All Questions
262 questions
35
votes
19
answers
9k
views
Interesting applications (in pure mathematics) of first-year calculus
What interesting applications are there for theorems or other results studied in first-year calculus courses?
A good example for such an application would be using a calculus theorem to prove a ...
CommunityBot
- 1
8
votes
2
answers
2k
views
What is the best *general triangle*?
During courses on geometry it is sometimes necessary to draw a triangle on the blackboard that can easily be recognized as a general triangle. It must not be rectangular and must not have two or more ...
CommunityBot
- 1
4
votes
1
answer
440
views
How to teach generalizing the induction hypothesis? [closed]
I just finished teaching a class on using proof assistants (in this case, Agda) to write provably correct programs. Reflecting on how it went, the biggest difficulty I noticed the students having was ...
- 82.7k
16
votes
12
answers
10k
views
How seriously should a graduate student take teaching evaluations? [closed]
Pretty much the question in the title. If a grad student gets bad reviews as a TA, how much does that hurt them later? How much do good reviews help? What if the situation is more complex? (For ...
- 50.6k
27
votes
3
answers
3k
views
Is “problem solving” a subject to be taught?
I am witnessing a new curriculum change in my country (Iran). It includes the change of all the mathematics textbooks at all grades. The peoples involved has sent me the textbook for seven graders (13 ...
CommunityBot
- 1
6
votes
3
answers
1k
views
An application of Maschke's theorem
I've been teaching some elementary representation theory to undergraduates, and want to provide applications of Maschke's theorem to complex group algebras to present in class. In particular, I'd like ...
7
votes
4
answers
2k
views
Help me find good math questions for my students [closed]
I am a teacher at 西铁一中。 I teach mathematics in English for students going abroad.
Now this is my problem, there are few mathematics books written in English that are at the level of high school, ...
CommunityBot
- 1
16
votes
5
answers
1k
views
Permission to use Online Notes
I am a new professor in Mathematics and I am running an independent study on Diophantine equations with a student of mine. Online I have found a wealth of very helpful expository notes written by ...
CommunityBot
- 1
28
votes
6
answers
2k
views
Means of Promoting Mathematics in Young Countries!
We all know mathematics is life, this question is for Mankind. It's mathoverflow here when some parts of the world we have mathunderflow! I think we can do something through ideas. A similar ...
CommunityBot
- 1
11
votes
5
answers
4k
views
Applications of Liouville's theorem
I'm looking for "nice" applications of Liouville's theorem (every bounded entire map is constant) outside the area of complex analysis.
An example of what I'm not looking for : a non-constant entire ...
- 7,902
11
votes
3
answers
729
views
Why does inconstructibility of $\sqrt[3]{2}$ imply impossibility of cube doubling? [closed]
In this question "constructing" and "doubling" is meant in the compass-and-straightedge sense.
On my desk I have five Basic Algebra texts treating constructability in the plane $\mathbb{C}$ or $\...
- 105k
19
votes
6
answers
6k
views
an engineering Ph.D. teaching math in college
I have a friend who has been teaching college-level math (e.g., all levels of calculus)
for about 4 years, although all of his education, including his Ph.D., was in engineering.
Now he is ...
CommunityBot
- 1
51
votes
6
answers
5k
views
What does it take to run a good learning seminar?
I'm thinking about running a graduate student seminar in the summer. Having both organized and participated in such seminars in the past, I have witnessed first-hand that, contrary to what one might ...
CommunityBot
- 1
27
votes
2
answers
3k
views
Teaching the fundamental group via everyday examples
This question is a "prequel" to a similar question about homology. Both questions were inspired by seeing a talk, by Tadashi Tokieda, about the interesting physics that appears in toys.
What ...
CommunityBot
- 1
7
votes
3
answers
3k
views
The etale fundamental group of a field
Background and motivation:
I am teaching the "covering space" section in an introductory algebraic topology course. I thought that, in the last five minutes of my last lecture, I might briefly sketch ...
CommunityBot
- 1
8
votes
2
answers
2k
views
Which universities teach true infinitesimal calculus? [closed]
My colleague and I are currently teaching "true infinitesimal calculus" (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by ...
- 16.6k
20
votes
2
answers
4k
views
Teaching stochastic calculus to students who know no measure theory (or PDE, or...)
I've got quite a challenge as my teaching assignment for the next Fall (not that I want to get rid of it, quite the contrary, but I still feel like asking for advice won't hurt :-)).
I'm to teach the ...
