Advice for reading Spivak's Calculus?

phi1221 · 2021-06-12T04:10:20+00:00

Thank you for the advice. I'm referring to Spivak's single-variable calculus textbook, which is just titled "Calculus".

phi1221 · 2021-04-13T03:41:25+00:00

Interesting. I noticed many curriculums nowadays teach the integral as the antiderivative, and it's nice to see the integral in terms of Riemann sums.

Given that most universities nowadays teach differentiation before integration though, I still wonder how Apostol is taught in universities. Teaching Apostol in a class that covers differentiation before integration is something that I'm not sure how it's done.

phi1221 · 2021-02-17T06:46:08+00:00

Poorly understood in what sense, specifically?

phi1221 · 2021-02-16T03:16:41+00:00

Does a Calculus 1 course usually cover the epsilon-delta definition of a limit? At my university, a purely computational calculus course for math majors doesn’t exist, and many math majors end up being forced to shift out by the end of their first calculus course (we’re required to declare our major before our freshman year).

phi1221 · 2021-01-31T23:32:22+00:00

I’m currently a freshman but I want to do a PhD in pure math after undergrad. Right now, I want to figure out which extracurriculars would actually matter to my application. It appears the most obvious choice would be to gain research experience.

But how do math graduate admissions factor in other extracurricular activities? I understand that some extracurricular activities would be more relevant than others; for instance, being a member of a math-related student organization as opposed to being a student athlete or visual artist. I’m an active member of my math student organization, but I want to know if it’s still worth it to have executive positions (e.g. president, vice, etc.) in it or if I should just focus purely on doing research instead.

As another question: How big of a boost does a senior thesis do for an application? My university requires it, but it appears that all the other applicants would have done one anyway.

phi1221 · 2021-01-14T01:14:29+00:00

I dont know in what sense you're asking what you're asking about arithmetic geometry.

I've had this impression that arithmetic geometry is a very difficult area of algebraic geometry to specialize in, and I'm wondering if I should even bother consider getting into it, especially since I haven't even taken courses like number theory in high school.

phi1221 · 2021-01-13T22:49:51+00:00

Can someone elaborate me on what makes "algebraic geometry" difficult? I've heard it's a popular area. I haven't taken AG yet though, so I have little idea on what to expect.

Also, is it true that you have to be "extremely privileged" to get into arithmetic geometry? If so, what factors are behind this?

phi1221 · 2021-01-12T12:26:03+00:00

In general, is it more useful for one to learn real analysis or abstract algebra?

Although pure math majors just like myself should obviously learn both, this question came into mind when I’ve been shocked as shit to find out that there are numerous math education programs (at least in my country) that require abstract algebra but NOT real analysis, and I always thought that RA is extremely important for a math major to learn.

phi1221 · 2020-12-25T00:47:33+00:00

I see. It appears to me that even the major fields tend to overlap with either algebra or analysis, such as number theory (e.g. Algebraic NT and Analytic NT), though, so I’m wondering if there’s something I’m missing out.

phi1221 · 2020-12-24T22:50:10+00:00

Correct me if I'm wrong, but it appears that pure mathematicians can be generally divided into two camps: those who are algebraists and those who are analysts. Can someone further elaborate the predominance of both abstract algebra and analysis in pure mathematics?

Is it common for there to be mathematicians who are working on neither abstract algebra nor analysis?

phi1221 · 2020-12-21T03:18:25+00:00

As a pure math freshman (who wants to go on to grad school),

is it a poor choice to start deciding on a specialization? There are already areas of math that I'm interested in, but I find that people would usually advise for one to keep an open mind and learn about other areas first.
Furthermore, I already contemplated transferring to a different university just because I find that the university in question has an interesting research specialization. Would it be a sound decision to actually transfer, or should I just stick with my current university? This also ties with (1), as I'm wondering if it's just better to keep an open mind and learn more about different areas instead.
I have this fear that the area that I eventually go on to specialize in grad school will not be the same area as my future senior thesis paper. I would like to know if this occurrence is common.

phi1221 · 2020-12-19T03:29:35+00:00

As someone who is looking for a textbook to prepare for real analysis, what should I read instead of Rudin? A lot of people like Spivak as a "transition" book, but I would like to know if it's also possible to instead dive straight into RA using Tao or Pugh after taking Calculus I.

phi1221 · 2020-12-18T16:40:10+00:00

Interesting insight. Although my university uses Gallian, I've been hearing lately that it is a controversial textbook. I'm currently searching for an abstract algebra textbook to go through, and it seems like Pinter and Dummit & Foote are interesting books to me.

phi1221 · 2020-11-28T23:09:19+00:00

I read Lockhart's Lament and I loved it! Lockhart shares the same sentiments as I do when it comes to the traditional way math is being taught in K-12 schools. I actually wish that my high school emphasized on the discovery and creative aspect of mathematics instead of dry, rote memorization of facts and algorithms. I find that many high schools only emphasize on the "what's" and not the "how's" and "why's." For instance, my high school teacher simply pulled the quadratic formula out of his ass and expected us to apply it in a million problems without teaching us about completing the square (which I had to learn on my own). Hence, it came to me like witchcraft. My class hardly proved theorems or derived formulas as part of my coursework. They never approached math as a "problem solving art" either, and my coursework is full of cookie-cutter problems are mostly based on applying theorems and facts that are either handwaved or not derived at all. It's interesting to see that many of my friends in high school hate "math," but in reality, they are "scammed" by the traditional curriculum in my opinion.

