Questions about the process of studying mathematics without formal instruction.

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1answer
313 views

Effect of pullback of differential forms on an ideal

Say that the exterior differential system (EDS) corresponding to a PDE system is: $$df-f_x\,dx-f_y\,dy-f_w\,dw-f_z\,dz=0,\\ a_1\,f_x+a_2\,f_y=0,\tag{sys}$$ Of course we also require the independence ...
8
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1answer
273 views

Journals or Magazines on Study Skills or How to Study Math

I am trying to find only journals or trustworthy magazines which can help math students to study math more efficiently and productively. I am not asking about books in this thread. In particular, I am ...
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2answers
2k views

How do I get good at Math?

How do I get good at math? I'm a freshman in college, and I've always done OK in math. I never had any good teachers in High school, and I always have done the bare minimum. Over the course of this ...
8
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1answer
321 views

“Easy” (maybe not) question about dual spaces (Lineal Algebra).

Hi everyone is my first time reading about dual spaces and in one part of the notes that I read, says: The dual of the quotient space $V/U$ is naturally a subspace of $V$, namely the annihilators of ...
8
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1answer
76 views

Importance of Exercises in Mathematics for Self-Studying

I am a high school student wanting to major in Mathematics in the future. I started to like Mathematics recently, starting a year ago and I watched some interesting math videos on YouTube for fun (ex: ...
8
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1answer
158 views

Math competitions for hobbyists?

Are there any math competitions for hobbyist / amateur mathematicians? Something like the Putnam or the International Mathematical Olympiad, but open to regular people who are not full-time students?
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3answers
149 views

What to look for in a proof?

I am a physics undergrad, wishing to pursue a PhD in Math. I am mostly self taught in the typical math undergrad curriculum. I am looking for more input, in ways I can improve my mathematical ...
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2answers
164 views

Given $N$, count $\{(m,n) \mid 0\leq m<N, 0\leq n<N, m\text{ and } n \text{ relatively prime}\}$

I'm confused at exercise 4.49 on page 149 from the book "Concrete Mathematics: A Foundation for Computer Science": Let $R(N)$ be the number of pairs of integers $(m,n)$ such that $0\leq m < N$, ...
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4answers
3k views

Books, Video lectures, other resources to Teach Yourself Analysis

So my limited mathematics education has been especially ignorant of analysis. In this vein, I'd like to teach myself some of the introductory basics. I'm intrigued by sources that might contain ...
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8answers
2k views

Self-study resources for basic probability?

I am taking a Computer Science class soon that requires a solid knowledge of the basics of probability. I've only had minimal exposure to probability in classes I've taken in the past, so I need to ...
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4answers
1k views

Expanding problem solving skill

I have a great passion for Math but my lack in problem solving skill always keeps me away from the "good stuff". I always wanted to be better at Math and one of the things I figured out was to keep ...
7
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2answers
721 views

Prerequisite reading for Weil's Basic Number Theory

I'm interested as to what would constitute prerequisite reading for André Weil's book Basic Number Theory. For simplicity (and generality), you can assume that the reader can read anything that ...
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4answers
6k views

Just how much proof knowledge is necessary to begin Spivak's Calculus?

I bought Spivak's classic Calculus a month or so ago, and after doing a few problems from the first chapter, it's become apparent that some type of foundational knowledge in formal maths and proofs ...
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3answers
322 views

How do i visualize Cosets of a group

The Lemma asserted in Herstein as given by $[a] = Ha$ seems very non intuitive to me. How do I think in order that this thing makes sense to me? LEMMA 2.4.4 For all $a$ in $G$ , $$Ha = \{ x \in ...
7
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2answers
101 views

How to show this inequality?

Show that $$-2 \le \cos \theta(\sin \theta+\sqrt{\sin^2 \theta +3})\le2$$ Trial: I know that $-\dfrac 1 2 \le \cos \theta\cdot\sin \theta \le \dfrac 1 2$ and $\sqrt 3\le\sqrt{\sin^2 \theta ...
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2answers
5k views

Rigorous Text in Multivariable Calculus and Linear Algebra

So I'm wanting a solid math book for Christmas. I have a solid background in Calculus and am currently working through baby Rudin. I really want a rigorous book dealing with multivariable calculus ...
7
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3answers
957 views

Help with proof that $\mathbb Z[i]/\langle 1 - i \rangle$ is a field.

