Questions tagged [mathematical-analysis]
For questions applying to analysis courses: Real and complex analysis. Typically a higher and more proof-based level than calculus.
70
questions
4
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2answers
253 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 ...
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 ...
4
votes
2answers
111 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
142 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
136 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 ...
11
votes
2answers
329 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 ...
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 ...
2
votes
2answers
208 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, ...
13
votes
4answers
440 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 ...
3
votes
0answers
189 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
1answer
171 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 ...
7
votes
4answers
373 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 ?
3
votes
2answers
152 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
223 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
230 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
315 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
170 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
720 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
100 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 ...
6
votes
1answer
161 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, ...
0
votes
1answer
84 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
566 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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
2
votes
1answer
105 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
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?...
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 ...
4
votes
2answers
562 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
547 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
926 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
411 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
385 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
723 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
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 ...
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 ...
14
votes
1answer
274 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
244 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
470 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
150 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 ...
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 ...
4
votes
2answers
214 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
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 ...
4
votes
2answers
237 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
48 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
614 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
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, ...
