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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Why do most Analysis textbooks overlook, and fail to teach delta-epsilon proofs, using the K-ε principle?
When writing $\delta$-$\varepsilon$ proofs, it's common that the ''natural'' choice of $\delta$ leads to the final inequality in the form, say, $|\ldots| < \varepsilon+\varepsilon+\varepsilon$ ...
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Do undergraduates struggle with δ-ε definitions because they lack a habit of careful use of their native language?
I transcribed this excerpt starting at the 22-minute mark, of Okinawa Institute of Science and Technology’s May 19 2020 podcast with Professor Tadashi Tokieda:
For example, this is a bit too ...
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Should an undergraduate math program contain a course on Lebesgue integration?
Is it standard for a math undergraduate program to have a course on Lebesgue integration?
Does Riemann integral suffice for undergraduates?
The reason of my question is I read a paper by Bartle titled ...
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What is important to keep in mind in grading proof-based courses?
I am an undergraduate grader at my institution where I have been entrusted with grading a section of an undergraduate analysis course; it's usual for this role to be offered exclusively to graduate ...
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Why are hand waving arguments made in textbooks of undergraduate analysis and how should readers deal with them?
Having read several undergraduate textbooks in complex analysis (Stein-Shakarchi, Gamelin, etc.), I find that some "hand-waving" arguments are frequently used. An example (the proof of the ...
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Should one study Laplace Transformation before Fourier Transforms?
(Im sorry if the question is out of the scope of the forum)
Hi, Im currently a Physics student. I have studied most of the Calculus. Now, according to the book Im using, there is chapter on "...
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If I take Modern Analysis next year, will I be prepared to teach multivariable/vector calculus?
I’m currently getting my Master’s in Math at Portland State University so that I can teach community college mathematics. I’m specifically hoping to teach calculus, statistics, and linear algebra, so ...
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Suggestion for IB program Analysis and Approaches SL book?
What is the most suitable book for the IB program Analysis and Approaches SL for a student with significant weaknesses?
I had suggested the book from HAESE Mathematics yet he finds it particularly ...
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What is the text for "the other second-term course in analysis at MIT?"
My question comes from first few paragraphs of preface of "Analysis on Manifolds" by James R. Munkres, as excerpted below:
A year-long course in real analysis is an essential part of the
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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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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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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 ...
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What are your experiences with Buck’s Advanced Calculus?
I stumbled across the book when searching for rigorous alternatives to Rudin with some solutions. It’s an “old school” (1965) calculus text but, I think, covers similar material to Rudin in a more ...
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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 ...
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Are the following topics usually in an introductory Complex Analysis class: Julia sets, Fatou sets, Mandelbrot set, etc?
I'm an nntaleb 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 '...
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What are some good examples to motivate the implicit function theorem?
I always had problems teaching the implicit function theorem in advanced analysis courses. This result is motivated by later applications, but it would be great to provide easily accessible examples ...
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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 ...
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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 ...
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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:
$$V_a(x)= x(a-2x)^2=a^2 x ...
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Grade on proving |$a_1 +a_2+...+a_n| \le |a_1|+|a_2|+... +|a_n|$
In an Advanced Calculus course, students were asked to prove $$|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 ...
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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 ...
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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 ...
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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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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 "...
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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. ...
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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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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 ...
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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 ...
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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 ...
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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, ...
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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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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 ...
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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 ...
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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, ...
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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 ...
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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, ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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, ...
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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 ...
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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 ...
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