For questions applying to analysis courses: Real and complex analysis. Typically a higher and more proof-based level than calculus.
3
votes
2answers
98 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
175 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
142 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
280 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 commonly ...
5
votes
2answers
125 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|...
7
votes
3answers
337 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
89 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 ...
4
votes
0answers
103 views
Are there any high school level summer program that teaches Analysis?
All summer programs I know for highschool students focuses on number theory, combinatory, graph theory, logic, and all kinds of stuffs in discrete mathematics. (I am mainly interested in UK, US, ...
0
votes
1answer
74 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.
-3
votes
3answers
517 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 ...
6
votes
2answers
190 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 ...
16
votes
10answers
3k views
Complex numbers in high school
Are complex numbers taught in high school in other countries? I am from Germany and complex numbers are next to never touched in high school with the exception of extra-curricular activities, for ...
7
votes
1answer
155 views
Lipschitz continuity before standard definition of continuity
In Practical Analysis in One Variable, Donald Estep introduces Lipschitz continuity early on, delaying the standard definition of continuity, along with uniform continuity, until the beginning of his ...
14
votes
4answers
490 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 ...
10
votes
9answers
2k 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
1answer
95 views
Proving convergence or divergence of series: tips and recommendations
This is a follow up of my question on MSE. Which tips and recommendations would you give students who want to investigate series about convergence or divergence?
So far we have collected:
It is ...
11
votes
6answers
1k views
Why should we study continuity?
This question is related to How can I motivate the formal definition of continuity? Imagine a student asks the question why it is worth it to study continuity. What is a good response to this question?...
21
votes
10answers
5k 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 ...
4
votes
2answers
438 views
Why is continuity only defined on its domain?
As mentioned in this question students sometimes struggle with the fact that continuity is only defined at points of the function's domain. For example the function $f:\mathbb R\setminus\{0\} \to \...
3
votes
1answer
364 views
Why is continuity defined as a local property?
The formal definition of continuity is a local property (the definition of continuity at a point is a property of the germ of the function at this point). Why is it a good decision to make the ...
18
votes
7answers
764 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.
...
8
votes
5answers
343 views
Rubric for grading an undergraduate real analysis course
I find it extremely time consuming to grade a homework in an undergraduate real analysis course without a rubric. Several instructors I worked with did not have a clear rubric in their mind at all. ...
6
votes
1answer
311 views
Why is multivariable analysis often omitted?
Related but not duplicate: What courses require multivariable analysis?
By multivariable analysis I mean the rigorous version of multivariable calculus (something equivalent to Ch.9-10 in baby Rudin ...
5
votes
4answers
463 views
What is the intuition behind the limit superior?
I want to write an article which explains the limit superior. I also want to present the intuition behind this concept. Currently I would describe the limit superior as the "least upper bound of a ...
6
votes
2answers
301 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 ...
24
votes
5answers
3k 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 ...
14
votes
1answer
245 views
Spiral learning in real analysis
Has there been any attempts at developing a curriculum for teaching analysis (here let us be narrow and say real analysis in the sense of rigorous integral and differential calculus) in a multipass, ...
4
votes
1answer
228 views
Brief book on calculus to read before studying the analysis
I am going to start studying the analysis texts (Rudin-PMA, Apostol-MA, Pugh-RMA) on the first week of August. I have a good proof skills through working on Artin's Algebra and Hoffman/Kunze's Linear ...
5
votes
2answers
366 views
Inquiry about My Self-Study Plan for Real Analysis (for my research and self-enrichment)
I am a college sophomore in US with a major in mathematics and an aspiring mathematician in the computation theory and cryptography. I recently got an undergraduate research in the computation theory, ...
3
votes
0answers
133 views
What is the point of using half range Fourier series for standard functions?
If we have a standard function, like $f(x) = x$ or $g(x) = x^2$, defined between $0$ and $\pi$, then why should we be interested in extending this function to give a Fourier series that resembles this ...
18
votes
7answers
2k 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 ...
1
vote
0answers
32 views
References help [duplicate]
a couple of days ago i asked question about how to study mathematical analysis in mathematics
educators.stack exchange and concerning my desire to learn mathematics i kept searching and found a ...
4
votes
2answers
204 views
Mathematics curriculum and book titles to study mathematical analysis for post-grad studies
I am an engineering student trying to study mathematical analysis because it will help me in my post graduate studies.
My problem is that when I searched the internet I found that some university ...
8
votes
6answers
303 views
Any metaphors/intuitions for a limit of a sequence?
I'm writing (together with a colleague) a minicourse on mathematical analysis (currently we want to cover the Weierstrass theorem on functions on compact intervals, so the aim is to present only the ...
4
votes
2answers
228 views
Real analysis: why usually first limits of sequences and then limits of functions?
I notice that all of the analysis books that I've studied start from dealing with limits of sequences and only then move on to limits of functions.
Does this kind of approach have any particular ...
1
vote
0answers
47 views
Curriculum Standards [duplicate]
Please, Is there any standards for mathematical curriculum and their corresponding book titles and standard prerequisite graphs for fields of mathematics, as am trying to study analysis, I have ...
1
vote
4answers
2k views
Prerequisites of mathematical analysis [closed]
What topics should I read before studying mathematical analysis?
I want to have a solid foundation in terminology, notation and concepts in general.
Please suggest titles for books.
6
votes
4answers
544 views
Complex analysis (Applied versus pure)
I am studying Electrical Engineering and I want to specialize in signal processing.
However, I have to study complex analysis first (I am an undergraduate, so I lack some terminology). In your opinion:...
11
votes
2answers
1k views
Advanced Calculus vs. Analysis for a first proof-based course
Question: Why was advanced calculus removed as the first proof-based course in favor of real analysis in most curriculums?
I regularly see in advanced calculus books either that:
its purpose is, ...
13
votes
2answers
680 views
Introducing the Lebesgue integral before Riemann's
Has anyone attempted to introduce, or has data on such endeavor, Lebesgue integration before Riemann? I've seen many discussions about how the Riemann integral is obsolete and that it is presented ...
2
votes
0answers
121 views
Searching for an analysis textbook [closed]
Particularly, I'm looking for an undergraduate text with an excellent explanation of $\delta$-$\varepsilon$ proofs, and many example questions related thereto.
16
votes
3answers
553 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 ...
14
votes
3answers
590 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 ...
8
votes
3answers
309 views
What courses require multivariable analysis?
For which undergraduate and graduate mathematics courses is multivariable analysis* an essential prerequisite?
$\text{*}$ That is, a rigorous follow up to a first real analysis course at the level of ...
19
votes
8answers
956 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 ...
24
votes
4answers
1k 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 ...
18
votes
4answers
1k 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 ...
11
votes
3answers
153 views
Introducing (Borel) measures via Riesz representation theorem
I think, the most standard way is to introduce measures in real analysis is to define them via the usual properties like $\sigma$-additivity, etc.
However, if the students are familiar with ...
13
votes
3answers
351 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 ...
11
votes
2answers
280 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 ...
