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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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A visualization for the quotient rule

Context: first year didactics of mathematics course for middle school teacher students (in Norway). I have a reasonable visualization for the product rule of derivatives: Consider a rectangle with ...
Tommi's user avatar
  • 7,202
6 votes
1 answer
355 views

How to assess students in real analysis?

Terence Tao says the following in the preface to his book Analysis I: With regard to examinations for a course based on this text, I would recommend either an open-book, open-notes examination with ...
Pedro's user avatar
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2 votes
2 answers
114 views

Real-World Problems for Teaching Extrema and Derivative Tests in STEM Education

For educational purposes, I am seeking example problems in the realm of natural sciences, engineering, and business that satisfy the following criteria: Consider a one-dimensional real function $f$ (...
Julia's user avatar
  • 1,265
-1 votes
4 answers
548 views

Motivating a definition of "gap" in a line just barely more advanced than the one used in the typical first-year calculus course

Imagine a course barely getting into some topics more theoretical than what is done in the typical very staid first-year calculus course, and the kind of students for whom such a course is appropriate....
Michael Hardy's user avatar
2 votes
3 answers
170 views

Flashcards and Study Methods for Undergraduate and Masters Degrees

I need advice on methods and techniques for studying mathematics that are commonly used at undergraduate and masters level in mathematics. What are some strategies that you find useful in coping with ...
Ad_M's user avatar
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7 votes
6 answers
2k views

How can we motivate that Newton's method is useful?

If you teach Newton's method for finding roots of real functions on the high school (or freshmen) level, I think some students may reason like a variant of the following: Why do I need learn such a &...
Julia's user avatar
  • 1,265
9 votes
8 answers
4k views

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, $|\ldots| < \varepsilon+\varepsilon+\varepsilon$ ...
user27289's user avatar
  • 139
15 votes
5 answers
14k views

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 ...
user95017's user avatar
  • 429
15 votes
6 answers
2k views

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 ...
kodiak's user avatar
  • 253
2 votes
2 answers
444 views

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 "...
Rohit Shekhawat's user avatar
7 votes
4 answers
2k views

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 ...
blakedylanmusic's user avatar
0 votes
1 answer
95 views

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 ...
Athanasios Paraskevopoulos's user avatar
1 vote
0 answers
180 views

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 ...
zzzhhh's user avatar
  • 111
4 votes
0 answers
729 views

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 ...
akm's user avatar
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10 votes
5 answers
4k views

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 ...
yaa09d's user avatar
  • 203
0 votes
2 answers
495 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 ...
ZeroToZero's user avatar
2 votes
0 answers
108 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 ...
Rico1990's user avatar
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11 votes
3 answers
804 views

How to evaluate a proof that seems to be using induction informally?

In an Advanced Calculus course, students were asked to prove $$|a_1 +a_2+...+a_n| \le |a_1|+|a_2|+ \ldots +|a_n|$$ for $n$ real numbers $a_1,a_2,\ldots,a_n$ I am teaching assistant for this course, ...
JAEMTO's user avatar
  • 229
2 votes
3 answers
297 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 ...
GradStudent's user avatar
-1 votes
1 answer
117 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 ...
Gülümden Aktürk's user avatar
5 votes
3 answers
351 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. ...
Jaikrishnan's user avatar
3 votes
1 answer
247 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 ...
Tobi's user avatar
  • 41
6 votes
2 answers
455 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, ...
Consider Non-Trivial Cases's user avatar
2 votes
1 answer
166 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 ...
AlfredG's user avatar
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10 votes
2 answers
510 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 ...
David Elm's user avatar
  • 485
3 votes
1 answer
302 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, ...
Nights's user avatar
  • 183
1 vote
1 answer
195 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 ...
user avatar
2 votes
0 answers
159 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, ...
RandomStudent's user avatar
4 votes
2 answers
315 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 ...
Hana Puk's user avatar
  • 221
22 votes
16 answers
8k 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 ...
Zuriel's user avatar
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4 votes
2 answers
176 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 ...
Praphulla Koushik's user avatar
7 votes
2 answers
213 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 ...
Ma Joad's user avatar
  • 1,673
1 vote
1 answer
375 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 ...
Saravanan's user avatar
  • 123
13 votes
2 answers
3k 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 ...
J W's user avatar
  • 4,733
8 votes
3 answers
667 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 ...
Arctic Char's user avatar
2 votes
2 answers
269 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, ...
mathnoob123's user avatar
14 votes
4 answers
859 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 ...
SBK's user avatar
  • 301
3 votes
0 answers
688 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 ...
Sophia's user avatar
  • 31
9 votes
1 answer
309 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 ...
David Steinberg's user avatar
6 votes
4 answers
1k 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 ?
Vagabond's user avatar
  • 339
3 votes
2 answers
274 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 ...
AegisCruiser's user avatar
  • 1,427
6 votes
3 answers
275 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 ...
Tom Au's user avatar
  • 1,502
11 votes
1 answer
550 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 ...
benblumsmith's user avatar
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11 votes
4 answers
413 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 ...
Lee David Chung Lin's user avatar
5 votes
2 answers
1k 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|...
user37237's user avatar
9 votes
3 answers
2k 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 ...
benblumsmith's user avatar
  • 1,936
1 vote
0 answers
119 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 ...
Saaqib Mahmood's user avatar
6 votes
1 answer
252 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, ...
Ma Joad's user avatar
  • 1,673
15 votes
2 answers
1k views

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 ...
user avatar
0 votes
1 answer
132 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.
Jessie's user avatar
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