# What is Rigorous: Definition and 160 Discussions

Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. Rigour frequently refers to a process of adhering absolutely to certain constraints, or the practice of maintaining strict consistency with certain predefined parameters. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as mathematical proofs which must maintain consistent answers; or socially imposed, such as the process of defining ethics and law.

View More On Wikipedia.org
1. ### A Grassmann number: a more rigorous treatment?

Most textbooks on fermionic path integral only briefly introduce Grassmann numbers. However, I want a more systematic treatment to feel comfortable about this approach. For illustration, I have several examples here. Example 1: Consider a system with only one state, how to calculate ##\langle...
2. ### Rigorous Approach to limits

Lim x->c f(x)=L means that for a given ϵ we can find a δ such that when |x-c|<δ-> |f(x)-L|<ϵ. To satisfy the criterion m<f(x)<M we choose ϵ=min (L-m, M-L) and for that ϵ we determine a δ. m<f(x)<M m-L<f(x)-L<M-L |m-L|<|f(x)-L|<|M-L| |L-m|<|f(x)-L|<|M-L| |L-m|<|L|+|m| |f(x)-L|<|f(x)|+|L|...
3. ### Rigorous mathematical definition of sampling

Please could you help me find a rigorous mathematical definition of sampling as it is used in mathematical statistics? Let ##X:\Omega\rightarrow\mathbb{R}## be a random variable and ##X_{1},...,X_{n}## is a statistical sample. What is its mathematical meaning in probability theory? Are we...
4. ### I More rigorous Euler-Lagrange derivation

Sorry if there are other threads on this, but after a discussion with a friend on this (im in the mountains, so no books, and my googlefu isn't helping), I realize that my understanding of the variational principles arent exactly... great! So, maybe some one can help. Start with a functional...
5. ### I Rigorous: Where is the Quantum System Prior to Measurement?

In the other thread of a similar name it was stated, and probably rightly so, that I wasn't using rigorous terminology or that I wasn't using them in a rigorous way. While I was making certain assumptions about the ability to interpret the 'jargon' I was using, it seemed to be a serious...
6. ### Calculus Looking for a rigorous multivariable calculus book

Hello everyone. I'm about to take Calc 3 next semester and am looking for a rigorous book to work with on multivariable calculus. I've gone through Spivak's "Calculus" from cover to cover and am hoping to find something with the same degree of rigor, if possible, and preferably with a solution...
7. ### Trying to make a rigorous imaginary world

My fantasies: 1) There happens a rapid evolution every some thousand years in biological organisms like humans. The sleeping pattern of people changes and by some unknown means people are connected to each other. The number of cycles in the sleep changes and the brain rewires itself to change ...
8. ### Classical Physics from a Rigorous, Experimental Perspective

I have a physics B.S. It seems like I learned little more than how to solve math problems as an undergrad. I want to know how the physical world works in concrete physical terms as much as is possible, so a big problem I have with physics is that concepts are very often explained in terms of...
9. ### Rigorous Study of the Higgs Boson - Question

Summary:: I would like to ask about books and other materials to study in order to understand in a really, really rigorous manner the Higgs field, Higgs boson and other related topics. Answers that are detailed but at the same time precise and to the point would be highly appreciated (please...
10. ### I Why integrals took 2000 years to come up in a rigorous manner?

Archimedes Riemann integral is one of the most elegant achievements in mathematics, I have a great admiration for it. Mr. Patrick Fitzpatrick commented on it as Archimedes first devised and implemented the strategy to compute the area of nonpolygonal geometric objects by constructing outer...
11. ### Higher dimensional FTC in electrostatics: Mathematically rigorous or not?

I have two volumes ##V## and ##V'## in space such that: 1. ##∄## point ##P## ##\ni## ##[P \in V ∧ P\in V']## 2. ##V## is filled with electric charge ##q## 3. ##\rho = \dfrac{dq}{dV}## varies continuously in ##V## 4. ##V'## is filled with electric charge ##q'## 5. ##\rho' =...
12. ### Foundations Rigorous Alternatives to AoPS for PreAlgebra-PreCalculus?

