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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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33
votes
9answers
1k 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 ...
26
votes
4answers
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 ...
25
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 ...
25
votes
6answers
1k views

Good definition for introducing real numbers?

In the first lectures about calculus/analysis, you should introduce real numbers. Let's assume students know that rational numbers are. What are the advantages or disadvantages in the different "...
25
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 ...
23
votes
4answers
1k views

What are some good examples to motivate the implicit function theorem?

I always had problems when teaching the implicite function theorem in advanced analysis courses. This result is motivated by later applications, but it would be great to be able to provide easily ...
22
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 ...
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 ...
22
votes
4answers
1k 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 ...
20
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 ...
20
votes
4answers
818 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 ...
19
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 ...
18
votes
13answers
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 ...
18
votes
7answers
918 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. ...
16
votes
10answers
4k 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 ...
16
votes
2answers
458 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 ...
16
votes
4answers
733 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 ...
14
votes
4answers
520 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 ...
14
votes
3answers
228 views

Resources for teaching Riemann integration in higher dimensions and on submanifolds, with view toward Stokes' theorem?

Question I am looking for suggestions of good resources (textbooks or lecture notes preferably) for teaching Riemann integration in $\mathbb{R}^d$ with $d\geq 2$ and also for Riemann integration ...
14
votes
2answers
881 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 ...
14
votes
1answer
272 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, ...
13
votes
5answers
5k views

Best textbooks to introduce measure theory and Lebesgue integration?

What are the best textbooks to introduce measure theory and Lebesgue integration to undergraduate math majors? Many students in such a class will go on to graduate school and be required to take a ...
13
votes
4answers
436 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 ...
13
votes
3answers
412 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 ...
12
votes
2answers
370 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 ...
11
votes
9answers
4k 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 ...
11
votes
6answers
2k 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?...
11
votes
2answers
304 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 ...
11
votes
4answers
311 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 ...
11
votes
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, ...
11
votes
3answers
197 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 ...
11
votes
1answer
227 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 ...
8
votes
3answers
360 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 ...
8
votes
3answers
250 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 ...
8
votes
6answers
345 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 ...
8
votes
5answers
408 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. ...
7
votes
4answers
367 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 ?
7
votes
4answers
259 views

Multidimensional differentials for students with poor spatial imaging

When teaching multidimensional differentials (I'm assuming the students grasped the one-dimensional case), there are many useful parallels relating to spatial imagination. For example, when ...
7
votes
2answers
212 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 ...
7
votes
1answer
164 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 ...
7
votes
3answers
709 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 ...
7
votes
1answer
169 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 ...
6
votes
4answers
612 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:...
6
votes
2answers
326 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 ...
6
votes
1answer
379 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 ...
6
votes
2answers
138 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 ...
5
votes
4answers
707 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 ...
5
votes
2answers
164 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|...
5
votes
3answers
222 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 ...
5
votes
2answers
464 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, ...