- 24.8k
13
votes
1
answer
605
views
A funny factorization of the Jacobian coming from the lines on the Fermat cubic
Here is something which came up in my algebraic geometry class, and I'm wondering if it has a deeper explanation. Let $F(w,x,y,z) = w^3+x^3+y^3+z^3$ and let $X$ be the cubic surface in $\mathbb{P}^3$ ...
- 1,829
11
votes
0
answers
2k
views
Total spaces of tangent/cotangent bundles in a course where all varieties are quasi-projective
$\def\PP{\mathbb{P}}$In a course where all varieties are quasi-projective (as in Shafarevich Volume I), I am trying to figure out whether I can justify talking about the total spaces of the tangent ...
CommunityBot
- 1
8
votes
0
answers
554
views
Lower semicontinuity of naive fiber size
I would like to present the following result in my algebraic geometry class, but it is seeming much harder than I would expect. Since my class is working with closed points over an algebraically ...
- 156k
7
votes
3
answers
1k
views
Higher dimensional Bezout via Hilbert polynomials: a reference
For the purposes of teaching my elementary course in algebraic geometry I am looking for a reference (or notes) that contains a complete proof of a higher-dimensional weak Bezout theorem. I only want ...
- 156k
1
vote
2
answers
825
views
Simple yet interesting applications of Calculus or Linear Algebra to Economics [closed]
This is essentially a vast generalization of my previous question: Examples of separable ordinary differential equations in economics
I'm giving a talk to college-level math teachers on some ...
- 325
2
votes
0
answers
3k
views
What is the geometric meaning of the third derivative of a function at a point? [closed]
What is the geometric meaning of the third derivative of a function at a point?
This question is now asked on the sister site: https://math.stackexchange.com/questions/14841/what-is-the-meaning-of-...
CommunityBot
- 1
18
votes
12
answers
10k
views
Looking for an introductory textbook on algebraic geometry for an undergraduate lecture course
I am now supposed to organize a tiny lecture course on algebraic geometry for undergraduate students who have an interest in this subject.
I wonder whether there are some basic algebraic geometry ...
CommunityBot
- 1
3
votes
3
answers
515
views
undergraduate handle decomposition. Reference
As the title says, I'm searching for a nice textbook for introducing the theory of handle decomposition of manifolds to undergraduate students.
- 10.5k
5
votes
9
answers
2k
views
Suggestions for teaching advanced high school students
Hi all,
I'm a grad student and just joined a mentoring program in which I will visit a group of advanced year ten high school students (around 16 years old) from a group of schools in the area. I don'...
- 547
4
votes
0
answers
176
views
Are injective modules flabby on basic open sets?
In order to give a simple proof of a basic fact about quasi-coherent modules (see below), I'm interested in knowing whether the following statement holds:
Statement: If $A$ is a commutative ring and $...
CommunityBot
- 1
0
votes
1
answer
552
views
Teaching profession:Differential Equations and Mean Value Theorems
Usually I teach Algebra,Algebra and Geometyry, Topology, at various University levels. This semester (Spring 2014) I have to teach Differential Equations to University second year students (4th ...
- 1,422
4
votes
4
answers
971
views
Understanding reasons for best constants in inequalities
Why, in functional analysis, is so important to calculate best constant in an embedding inequality?
Cross-posted from "https://math.stackexchange.com/questions/727690/understanding-reasons-for-best-...
CommunityBot
- 1
17
votes
2
answers
3k
views
How useful/pervasive are differential forms in surface theory?
Every year I teach an introductory class on the differential geometry of surfaces, including numerical aspects (e.g., how to solve PDEs on surfaces). Historically this class has included an ...
- 108k
8
votes
3
answers
2k
views
The harmonic (series) beetle: live illustrations of mathematical theorems
In my analysis class I use the following problem to illustrate the divergence
of the harmonic series (consider this as a hint for solving it).
Exercise.
A beetle creeps along a 1-meter infinitely ...
- 129
7
votes
4
answers
841
views
Easy to state applications of dimension theory in algebraic geometry
Dimension theory is quite a sophisticated topic (at least for me), it is fully settled in Shafarevich's book on the first 100 pages.
Shafarevich gives two nice applications of the theory. 1) A proof ...
- 2,607
5
votes
1
answer
393
views
Not quite adjoint functors
What are standard and/or natural examples of pairs of functors $F:C\leftrightarrows D:G$ and unnatural bijections $\hom_D(Fx,y)\to\hom_C(x,Gy)$ for all $x$ and $y$? Can one do this so that the ...
- 2,754
11
votes
3
answers
729
views
Calculus Teaching: Is it possible or desirable to give a severely abbreviated treatment of series convergence tests?
I will be teaching Calculus 2 this fall at a large U.S. state university. Our incoming students tend to have a limited or inconsistent background, which limits the amount of material we can cover.
