Questions tagged [mathematical-analysis]

For questions applying to analysis courses: Real and complex analysis. Typically a higher and more proof-based level than calculus.

Filter by
Sorted by
Tagged with
4
votes
3answers
144 views

The Riemann integral vs Lebesgue integral in several variables for advanced undergraduates

I am about to teach a second course in analysis for advanced undergraduate students. The students have already studied roughly the first eight chapters of Rudin's Principles of mathematical analysis. ...
19
votes
8answers
3k views

Fourier Animation

Are there any resources which show Fourier series approximating a given waveform? I am looking for examples which have a real impact on students and provides motivation. I am trying to find something ...
9
votes
3answers
634 views

Are the following topics usually in an introductory Complex Analysis class: Julia sets, Fatou sets, Mandelbrot set, etc?

I'm a Nero fan so I'm glad I learned about the Mandelbrot set, but I notice that said topics are not in Brown-Churchill or 'A First Course in Complex Analysis' while they are in Coursera's '...
23
votes
4answers
2k views

How should one tutor a student in undergraduate real analysis?

I am an undergraduate. Other undergraduates sometimes ask me to tutor them in an introductory real analysis course that covers the equivalent of the first half-dozen chapters of Rudin's Principles of ...
3
votes
1answer
139 views

Learning strategies for high volume/pace learning?

Background: I am a graduate student in a mid-tier U.S. university, and I am struggling. I feel like I during my undergrad, I haven't aquired the neccesary skills to keep up with the high volume/pace ...
22
votes
4answers
2k views

Lesson plan to self-teach real analysis to student with comp-sci background

For my background, I'm a software engineer currently studying for his master's degree in information security. But when that's all done, I plan on going back to mathematics to keep me busy. But with ...
5
votes
2answers
129 views

Introductory book or other resource on $p$-adic numbers/number theory/analysis

I am having problems understanding $p$-adic numbers/$p$-adic number theory/$p$-adic analysis. I have tried some notes on the internet, but these notes were not helpful. Can anyone suggest a book, ...
29
votes
5answers
2k views

The best way to introduce trigonometric functions in a rigorous analysis course

This is something I have always had issues with. Generally, three approaches are used: The geometric path: this follows the standard way how you would introduce these functions in school. The problem ...
2
votes
1answer
112 views

Advanced textbook for vector fields [closed]

I am currently reading Spivak Calculus on Manifolds and Munkres Analysis on Manifolds. I am looking for a more advanced text, especially on vector fields as they relate to the great conserved fields ...
10
votes
2answers
458 views

Is there research to back up the claim that math classes help develop analytical skills?

When I teach math classes, one goal I have in mind is to help students develop the cluster of thinking skills usually called analytical skills or critical thinking skills. And I think that math ...
3
votes
1answer
115 views

Introductory Analysis lecture slides

I will be teaching an introductory analysis course (see topics below) and I need some source-code Latex slides or PPT slides, and am willing to choose my textbook based on these slides (rather than, ...
20
votes
14answers
7k views

Why is it possible to teach real numbers before even rigorously defining them?

In mathematics, one can hardly study any mathematical concept unless it is clearly and rigorously defined. For example, without the definition the fundamental group, it is almost impossible to teach ...
34
votes
10answers
2k views

Reasons for (not) distinguishing $f$ from $f(x)$

Formally, if $f$ is a function, $f(x)$ is a value. So for instance, $f$ can be continuous, but not $f(x)$. In teaching at school and university, notation is quite often mixed up, e.g. the function is ...
1
vote
1answer
131 views

Making epsilon-delta proofs not just precalculus

In trying to find lecture-length videos of epsilon-delta proofs, I've found that almost all of them just start with the definition and then work through the algebra to get the answer. In effect, it ...
2
votes
0answers
135 views

Why are a.e. defined functions rarely mentioned in elementary books?

In any standard development of measure theory in several well-known textbooks, the use of almost everywhere (a.e.) defined functions are first seen in the statement of Fubini's theorem, which states, ...
6
votes
1answer
191 views

Is there any high school level summer program that teaches Analysis?