Hi PF, If one were so inclined to self-study PreAlgebra - PreCalculus, with textbooks of the very same caliber and rigor of the Art of Problem Solving books, no matter the cost, which textbooks would be the best choice? Thank you in advance. Chandller
13. ### Classical Rigorous introductory books on Electromagnetism

I need some rigorous introductory books on Electromagnetism, by rigorous I mean detailed and mathematical. Many books that I have found don’t actually work out the field produced by current carrying toroid, solenoid or even some other simple electrostatic situations. They just write “by...
14. ### Foundations Precalculus: Sixth Edition by Warren W. Esty, is it Rigorous?

Hi PF, I happened up a textbook called Precalculus (Sixth Edition) by Montana State University Professor Emeritus, Dr. Warren W. Esty and I understand that it is supposed to be a great textbook book for Precalculus self-study. I was just wondering if anyone had heard of, or used it in the...
15. ### Courses What would you do if your school is not rigorous enough?

I'm currently a 2nd year physics student in Asia. My college is quite a good school (By nature index) , the science faculty is supportive and well organized ,my physics department is a cozy ,ever-forgiving and cooperative place. But I find "something off" -They don't put their own students...
16. F

### Is a formal, rigorous math education useful in problem solving?

I'm very curious as to know how a person who pursued a formal, strict and rigorous education in mathematics would fair in comparison with a person who learned applied math by "intuition" (that is without doing any proofs and relying more on the computational part), when confronted with problem...
17. ### Rigorous proof of isomorphism

Homework Statement *This is from a Group Theory class **My secondary aim is to practice writing the math perfectly because I tend to loose a lot of points for not doing so in exams... Let λ ∈ Q* fλ : Q → Q defined as fλ(x) = λx a) Show that fλ is and automorphism of the group of rationals...
18. ### A Rigorous transition from discrete to continuous basis

Hi all, I'm trying to find a mathematical way of showing that given a complete set $$\left |a_i\right \rangle_{i=1}^{i=dim(H)}∈H$$ together with the usual property of \left |\psi\right \rangle = ∑_i \left \langle a_i\right|\left |\psi\right \rangle\left |a_i\right \rangle ∀ \left...
19. ### I A rigorous definition of a limit and advanced calculus

i'm trying to review calculus and look a little deeper into proofs/derivations/etc. I'm doing this both for fun and to review before i go back to school. am i the only one who has difficulty understanding the "rigorous" definition of the limit? i found this web page...
20. ### Other Looking for a Detailed and Rigorous Combinatorics Textbook

Hello! I’m teaching myself mathematics and physics, and I’m looking for a clear, rigorous book on combinatorics. The reason being is that past books I’ve read that included some combinatorics were difficult to understand (for example, my first encounter with the subject was in a precalculus...
21. ### A Rigorous interacting effective QFT in d=4?

Sometimes one can read that constructive QFT has become somewhat superfluous with the advent of effective QFT, so there is no need anylonger to define a QFT on arbitrary small distances. But is there a rigorous interacting effective QFT in d=4 at all? If so, how is it constructed?
22. ### B Tips on writing a formula in rigorous way

Dear, I am writing a report of a tool I have developed where there is a chapter with all the formulas included in the tool. I have a formula like the following: P(l,1)=\sum_k PF(k,l) but P(l,1) contains the sum of all the positive PFs, while P(l,2) contains the sum of all the negative PFs. Is...
23. ### Atomic Introductory (yet rigorous) Organic/Inorganic Chemistry book

I'm a student in Pre-med looking to begin self-studying chemistry at a meticulous level. Since I've only recently graduated I haven't been too full on my course load, and I'm ready to change that immediately. However, as a student, I have no clue how I might choose great authors who give a...
24. ### Calculus Please recommend a textbook containing a rigorous proof of ....