I have been having a lot of trouble teaching myself rings, so much so that even "simple" proofs are really difficult for me. I think I am finally starting to get it, but just to be sure could some one ...
7
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3answers
116 views

How do I decide what problems and how many problems to do when I try to self study?

I am a math major at a relatively small college with barely any choice of classes to choose from so I have to supplement my studying with a lot of self studying. I usually have no problem getting ...
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3answers
320 views

Is formal logic necessary for pure/“higher” mathematics?

I'm asking this as an autodidact who wants to learn math rigorously for its own sake. And I was just wondering if understanding proofs could be achieved without a formal grounding in symbolic logic. I ...
7
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1answer
471 views

What pure mathematics foundations should an applied mathematician have?

I'm studying mathematics, with some statistics also, and I've always chosen applied courses. I'm getting to the point where I'm studying 3rd year undergraduate to graduate level material. My first ...
7
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1answer
113 views

Find $\sum_{k=1}^{\infty}\frac{1}{z_k^2}$

Let $z_1, z_2,\dots, z_k,\dots$ be all the roots of $e^z=z$. Let $C_N$ be the square in the plane centered at the origin with siden parallel to the axis and each of length $2\pi N$. Assume that ...
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5answers
702 views

Arithmetic mean is less than geometric mean (Spivak Calculus 3rd Chapter 2 Problem 22)

If $a_1, \ldots, a_n \ge 0$, the arithmetic mean $$A_n={a_1 + \cdots + a_n \over n}$$ and the geometric mean $$G_n = \sqrt[n]{a_1 \cdots a_n}$$ satisfy $G_n \le A_n$. As a first step to prove this ...
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3answers
95 views

What does one study to increase understanding of the $P \stackrel{?}{=} NP$ problem?

If one were to learn more about the $P \stackrel{?}{=} NP$ problem, where would one start? I understand what the problem is—but not enough to be able to read anything technical about it. ...
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3answers
129 views

Is there a better way to read proofs?

I'm finishing my undergraduate degree in 6 weeks and I'm pretty happy with how my education is coming along so far. I can write proofs, solve many different problems, and I even have some idea as to ...
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3answers
285 views

Self-study: what fractions of problems to solve?

I am self-studying measure-theoretic probability out of Billingsley's Probability and Measure. So far I have been trying to solve all the exercises. While the exercises are wonderful and I can ...
7
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2answers
159 views

How to get used to commutative diagrams? (the case of products).

I've returned to Aluffi's book (after getting the basics of groups from Herstein's) and I hit the same brick wall that made me put it aside. My general problem is this: I can't seem to get used to ...
7
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1answer
539 views

Self Study of Fractals

I am looking for a book to self-study fractals with a certain criteria. I have checked out Getting Aquainted with Fractals. Note that Getting Aquainted with Fractals does not include ...
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2answers
87 views

Are basic trigonometry functions ( sine, cosine, tangent ) intuitive or memorized?

First, I'm really sorry for this somewhat vague and possibly just silly question. I also apologize if the following context runs a bit long. But please trust me that I'm asking with total sincerity ...
7
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1answer
184 views

“Visualizing” Mathematical Objects - Tips & Tricks

It has been a while since I am kind of stuck with my skills concerning the visualization of mathematical objects. Here there is the problem. First of all, let me point out that I am completely ...
7
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2answers
3k views

Some basic practical applications of Calculus

I am currently studying Calculus on my own for fun. I enjoy different components of math and how they can be used to solve so many problems. Many people, however, think I am crazy because I am ...
7
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1answer
293 views

Basics of schemes and morphisms of schemes

I'm currently reading through Hartshorne, and have come across a few things that have left me wondering. (i) Somewhat pedantic, but also because I don't actually know the answer, (in Example 2.3.4) ...
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1answer
230 views

Intuition behind proof and verification partially ordered sets

Hi everyone in the book that I read I have trouble to understand the argument of the proof at the below proposition. There is a lot of point which are left as exercises, which is great. One of these ...
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3answers
4k views

