Questions tagged [mathematical-analysis]
For questions applying to analysis courses: Real and complex analysis. Typically a higher and more proof-based level than calculus.
83
questions
0
votes
2answers
264 views
Real before complex analysis or vice versa?
I used to learn Real Analysis before Complex Analysis in my bachelor study, but now the order is reversed in my university.
I would like to ask which order is better to learn the subjects, and which ...
2
votes
0answers
85 views
Locus of the maximal turning point and the point of inflection
Suppose you have a carton that has the form of a square with sides of length a. If we want to produce a box out of it whose height is x we might deduce the following formula:
$$V_a(x)= x(a-2x)^2=a^2 x ...
8
votes
1answer
345 views
Grade on proving |$a_1 +a_2+...+a_n| \le |a_1|+|a_2|+... +|a_n|$
In an Advanced Calculus course, students were asked to prove $$|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 ...
2
votes
3answers
219 views
Why do we typically only teach high-school students affine transformations of elementary functions?
A standard pre-calculus curriculum consists of the study of elementary functions: Polynomials, rational functions, (circular and hyperbolic) trigonometric functions, exponential functions, their ...
-1
votes
1answer
104 views
Math elementary textbooks [closed]
I’m a master student in Turkey. I’m researching math textbooks from different counties to compare them for my thesis. However, it is really difficult to find them. I need your suggestions. Or is there ...
5
votes
3answers
221 views
The Riemann integral vs Lebesgue integral in several variables for advanced undergraduates
I am about to teach a second course in analysis for advanced undergraduate students. The students have already studied roughly the first eight chapters of Rudin's Principles of mathematical analysis. ...
3
votes
1answer
164 views
Learning strategies for high volume/pace learning?
Background: I am a graduate student in a mid-tier U.S. university, and I am struggling. I feel like I during my undergrad, I haven't aquired the neccesary skills to keep up with the high volume/pace ...
5
votes
2answers
174 views
Introductory book or other resource on $p$-adic numbers/number theory/analysis
I am having problems understanding $p$-adic numbers/$p$-adic number theory/$p$-adic analysis. I have tried some notes on the internet, but these notes were not helpful.
Can anyone suggest a book, ...
2
votes
1answer
131 views
Advanced textbook for vector fields [closed]
I am currently reading Spivak Calculus on Manifolds and Munkres Analysis on Manifolds. I am looking for a more advanced text, especially on vector fields as they relate to the great conserved fields ...
10
votes
2answers
476 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 ...
3
votes
1answer
187 views
Introductory Analysis lecture slides
I will be teaching an introductory analysis course (see topics below) and I need some source-code Latex slides or PPT slides, and am willing to choose my textbook based on these slides (rather than, ...
1
vote
1answer
152 views
Making epsilon-delta proofs not just precalculus
In trying to find lecture-length videos of epsilon-delta proofs, I've found that almost all of them just start with the definition and then work through the algebra to get the answer. In effect, it ...
2
votes
0answers
142 views
Why are a.e. defined functions rarely mentioned in elementary books?
In any standard development of measure theory in several well-known textbooks, the use of almost everywhere (a.e.) defined functions are first seen in the statement of Fubini's theorem, which states, ...
4
votes
2answers
292 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
130 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
172 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
224 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 ...
13
votes
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 ...
8
votes
3answers
306 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
228 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
578 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
348 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 ...
9
votes
1answer
244 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
836 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
202 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 ...
6
votes
3answers
250 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
338 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
365 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 ...
5
votes
2answers
474 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|...
8
votes
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 ...
2
votes
0answers
105 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
222 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
102 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.
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votes
3answers
639 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
238 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 ...
9
votes
3answers
648 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 '...
16
votes
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 ...
7
votes
1answer
174 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 ...
15
votes
4answers
592 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
9k 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
123 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 ...
13
votes
6answers
4k 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?...
25
votes
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 ...
4
votes
2answers
1k 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 \...
4
votes
1answer
889 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 ...
19
votes
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.
...
8
votes
5answers
483 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
1answer
619 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 ...
7
votes
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 ...