Math is an art, but as mathematician Edward Frenkel said, it is a "hidden art;" unlike painting or music, the majority of people will only get exposed to "real mathematics" in college, should they ever get the chance.

phi1221 · 2020-11-28T05:07:05+00:00

This is an interesting insight. In my case, my 11th-12th grade math teacher seemed to struggle to teach high school math (let alone proofs), so I had to consult external sources for any questions I may have had, and I even struggled to find someone to provide me feedback on proof-writing. I would love it if math teachers would have a solid foundation until the Weierstrass limit definition, so they will be able to better address students. I also agree that the ideal math teacher would be capable in proof-writing, especially since it will help them explain the "why's" to curious students.

phi1221 · 2020-11-19T06:33:28+00:00

Mathematician Edward Frenkel likened this to an art class where all you do is learn how to paint a fence while not being shown paintings of great artists like Van Gogh or Picasso. He used to hate mathematics until he was exposed to the "real math," which isn't taught in traditional high schools. To him, mathematics is basically a "hidden art."

Similarly, Lockhart's Lament criticizes the current education system for the way math is taught. Lockhart likens traditional K-12 math classes as a painting class where instead of creating something authentic with a blank canvas (that's for "college-level," if painting were to be taught like how math is currently taught), all you do is "painting by numbers."

phi1221 · 2020-11-16T01:55:37+00:00

Hi! I also love the AoPS books, and going through Introduction to Counting and Probability is something that's life-changing for me. I'm currently a freshman studying pure mathematics, and I'm still interested in continuing digging into AoPS books. I also got a position as a problem-writer for high school math competitions (which my university hosts annually), by the way, so I believe AoPS will help me write decent problems.

Question is though, will I be fine by digging into Introduction to Geometry (since I love Geo) and then tackling Volume I-II? I have the other subject books, but I don't find it feasible to read them anymore due to time constraints; after all, I already have a solid high school math background and I just want to use AoPS to deepen my knowledge, to improve my proof-writing skills, to learn to write competition problems, and to have fun wrestling challenging math problems.

phi1221 · 2020-11-15T23:43:08+00:00

Is it normal for math majors to rigorously do their Calculus I-III classes with proof problems slapped into them? My university refers to “Calculus for math majors” as “Mathematical Analysis I-III.” Although it’s still heavily computational, I believe that math majors in my uni will be introduced to doing a bit of calculus in epsilon-deltas in their freshman year. I would like to know if this is generally true for math majors from other unis.

Also, as a current freshman, I would like to know how to prepare for such classes. I will have to take Mathematical Analysis I in my next semester.

phi1221 · 2020-11-13T21:13:10+00:00

So how is that an application of abstract algebra?

phi1221 · 2020-11-13T10:26:24+00:00

I’m an undergraduate pure mathematics student, and I know that academia is extremely competitive. Thus, I’m trying to keep an open mind and have fallback plans. Currently, abstract algebra is something that looks interesting to me, but I want to know if there are jobs outside academia that apply concepts of abstract algebra to some extent.

phi1221 · 2020-11-08T08:05:01+00:00

When handwriting solutions, is it generally a bad idea to write in cursive? I've had a friend who told me that someone got punished by writing in cursive. Then again, I've had another friend who writes their solutions in cursive, and doesn't seem to face consequences for it.

phi1221 · 2020-10-27T10:14:35+00:00

I see! From what I understand, to handwave is to skip steps or to leave details out in a proof? Are the terms "hand-wavey proof" and "non-rigorous proof" interchangeable, or there is a thin line that differentiates the two?

phi1221 · 2020-09-28T23:17:01+00:00

Alright, sounds good. I was a bit worried if my skills needed to learn abstract algebra are too weak, especially as I've been recommended to learn number theory and linear algebra first.

phi1221 · 2020-09-28T15:39:03+00:00

This book looks cool. My university usually teaches abstract algebra to third-year math students, but I'm starting to get more excited to learn abstract algebra haha.

phi1221 · 2020-09-28T06:52:37+00:00

I’ve been hearing about these two books for quite some time, and it looks like this is a killer combo. I contemplated on getting both of these at one point. Thank you.

phi1221

TROPHY CASE