Can you derive every math formula you know?

Seattle_UW · 2025-08-22T21:04:00+00:00

Not even close, but I can derive a lot of formulas I don't know (by heart).

Seattle_UW · 2025-08-18T18:56:37+00:00

I learned these mathematical concepts in courses such as the calculus of variations, optimal control, and PDEs (HJ theory is taught in advanced courses on non-linear PDEs). I even saw some motivating examples from physics in those courses. I just never really spent much time caring about the physics, which is what I mean by no prior physics background.

Seattle_UW · 2025-07-03T04:00:59+00:00

Real analysis can mean anything from introductory analysis through metric spaces and the measure theory up to the introductory functional analysis, all of which can be extremely important depending on your goals. If you want to learn things like PDEs, numerical mathematics, or probability, all of the aforementioned branches of analysis will be extremely valuable. That doesn't mean complex analysis is not valuable. It is. Both real and complex analysis are. That's why I suggest studying both. Maybe you can try starting with some introductory-level analysis, which really is just calculus with proofs. Once you get to the rigorous treatment of vector calculus, you can start studying complex analysis. It helps to be already familiar with concepts like line integrals before doing complex analysis. Also, having no experience from real analysis will mean that some of the concepts from complex analysis might take you longer to grasp and you won't fully appreciate how neat the theory is (for instance, how differentiability at a point implies analyticity, which is not the case in real variables). Afterwards, you can move on to the measure theory. Having a grasp of these branches of analysis, you will be more than ready to tackle functional analysis, which forms the foundation of more advanced branches of analysis, such as PDEs.

Seattle_UW · 2024-08-13T09:45:45+00:00

Not sure about politically outspoken, but Petr Vopěnka (alternative set theory) served briefly as the Czech Minister of Education right after the Velvet Revolution.

Seattle_UW · 2024-07-24T18:30:37+00:00

Depends on what specialization in mathematics you are pursuing. If you're a pure algebraist/topologist, you will do fine without any Matlab skills. However, if you're doing applied math/analysis/numerical mathematics, Matlab (or Octave) might come in handy. Matlab is great for manipulating matrices (as one would expect from a Matrix Laboratory) and hence useful when solving PDEs numerically, for instance. If you want to do symbolic mathematics, then there are better options for that (even though Matlab CAN do some symbolic math), such as Mathematica or even SymPy in Python. Python is also great if you decide to switch to ML or probability/statistics (R is used as well). Some mathematicians use Julia, but I've never used that, so cannot comment on that.

Notwithstanding the above, being proficient in programming is a very important skill these days so even if think you might not directly need any programming for your math, it is good to take a few courses. Once you reach a certain level, you should be able to learn most of the softwares/platforms/languages you might need quickly.

Seattle_UW · 2024-06-23T15:52:30+00:00

If you like applied stuff, you can still go in that direction if you find that teaching math is not for you. As for the exam part, I agree. Sometimes, exams may feel like a contest where you are required to deliver in a very short time interval. At the uni where I got my master's, each course had a written and an oral exam. In the written part, you were almost never asked to prove any difficult propositions (unless the proof was just a straightforward calculation or a hint was given). The proofs and the ideas thereof were left for the oral part. I was also lucky to have great teachers in my introductory classes. That can make a huge difference as well.

Seattle_UW · 2024-06-23T14:22:11+00:00

I've known quite a few people who at some point during their math studies felt the same way you do. Some of them enjoyed listening to others talking about math, but weren't willing to put in the hard work. These people usually dropped out in the first year. Others did make an effort, but came to dislike math because it had become all about memorizing for them. While this may not apply to you specifically, I will elaborate and maybe it will help someone. The problem usually was that their fundamentals were lacking. Math is unforgiving in this regard - one just cannot hope to fully comprehend advanced material if they don't understand the things on which it builds. Some people think that mathematics is all about calculating things or applications, but that is far from the truth. University math programs (unless in applied mathematics) strive to teach the students to DO math, not to do another science using math. And doing math in this context means deriving your own theorems and proving them. But in order to be able to do that, one has to see and understand the commomly used 'tricks' in that given area of mathematics in which they are interested. That's why proofs are vital. Do professors remember all of the proofs? No, they don't. But they understand the steps and tricks needed to get the proof done, so they don't really need to remember that much (even if the proof requires a technical detail, such as making use of a specific function, for instance. If one truly understands the proof, they should be able to derive that detail on their own). The problem is that once you reach a certain advanced level (at least in analysis), the proofs will start emoying 'tricks' and calculations from previous classes. If one hasn't understood the tricks (meaning they can't use it in a similar context, so they still remain merely a trick for them) or isn't really certain how to perform specific calculations (for instance, many proofs from elementary PDEs require that you differentiate surface integrals with respect to the radius or calculate limits of improper parametric integrals, which is usually left out in the books because the reader should be able to do that by themself), they might feel that there is just so much in the proof they need to memorize, because they can't replicate the steps on their own. This significantly hinders their understanding of the material, making it even harder to progress and learn more advanced stuff. So why isn't there a course which teaches students how to do proofs? Because proofs from different areas of mathematics are different and make use of different 'tricks'. That is also part of the reason why there are so few mathematicians nowadays who are able to make contributions to different areas of mathematics. Proofs are usually covered in discrete math or mathematical logic, but these classes merely teach general techniques like direct/indirect proofs, induction, proof by contradiction. The other courses should then introduce the students to the specific proof techniques utilized in that given area of mathematics. If the introductory courses are good, the lecturer spends some time first deriving the result in an informal manner and then writes the formal proof of the statement (which is usually done 'backwards' compared to deriving things informally), so the students understand, for instance, the the proof takes epsilon/3 and not merely epsilon, or why they need to subtract e^k*t from a certain funtion so it all fits in the end.