All summer programs I know for high-school students focuses on number theory, combinatorics, graph theory, logic, and all kinds of topics in discrete mathematics. (I am mainly interested in UK, US, ...
4
votes
2answers
282 views

The trick didn't like me (teaching Fourier transform)

I was teaching Fourier transform for engineering students. Since I didn't want to go into rigourous proofs during class, I often use intuition, just give students an idea to persuade them with the ...
18
votes
2answers
534 views

Comparison of different concepts of integral

As the following math stack exchange question (and answers) show: https://math.stackexchange.com/questions/703212/is-dxdy-really-a-multiplication-of-dx-and-dy There are a lot of different ways to ...
4
votes
2answers
117 views

Road map to teach undergrads a first course in real analysis that concludes with convergence of fourier series

I am planning to teach (unofficially, I am a Grad student) a course in real analysis. Aim of the course is to understand the convergence of Fourier series. I want to start with the notion of ...
6
votes
2answers
154 views

Why is there an emphasis on analysis courses in undergrad progams?

In undergraduate maths study, there are three main areas: analysis, algebra, and geometry. (There are of course other small topics as well, but they don't have to be learnt by every student.) I have ...
1
vote
1answer
166 views

How to improve mathematical skills(University level)?

I am doing Ph.D in Mathematics, I feel I lack few of the skills, if I can improve those skills I think I can do better as a Math scholar. I need some suggestion on these following(below I am talking ...
20
votes
4answers
912 views

Evaluating the reception of (epsilon, delta) definitions

Both education researchers and mathematicians discuss the challenge of (epsilon, delta) type definitions in real analysis and the student reception of them. My impression has been that mathematicians ...
12
votes
2answers
944 views

Introductory real analysis before or after introductory abstract algebra?

What are the pros and cons for students of taking introductory real analysis before or after introductory abstract algebra, assuming they are going to take both? I recognize that the overlap between ...
26
votes
8answers
2k views

Good motivation for the introduction of Lebesgue integral?

When students take a course on real analysis, they have likely learned about Riemann integrals. What is a good motivation why they have to learn a new way to integrate? A student don't want to hear ...
8
votes
3answers
284 views

How to make students comfortable with the use of axiom of choice in analysis

I am teaching introductory real analysis this term and realize that my students have problem coming up with sequence in some arguments in real analysis. Let's take this example: Theorem: Given a ...
11
votes
9answers
6k views

Why do we study ordinary differential equations?

What is a good answer to the question: Why should one study ordinary differential equations? I would give the answer: ODEs are used in many models to determine how the state of this model is changing ...
2
votes
2answers
222 views

Learning proofs in introductory analysis courses

I have browsed the website a lot and I encountered many similar questions but not a question that asks the same question as I intend to. In introductory undergraduate classes in Analysis, usually, ...
12
votes
2answers
384 views

How to test knowledge on the real numbers in a written exam?

In German universities, the first-year students typically start their analysis courses with introducing the real numbers. Most commonly, the incompleteness of $\mathbb{Q}$ is discussed using the ...
13
votes
4answers
494 views

How to deal with “Why can't I just do …” in real analysis?

I'm teaching introductory real analysis at a large public university in the US. A common question from students is of the form "Why can't I just do it like this?". i.e. Often a student has come up ...
8
votes
1answer
214 views

Flipped introductory real analysis resources?

I am going to teach a flipped real analysis class next term, using Abbott's book. Does anyone know of resources for such a class? I have found the article: "Flipping the Analysis Classroom" by ...
3
votes
0answers
266 views

A proof based Multivariable Calculus and Linear Algebra

May I know how can I teach a proof-based Multivariable Calculus and linear algebra as a single course? While there are quite a few known books in the field such as: 1) Vector Calculus, Linear Algebra ...
7
votes
4answers
529 views

Why do we study Cantor Set?

For finding counter examples. That does not sound convincing enough, at least not always. Why as a object in its own right the study of Cantor Set has merit ?
16
votes
4answers
898 views

What is the motivation for characterizing second order linear PDEs as hyperbolic, elliptic, or parabolic?

I'm teaching an Intro to PDEs course (I'm an analyst, but PDEs are a bit outside my bailiwick) and I'm covering the basic examples: Heat, Wave, and Laplace. How should I move from these examples to ...
19
votes
7answers
1k views

How can I motivate the formal definition of continuity?