...
CommunityBot
- 1
3
votes
2
answers
395
views
Integration in several variables and elementary applications
This fall I'm teaching the "second half" of the standard entry-level undergraduate multivariable calculus course: the focus is on double and triple integrals, path integrals, Green's theorem, Stokes' ...
CommunityBot
- 1
5
votes
3
answers
647
views
Looking for ideas concerning the teaching of lower-division differential equation courses...
I'm looking for problems/lessons plans that could be used in a lower-division differential equations course that involve discerning properties of solutions of an equation, IVP, or BVP, without looking ...
- 2,437
11
votes
1
answer
2k
views
Good chalk in the UK
Sometime ago it was asked in Mathoverflow about good chalk in the US Where to buy premium white chalk in the U.S., like they have at RIMS?. I will be grateful for any recommendations on good chalk in ...
CommunityBot
- 1
3
votes
2
answers
651
views
Can this informal argument (for the fact that almost all reals in the unit interval are irrational) be saved?
In the textbook from which I am teaching a Discrete Math course, the authors propose randomly generating an infinite sequence of decimal digits $d_1, d_2, \dots$. We are to think of this as the ...
- 13.6k
4
votes
4
answers
4k
views
Variation on the Sobolev space $H^1_0$
Let $\Omega\subset\mathbb{R}^n$ be a bounded open set, let
$$
C^1_0(\overline\Omega) = \{u\in C^1(\Omega)\cap C(\overline\Omega):u|_{\partial\Omega}=0\},
$$
and let $C^1_c(\Omega)$ be the space of ...
- 2,834
2
votes
4
answers
4k
views
Best way to introduce the Chinese Remainder Theorem (to a high school student)
What do you think to be the most effective way to teach the Chinese remainder theorem to a smart high school student, which is supposed to only have a soft idea about how modular arithmetic works, and ...
CommunityBot
- 1
2
votes
0
answers
1k
views
Linear Algebra Text Book [closed]
In our department we do not like our current linear algebra book and so we would want to find a better book. This is for the first course in linear algebra and the title of the course is
Elementary ...
- 44.4k
5
votes
5
answers
2k
views
Topics for a matrix analysis course
I recently taught a new (to my department) course titled "Matrix Analysis". For various reasons that I won't go into here, I was dissatisfied with the textbook I (loosely) followed, and with every ...
- 3,699
18
votes
14
answers
3k
views
Teaching a pedagogy course
At my institution incoming graduate students must take a semester long course on pedagogy taught by current grad students. I may soon be in the position of having to teach this course and I'm looking ...
CommunityBot
- 1
17
votes
10
answers
109k
views
What are the qualities of a good (math) teacher? [closed]
In forming your answer you may treat the qualifier math or maths as optional, since part of the question is whether there is anything peculiar to the subject of mathematics that demands anything ...
CommunityBot
- 1
2
votes
2
answers
1k
views
Decomposition of $K_{10}$ in copies of the Petersen graph
It is a well-known and cute exercise in algebraic graph theory to show that $K_{10}$ cannot be written as the edge-disjoint union of three copies of the Petersen graph $P$. Indeed, the graph $G$ whose ...
- 10.9k
2
votes
4
answers
1k
views
Eigenvalues of powers of linear mappings
Let $\tau$ be a linear map on a finite dimensional complex vector space. Clearly, if $\lambda$ is an eigenvalue of $\tau$ then $\lambda^n$ is an eigenvalue of $\tau^n$, for any natural (integer, on ...
CommunityBot
- 1
21
votes
7
answers
2k
views
Pros and cons of math teaching using smartboards
Currently, there is some talk in my university concerning a change in our lecture rooms from blackboards to smartboards (or other alternatives, such as a smart podium). For that reason, I'm interested ...
- 345
30
votes
3
answers
4k
views
Nearly all math classes are lecture+problem set based; this seems particularly true at the graduate level. What are some concrete examples of techniques other than the "standard math class" used at the *Graduate* level?
In the fall, I am teaching one undergraduate and one graduate course, and in planning these courses I have been thinking about alternatives to the "standard math class". I have found it much easier ...
13
votes
3
answers
2k
views
History surrounding Gauss Theorema Egregium and differential geometry
I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Gauss Theorema Egregium, that is the Gaussian ...
- 8,842
0
votes
1
answer
860
views
Sierpinski Triangle and the Chaos Game
The chaos game is a way to construct (an approximation) of Sierpinski triangle. It's clear (using Thales' theorem!) that if we begin with a point on the sierpinski triangle, then we will never leave ...
CommunityBot
- 1