Functional analysis textbook (or course) with complete solutions to exercises

I am a Ph.D. student in economics and I plan to study functional analysis by myself either this winter or the next summer. I am currently looking for a textbook, and since I am studying it by myself, ...
7
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1answer
647 views

Quick questions about studying math and physics? [closed]

Just curious about how people usually self study these subjects. 1) Is it the most efficient to read through the chapters, and write down theorems, definitions, and take notes of important parts and ...
7
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1answer
620 views

Philosophy of learning mathematics (combinatorics)

How does one learn mathematics in depth and well? I think I may be studying mathematics incorrectly, or I could be doing it a lot more efficiently/effectively. Sometimes I feel like I need every ...
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0answers
1k views

Learning higher-mathematics on your own

I was hoping someone had an opinion on how to learn higher-mathematics (specific fields that could be of use to me) outside of a classroom setting. I graduated with an M.S. in Computer science about ...
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6answers
2k views

What math is necessary to learn manifolds?

I want to learn about manifolds, but I'm only a senior in high school and obviously have a while to go. I'm in AP Calc BC. What should I study to eventually learn manifolds? Linear Algebra? What ...
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7answers
798 views

A book for abstract algebra with high school level

Any book that I find on abstract algebra is somehow advanced and not OK for self-learning. I am high-school student with high-school math knowledge. Please someone tell me a book can be fine on ...
6
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4answers
157 views

Evaluate $\int_0^{\infty} x p^xe^{\Large-\frac{x}{a}}\ dx$

I need to evaluate the integral: $$\int_0^{\infty} x p^xe^{\Large-\frac{x}{a}}\ dx$$ for $0<p<1$. Unfortunately I do not know where to begin. I tried integration by parts but got nowhere ...
6
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7answers
590 views

Good Number Theory books to start with?

I'm in Grade 11. I'm interested in elementary number theory and would like properly study it. I'm not intending to enter any competitions.
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3answers
845 views

Why are we interested in irreducible representation but not faithful representation?

I am reading some materials of representation theory (of a group). The motivation of representation theory is to represent (by a homomorphism $h: G \to GL(V)$, from the group $G$ to a vector space ...
6
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3answers
512 views

The Importance Of Good Teachers and Guidance In the Academics

I'm a first year student for a math degree. I'm very curious on how good students overcome their bad teachers in the journey of learning and grasping the courses material fully, all in the pressure of ...
6
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3answers
738 views

Mastering Math - Grade school- to College-level?

I'm having difficulty with my math, fractions and up. I used to understand it all, but it's been so long since I've touched the book (I finished it a couple of months ago, picked it up to review ...
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6answers
5k views

Steps to Re-Learn Mathematic the right way

I was always branded weak mathematics back at school though i loved it. The reason was that I was not having many basics in maths. Due to this problem I stopped learning Maths after my 10th grade. But ...
6
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3answers
289 views

Beginning with math

I am studying computer science since 3 years now. It is really math heavy and I like it. However the problem that I have is that I never really had math in school it was too basic and I lack some ...
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2answers
3k views

Self-study on probability.

What book do you recommend for self-study of probability theory? I have a rather significant gap in that area (in lame terms sometimes I feel I don't get it) and need to try (strugle more likely) ...
6
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4answers
883 views

Can a function with just one point in its domain be continuous?

For example if my function is $f:\{1\}\longrightarrow \mathbb{R}$ such that $f(1)=1$. I have the next context: 1) According to the definition given in Spivak's book and also in wikipedia, since ...
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5answers
832 views

Materials for self-study (problems and answers)

I'm hoping to self-study Geometry, Algebra, Calculus, Vector Calculus, Linear Algebra, Probability and Statistics, and other intermediate maths. I've found the best way for me to learn is to work on ...
6
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3answers
330 views

Equivalence Relation between Derivative Being Odd and Function Being Even

In the exercise, I am required to prove that $f'$ is odd $\iff$ $f$ is even Moving from right to left was pretty trivial, however, I couldn't move from left to right. Note that we can only use very ...
6
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3answers
203 views

How can I find all the solutions of $\sin^5x+\cos^3x=1$

Find all the solutions of $$\sin^5x+\cos^3x=1$$ Trial:$x=0$ is a solution of this equation. How can I find other solutions (if any). Please help.