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Questions tagged [mathematical-analysis]

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

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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 ...
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0answers
117 views

Writing Up Solutions To G.H. Hardy's A COURSE IN PURE MATHEMATICS?

Ok, this may be a ridiculous question and if so, you guys will shut it down. But I didn't know where else to ask it-it certainly doesn't belong on Math Overflow. Does anyone know if anyone ever ...
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4answers
285 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 ?
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2answers
109 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 ...
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3answers
183 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 ...
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1answer
153 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
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4answers
286 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 ...
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2answers
128 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|...
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3answers
423 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 ...
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0answers
90 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 ...
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105 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, ...
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1answer
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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.
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525 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 ...
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2answers
195 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 ...
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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 ...
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1answer
158 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 ...
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4answers
494 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 ...
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9answers
3k 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 ...
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1answer
98 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 ...
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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?...
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10answers
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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 ...
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2answers
467 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
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1answer
398 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 ...
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7answers
784 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. ...
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5answers
356 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. ...
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1answer
325 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 ...
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4answers
503 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 ...
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2answers
303 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 ...
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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
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1answer
251 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
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1answer
229 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
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2answers
372 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
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0answers
138 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 ...
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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 ...
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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 ...
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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 ...
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6answers
313 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 ...
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2answers
230 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 ...
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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 ...
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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.
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4answers
560 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:...
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2answers
2k 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, ...
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2answers
719 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 ...
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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.
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3answers
563 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 ...
16
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4answers
649 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 ...
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3answers
319 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 ...
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8answers
988 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 ...
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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 ...
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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 ...