This is excerpted from George E. Hay's "Vector and Tensor Analysis". The author gives the statement that the limit is equal to 1 without any explanation, perhaps because he thinks it does not belong to the contents of vector analysis. I can see it intuitively, but I want a rigorous mathematical...
25. ### Mathematically rigorous Calculus 2 book

So, i am currently studying physics in a brazilian university. I am going to have a Calculus 2 course which, in Brazil, covers Ordinary Differential Equations and multi-variable differential calculus. So which challenging and rigourous books would you guys recommend for that? Thanks for the...
26. M

### Analysis Looking for a rigorous analysis textbook

Hello all. I'm a mathematics undergrad student who finished his first university year succesfully. I got courses of calculus, but these weren't very rigorous. I did learn about stuff like epsilon and delta proofs but we never made exercises on those things. The theory I saw contained proofs but...
27. M

### Calculus Rigorous calculus textbooks from intro to advanced

My ultimate goal is to become a theoretical physicist, a great one at that. I have mastered the prerequisites and I am now looking for rigorous calculus textbooks that make some references to physics, or are more orientated for people who want to become physicist.Thanks for your help.
28. M

### Intro Physics Most rigorous and challenging physics guide

My plan is in the following order 1.The theoretical minimum by Leonard Susskind 2.Feynman lectures, with the exercises 3.A course of theoretical physics, Landau and Lifshitz Now, my intent is to not only become a physicist, but to be one capable of unveiling fundamental ideas and revolutionary...
29. ### How is this mathematically rigorous?

In the solution of this problem(121), dW = (kmgcosΦ + mgsinΦ) ds, where ds is the differential element along the curve. Now they have done kmg dscosΦ + mg ds(sinΦ) = kmgdx +mgdy. Makes sense intuitively, but I want to know how this is rigorous. What I'm thinking is, the curve is broken into N...
30. ### I Rigorous Explanation of dW in Problem 121

In the solution of this problem(121), dW = (kmgcosΦ + mgsinΦ) ds, where ds is the differential element along the curve. Now they have done kmg dscosΦ + mg ds(sinΦ) = kmgdx +mgdy. Makes sense intuitively, but I want to know how this is rigorous. What I'm thinking is, the curve is broken into N...
31. ### MHB Rigorous definition of "Differential"

First of all I want to clarify that I posted this question on many forums and Q&A websites so the chances of getting an answer will be increased. So don't be surprised if you saw my post somewhere else. Now let's get started: When it comes to definitions, I will be very strict. Most textbooks...
32. ### Rigorous definition of magnetic poles?

Hi. Since there haven't been observed magnetic monopoles so far, what exactly do we mean when we talk about the north/south pole of a magnet? Is it something like "north is where the field lines exit a solid body" and "south is where they enter" or is there a more rigorous definition?
33. ### Rigorous Multivariable Limit Definition Problem

Homework Statement Hey I'm trying to prove the rigorous definition of limit for the following function: Lim (x,y) approaches (1,1) of f(x,y)=(y*(x-1)^(4/3))/((x/1)^2+abs(x)*y^2) Homework Equations abs(x^2)<abs(x^2 +y^2) The Attempt at a Solution I know the rigorous definition of limit. I...
34. ### Looking for Mathematically Rigorous QM Textbooks? Any Suggestions?

Upon searching in this forum, i have found discussions about the standard undergraduate textbooks on QM not being so good in teaching you the foundations properly. A good example is the difference between Hermitian and self-adjoint operators. Some people are saying that we should study QM from a...
35. ### Calculus What is a good book for learning rigorous calculus?

