new, to get a good solution. Can get "significant" from the importance of the real problem. Can get much of "new" if have the first or better attack on the real problem. Can get "correct" if do the math with careful theorems and proofs. Can publish if pick an appropriate journal. Might get to regard the math as significant based on what it does for the real problem and not just its contribution to pure math. Even the best US grad schools are hungry for good students. With a good ugrad math major and good work on independent study, should get a good reception at grad schools and at least a tuition scholarship. In that way I got accepted to grad school at math departments at Cornell, Brown, Princeton, and more. In a sense, independent study is recommended: (A) That is basically what research profs have to do for all their careers. (B) At least at one time the math department at Princeton stated that no courses were offered for preparation for the qualifying exams, students were expected to prepare for the exams on their own, courses were introductions to research by experts in their fields, and students should have some research underway in their first year. For a little on how to do the original parts of math research, there is the A. Wiles comment: "Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it's dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly it's all illuminated and you can see exactly where you were. Then you go into the next dark room ...." That's some of how to learn some math, largely with independent study, and maybe to get a Ph.D. But you might want more: You might want your Ph.D. work to get you a good start on a good career in pure/applied math research/applications. For that, look around, pick up what you can in seminars and conferences, and get what your profs can explain to you. Broad point: Long the best opportunities in applied math, even with some advanced pure math prerequisites, have been in US national security, especially within 100 miles of the Washington Monument.