This might be a little personal, but it’s still related to maths.

I honestly don’t feel like I’m fit to be a math major. For starters, I struggle to form a train of thought when solving math problems. It’s like my brain refuses to focus on problem, and I just end up staring at it. If it’s not immediately obvious, then the chances of me coming up with a solution are dim.

This happened today in class. We were learning about double integrals, and we got to a problem involving trigonometry. I sat there and tried solving the problem, before I did something wrong. I didn’t know what exactly I did wrong, but I just knew something wasn’t right. I tried scanning the problem for any mistakes, but I couldn’t form any type of thought process on the problem. I eventually gave up and waited until my professor solved it.

I also have pretty crappy memory, which is unfortunate because I relied on memory for the entirety of precalculus - calculus 3. I often times forget very elementary things. For example, when were doing double integrals, we got to a problem with exponents. It was like I forgot that e^r + 2s can be separated into e^r • e^2s. I also still don’t remember the cosine and sine of basic trigonometry angles. I still keep a formula sheet for that.

Another example pertains to my exam. I spent all of yesterday cramming, and a part of this morning as well, just to open up the practice test, and forget how to solve the first question. I had to look at the example and got it immediately. So now I’m going through the entire test of 37 questions, and will probably have to relearn everything.

Stuff like this is so demoralizing. I wanted to major in pure mathematics because I thought I liked it and it was the only thing I’m semi good at. Is there just a problem in my thought process or is it something else?

I also have ADHD (I am medicated, but it’s a really low dose because I just started it.), but I’m sure many people with ADHD (and more) still do amazing in math.

I'm at chapter 7 of the book, in which Tao introduces the notion of series, I'll introduce 2 definitions below (as context) before getting to the problem.

First, Tao defines a finite series on a sequence as follows: (See [1] in the attached image)

And then he defines summations on a finite set X and a function f: X -> R as such: (See [2] in the attached image)

[It's also proven later that this definition is well-defined and gives off the same value for every bijective g]

So far so good, the problem is that on a lemma, he claims the following: (See [3] in the attached image)

I get the maths Tao is trying to convey, but shouldn't the left hand side of the equation be syntactically invalid? Regarding the inner sum, y in Y isn't a valid input for the function f(x,y) as it isn't an ordered pair (x,y).

I think a way to resolve this would be to define double-summation on a function through another 'intermediary' function (f_x)(y) (See [4] in the attached image)

and then we can proceed with proving the proposition by using the original definition of summations on a finite set.

Edit: Also the x subscript in f_x is (meant to be) purely notational, it has nothing to do with the actual input x. Note also that [3] is not a definition, it's presented as a lemma (so the fact that the LHS is syntactically invalid does matter here) and that the image above is not Tao's book, I only wrote the necessary definitions and the lemma (that I have a problem with) down for the sake of discussion.

In topology, a pair of pants is a surface with three boundary components: one waist opening and two leg openings.
Suppose we take two identical pairs of pants and sew them together by matching the two leg openings of the first pair with the corresponding two leg openings of the second pair. The two waist openings remain unsewn.
What is the genus of the resulting surface? Is it a torus with two boundary components (genus 1), or am I misunderstanding how the sewing affects the topology?
I’m particularly confused about whether the large opening in the middle, formed by joining the leg openings, counts as an additional hole.

Hello all, I just started my MSc in statistics, after a background in Economics where I desperately tried to bridge the gap through self study. In my thesis I had the chance to read some chapters in Optimal Transport and manifold theory. It was really a great opportunity but still my understanding was very limited cause I have never done even an analysis course (just basic calculus and linalg, and measure theory which I struggled with during the proofs and exercises).

Now I decided for first semester to follow mathematical statistics and probability theory, cause I really really wanted to follow maths-theoretical modules and also have the right foundation for further courses. I thought that taking only these two courses would be beneficial because I'd have more time to study them deeply, but I feel like I am just taking the wrong choices: I am so slow at understanding how to do the Prob theory exercises because I feel like they use analysis tools stuff which I am not familiar with, and also I am 'familiar' with the measure theoretical tools but I keep going back to re read stuff like continuity of measures, or the fubini theorem which I didn't even know the existence of. So yeah maybe not the brightest choice, I was thinking of going through abbott's understanding analysis WHILE taking these two courses but still it feels like I am so behind in terms of mathematical knowledge.

So my question is basically if maths should always be studied in a linear order, or if some things can be done while also taking more difficult courses; I understand that it makes sense to build knowledge starting from basic stuff like analysis and only after build up on it, or e.g. take probability theory only after having studied measure theory, but is it sometimes somehow possibile to learn some things backwards or in parallel with the courses that actually require them? I mean like taking a course we don't have the background of, like prob theory, and learn the analysis tools used in the proofs and exercises from it? Or is it just more correct to give up and take certain courses only after making sure that we have the prerequisites?

i know the basel problem proof with the sin(x)/x product but it feels like magic that a circle constant shows up in a sum of integers. is there a more intuitive reason or is it just that sin has those zeros at multiples of pi

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Hi everyone!

The general idea is to investigate whether wave packet constructions can be adapted to the geometry of a sphere, potentially connecting spherical harmonics, discrete angular modes, Fourier methods, and complex coordinates.

