# 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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10answers
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### 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 ...
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 ...
7answers
2k 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 "...
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 ...
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 ...
4answers
2k 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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
4answers
944 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 ...
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 ...
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. ...
2answers
558 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 ...
10answers
5k 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 ...
5answers
7k 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 ...
4answers
988 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 ...
4answers
571 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 ...
3answers
272 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 ...
2answers
1k 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 ...
1answer
317 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, ...
6answers
3k 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?...
4answers
534 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 ...
3answers
470 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 ...
2answers
1k 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 ...
2answers
386 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 ...
9answers
7k 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 ...
4answers
351 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 ...
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, ...
3answers
232 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 ...
1answer
319 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 ...
2answers
470 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 ...
3answers
642 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 '...
3answers
444 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 ...
3answers
290 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 ...
6answers
397 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 ...
4answers
269 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 ...
5answers
463 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. ...
1answer
224 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 ...
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 ...
1answer
308 views

### Grade on proving |$a_1 +a_2+…+a_n| \le |a_1|+|a_2|+… +|a_n|$

In an Advanced Calculus course, students were asked to prove $$|a_1 +a_2+...+a_n| \le |a_1|+|a_2|+... +|a_n|$$ for $n$ real numbers $a_1,a_2,...a_n$ I am teaching assistant for this course, and one of ...
4answers
686 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 ?
4answers
843 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:...
4answers
1k 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 ...
2answers
228 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 ...
1answer
171 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 ...
1answer
561 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 ...
2answers
359 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 ...
2answers
163 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 ...