In order to teach continuity of real valued functions $f:D\to\mathbb R$ one may start with the (in some sense wrong) intuition $f$ is continuous when its graph can be drawn without lifting the pen. ...
6
votes
2answers
355 views

A Plan for a Treatise Study of the Classical Theory of PDEs

The Plan In the study of any special issue in mathematics, two things may be of importance, namely, subjects and order of them. I just wrote down a plan to study the theory of partial differential ...
3
votes
2answers
182 views

Supplemental text for undergraduate real analysis

Context: I am an assistant professor at a small college in the US. Next semester I am teaching real analysis for the first time, and we are using Steven R. Lay's book. (It also happens to be the ...
5
votes
3answers
233 views

What made (abstract) algebra grow in relative importance?

Nowadays, when I look at mathematics programs of study, "algebra" (at the abstract level) and "analysis" are treated as equally important. I'm "dating" myself, but this did not appear to be true in ...
11
votes
1answer
290 views

Motivation for uniform continuity

What are some problems or theorems that motivate the distinction between continuity and uniform continuity? In particular, I would like: a) A useful, appealing theorem that applies to uniformly ...
11
votes
4answers
341 views

How would you introduce Frullani integral to students?

Some integration techniques are just "tricks", while some integrals are analytically significant in that they connect different fields of math or they embody higher level concepts. In the ...
5
votes
2answers
282 views

An application of the Cauchy criterion for undergraduates?

The Cauchy criterion is used to prove the convergence of sequences $(a_k)$ with unknown or irrational limit: If for every $\epsilon > 0$ there is a $k$ such that for $m, n > k$ we have $|a_n-a_m|...
8
votes
3answers
1k views

Interesting but very easy epsilon-delta problems?

I am teaching a real analysis class. Students in the class have inconsistent high school algebra skills. They now have a complete but tenuous understanding of $\varepsilon$-$\delta$ limits. I want to ...
2
votes
0answers
103 views

Which book to use concurrently with each of these mathematics texts?

I'm in search of a good book that I can read --- and recommend to my proteges to read --- along with each one of the following books. Topology by James R. Munkres, 2nd edition Introductory ...
0
votes
1answer
96 views

Text book on real analysis for undergrad in statistics

May I get some recommendation on text book on real analysis for undergrad in statistics? Thank you.
21
votes
8answers
1k views

Counterintuitive consequences of standard definitions

Let me motivate my question with the following situation. While teaching the concept of continuity, I usually start with motivating the concept. Then, when we see that there is an important and ...
23
votes
10answers
6k views

Why would you teach Calculus before teaching Real Analysis?

Let's assume our students are actual aspiring mathematicians. Why would we introduce our students to Calculus rather than Real Analysis? After all, "Calculus is a subset of Real Analysis". He will ...
15
votes
4answers
549 views

How would you explain what a PDE is to a very educated layman with no math background?

Is every mathematical concept, even the complex ones, explainable? As someone who will be needing to explain my line of work for a position to a committee who is very, very, educated, just not in ...
-3
votes
3answers
596 views

How can I convince my brightest student of Cantor's theory?

At the end of the mathematical high-school education I usually introduce the easiest facts of set theory: counbtability and Cantor's proof as the basis of modern mathematics. Now my brightest student ...
25
votes
5answers
4k views

When did US mathematics programs start failing to prepare incoming students for books like “Baby” Rudin?

I've seen in a lot of questions about "which textbook to use for intro analysis", and inevitably Rudin's Principles of Mathematical Analysis comes up, with the (almost cliche) rejoinder that "today's ...
13
votes
3answers
451 views

Teaching Infinitesmals and Non-Standard Analysis

This question is asked from a self-teacher standpoint(I am currently trying to learn more about non-standard analysis on my own), but I'd think it could be applicable to educators also. What are good ...
7
votes
2answers
223 views

How to catch students from different subjects' interest to math?

I have just started to teach Calculus to freshmans and sophomores who study non-mathematical subjects, e.g., international relations, psychology. They have to take few mathematics classes -including ...