At the moment, I'm exploring how a discrete spectrum might affect the structure of a wave packet and how its angular localization could be described. I'm also interested in the possible role of stereographic projection and Möbius transformations in representing these states.

This is only an early draft of an idea, not a finished theory or a claim of an original result. I'm still studying the mathematical foundations and figuring out which parts of the approach are meaningful and mathematically consistent.

I'd really appreciate recommendations for books, papers, or related research that I should look into. If anyone has suggestions, sees a fundamental issue with the approach, or would like to discuss or contribute to the project, I'd be happy to hear from you.

Thanks for reading!

I'm debating studying applied math at my university, but I'm worried about not finding work. So I'm curious if any people who recently graduated have been able to find employment.

Edit: Having graduated in the last five years.

My engineering course requires to study the above mentioned topics. The books they recommend in college only tells me how to solve a problem and not how that method works. I need books that can clearly explain all the methods involved in the aforementioned topics. I would also appreciate some tips to study engineering mathematics in general.

I was solving a nice NT problem while trying to generate a formula from scratch and solve it... i spent 40 mins solving it and I just realized I have generated the wrong formula...

I think the last few weeks have raised some important questions on what Mathematics is, and what it's for, that might be raised about a lot of other fields in the future, so I wrote about it!

I'm 24 and trying to rebuild my math skills from the ground up so I can pursue mechanical engineering. Where should I start?

Hi everyone,

I'm 24 years old, and I'm trying to improve my math skills after struggling with math throughout much of my education. I'm hoping to get some advice from people who understand mathematics and might be able to point me in the right direction.

I have an autism diagnosis and had an IEP in school. Looking back, I feel like I was moved through grades without fully understanding some of the material. As a result, I have gaps in my mathematical foundation that I'm now trying to address.

I would estimate that some of my foundational skills are around the elementary-school level, although I can solve certain problems beyond that level. I can handle basic arithmetic, decimals, simple equations, and some geometry, but I struggle with other topics, especially when several steps are involved.

Some of the things I have difficulty with include:

• Simplifying fractions and finding common factors mentally.
• Adding, subtracting, multiplying, and dividing fractions.
• Long division and doing calculations efficiently.
• Ratios, proportions, and percentages.
• Negative numbers and algebra.
• Unit conversions and more advanced geometry.

For example, I might understand how to solve a straightforward problem but struggle when the numbers become less familiar or I have to figure out which method to use on my own.

My long-term goal is to study mechanical engineering, so I know I have a lot of work ahead of me. I'm interested in aerospace, robotics, aircraft, and hands-on engineering projects, which is part of what motivates me to keep learning.

I'm not looking for shortcuts or a way to avoid the work. I'm willing to start at the beginning and put in the time. What I need is a sensible path that helps me build a genuinely solid foundation instead of constantly discovering gaps in what I thought I knew.

My questions are:

1. If you were in my position, what exact sequence of math topics would you study, starting from the basics and working toward calculus?
2. What textbooks, free courses, websites, or other resources would you recommend for an adult rebuilding their mathematical foundation?
3. How can I get better at recognizing which method to use when solving a problem, rather than simply memorizing procedures?
4. Is it realistic to work toward a mechanical engineering degree starting from this point, and what should I prioritize to prepare myself?
5. For anyone who struggled with math earlier in life, what helped you finally understand concepts that previously seemed difficult?

I'd especially appreciate advice from mathematics teachers, tutors, engineering students, or people who have successfully rebuilt their math skills as adults.

I know this will take time, but I genuinely want to learn and understand the material rather than just get the right answers.

Thank you for reading!

Hi all, I'm currently an undergraduate student studying applied math, and I'd like to conduct research but I'm struggling to figure out how, and in what areas. Actual math research itself seems to be really hard to conduct for undergraduates because unlike other fields, there's no "busy work" for you to do. Would research in computer science, computational mathematics, or applied mathematics generally be more accessible for undergraduates?

I'd love to hear from other undergraduates who have conducted research: How did you find your research opportunities, what area did you work in, and what did your experience look like? Thanks!

There’s this law called the 10000-hour rule, which states that to truly master a complex skill one needs to spend 10000 hours on it.

I know that there are a few criticisms on it (which mainly says that one needs 10000 of intentional practice to master a skill), but I figure that 10000 hours could be a good bench mark to see if I have invested enough time in math.

For context, I just graduated from a 4-year programme in a math major. So 10000 hours is equivalent to in average 10000/(365 x 4)=6.849 hours per day, for the past four years.

I definitely didn’t spend 7 hours per day to study math, not to mention the time spent for intentional study. This changes my mindset completely. I have been doubting myself for not being ‘fluent in math’, but as of now it’s most likely that I did not give enough time for myself to grow.

EDIT: People seem to think that I believe the statement that "after fulfilling 10000 hours of practice, then one will magically be good at something". That's pure BS and I don't buy it, at least for maths. I am just using this "10000-hour rule" to examine my time invested in math. The main point is that I shall and I will work harder, and not to expect be good in such a short time, such as 4 or 5 years, despite this is what many people considered to be a lot time.

This one I made with elliptic integrals, transformations... Hope you like this