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 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 \...
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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 ...
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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.
...
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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. ...
user5897
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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 ...
user2139
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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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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 ...
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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 ...
user5661
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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, ...
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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 ...
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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, ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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.
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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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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, ...
- 2,990
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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 ...
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Searching for an analysis textbook [closed]
Particularly, I'm looking for an undergraduate text with an excellent explanation of $\delta$-$\varepsilon$ proofs, and many example questions related thereto.
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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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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 ...
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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 ...
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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 ...
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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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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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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 ...
- 9,122
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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 ...
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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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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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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 ...
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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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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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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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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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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 ...
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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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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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