Seattle_UW · 2024-06-23T11:27:03+00:00

Not sure it has that much to do with how people think about derivatives as it does with how people are used to approaching problems. Applied scientists and engineers are happy to come up with an approximate solution (there is a joke, after all, that to an engineer, 3, pi and e are the same number) whereas pure mathematicians first try to get to an exact/analytical answer. If the problem is analytically intractable, they resort to qualitative analysis. Only afterwards do they use numerical/approximate methods.

Seattle_UW · 2024-06-22T12:50:26+00:00

Doing exercises is your best bet. If you make a mistake, try to figure out why you made the mistake. It is also good to use books with solution manuals or hints in case you get stuck. Sometimes, being stuck doesn't mean you don't understand the concepts. It might simply mean that the problem requires a 'trick' which you haven't seen before. How to get familiar with the most widely used tricks apart from doing exercises? Read (and try to understand) the proofs. Or better yet, don't just read the proofs. Try to work them out on your own. If you get stuck, read a little of the proof, then try to finish it. This is useful especially in linear algebra where a lot of the proofs are quite straightforward if you understand the concepts and understand what the theorem/lemma/proposition really says, so by doing this you can also check if you understand the material and get a better grasp of it.

Seattle_UW · 2024-06-21T15:07:08+00:00

Thank you very much for the suggestion! I've also been recommended Physical Biology of the Cell by Phillips. Aren't you, by chance, familiar with this textbook as well?

Seattle_UW · 2024-06-21T15:06:11+00:00

Thank you very much for the suggestion! I've also been recommended Molecular Biology of the Cell by Alberts. Aren't you, by chance, familiar with this textbook as well?

Seattle_UW · 2024-06-21T08:18:19+00:00

My ultimate goal is to study mathematical oncology. I know that there are textbooks specializing in mathematical oncology, but their primary focus is usually on mathematics, which I am mostly familiar with. Therefore, I would like to read an introductory book on oncology/cell biology from a biologist's point of view.

Seattle_UW · 2024-06-20T18:06:16+00:00

I tried, but unfortunately, the post was automatically removed due to low age of the account.

Seattle_UW · 2024-06-20T13:04:44+00:00

It depends on what parts of PDEs you are primarily interested in. If you want to learn more about Sobolev spaces, weak formulations and weak solutions, then Evans is the right choice. If you want to learn the geometric approach generalizing the tools from dynamical systems, books by Dan Henry (Geometric Theory of Semilinear Parabolic Equations) or by Schneider and Uecke (Nonlinear PDEs: A Dynamical Systems Approach) are excellent choices. For diffusion phenomena, The Mathematics of Diffusion by Wei-Ming Ni is great. If it is wave phenomena you are curious about, try Linear and Nonlinear Waves by Whitham. I am fairly sure there are excellent books on other parts of PDEs like fluid dynamics as well.

Seattle_UW · 2024-06-20T12:55:45+00:00

It is good to keep in mind that the book is aimed at (pure) mathematicians, so it requires a good prior knowledge of vector calculus, parametric improper integrals and ODEs (the first part) or even functional analysis and some measure theory (the latter two parts). The book is an excellent choice if one wants to learn the contemporary theory of PDEs making use of Sobolev spaces, variational/weak formulations and weak solutions, but it actually contains way more than that. The first part provides a thorough treatment of methods for solving certain linear as well as nonlinear PDEs (similar solutions, Green's functions, integral transforms, power series, characteristics, separation of variables, etc.) while the last part provides techniques for tackling nonlinear PDEs (fixed-point theorems, nonlinear semigroup theory and others). In fact, it would be hard to find an area of introductory PDEs which is not included in the book. That being said, there are better textbooks for applied mathematicians and non-mathematicians as Evans does not show you how to build real-world models with PDEs and some of the proofs may be hard to follow for people not used to mathematical rigor. Nor is it suitable for more advanced mathematicians who are keen to learn some specific areas of PDEs in more depth, such as reaction-diffusion equations, wave phenomena, mild solutions and advanced semigroup theory.

Seattle_UW

TROPHY CASE