I'm not sure if the title correctly says what I am looking for. I'm a few years out of college and I'm trying to review some electromagnetics topics. A lot of the "proofs" in my EM book seem to take a lot of shortcuts, or use "intuition" to explain why some calculus operation can be simplified...
36. ### Discrete Rigorous Combinatorics Book

Hi, I am trying to find a completely rigorous book on combinatorics. For example, one that states the sum and product counting principles in terms of set theory and proves them, treats permutations as a bijection from a set onto itself, etc. Many don't even explain the reasoning behind those...
37. ### How rigorous is this program

Hi! I'm currently looking to do a bsc with honours in physics at the University of Auckland. I expect to do most of the postgraduate (700) courses by year 3/4 and the 100, 200, 300 courses in years 1 2 and 3. I will also be taking some 100 courses in computer science and 200 and 300 ones in...
38. ### Classical Recommendation for rigorous intro books to mechanics and E&M

Hey guys I'm a sophomore in college currently taking physics 2(intro E&M), Multivariable calculus, and Differential Equations. I was hoping some of you guys could recommend some good books for intro mechanics and E&M. I'm currently using University Physics by Young in my E&M class, and I used...
39. ### Schools How Rigorous are High School Calculus classes?

Hello everyone, I'm a high school student and wondering about the rigor of the AP Calculus classes that U.S. high school students often take. When I say rigor here, I'm talking about mathematical rigor as is commonly talked about with regard to math classes, and also whether or not the class is...
40. ### Why is induction rigorous?

I understand how to construct a proof by induction. I've used it many times, for homework because it was clearly what the book wanted, but when I've tried it in a research setting, it's because I have so little control of the objects in working with. So it has become my impression that since...
41. ### Classical Rigorous Treatment of E&M and Thermodynamics

So, I am deeply unsatisfied with the way I've been taught E&M and thermodynamics in school. Despite having done AP Physics C: E & M and having got a 5 in it, I feel my understanding is lacking. Could you guys ( and girls) please recommend something rigorous for me to use to teach myself.Books...
42. ### Insights Is there a rigorous proof of 1 = 0.999....? - Comments

micromass submitted a new PF Insights post Is There a Rigorous Proof Of 1 = 0.999...? Continue reading the Original PF Insights Post.
43. ### Geometry Beginning Trigonometry textbook

Good morning everyone, I have written a free math textbook, and I'd appreciate some feedback on it. It's about the basics of Trigonometry, including sine, cosine, tangent, radians, the unit circle, a bit on identities, and the Law of Sines, Cosines, and Tangents. I wrote it in a rigorous...
44. ### Quantum Good books on QM with rigorous math

I want to learn QM but in a way that i can learn both the deep mathematics, the physical intuition behind QM and the intuition behind the maths behind the QM(not many books have this). Note that this is my first attempt to learn QM and i will be watching some video lectures online,so i won't be...
45. ### Rigorous Differential Equations text

Hello, I am a math major and I was wondering if you guys knew what would be a good rigorous differential equations text. I really like rigor (like Rudin analysis style rigor or whatnot), instead of the typical books that just focus on the method. I want the proofs and everything. I also would...
46. ### Intro Math Rigorous math/physics self teaching material?

Hi guys, So, I'm a high school senior about to graduate, and because of reasons, I most likely will not be attending a college next year. However, I will apply to colleges next year for a Physics major. In the meantime, I would like to use my time wisely, and I was thinking that the smart thing...
47. ### Looking for rigorous General Chemistry textbook

I'm going to be tutoring next semester and I need a good general chemistry book to review. I used an old copy of zumdahl during general chemistry(not assigned text but worked fine) but I want a fresh look at the subject that is a little more rigorous mathematically especially. After a year of...
48. ### Classical Looking for rigorous text on dynamical systems

Hi, I'm looking for a modern rigorous text on (Hamiltonian) dynamical systems, perhaps with emphasis on perturbation theory. It should be in the same vein is Poincare's "methodes nouvelles", but modern.Thanks
49. ### MHB HIGHLY Rigorous Treatment of the Trigonometric Functions

I am looking for a rigorous (preferably HIGHLY rigorous) treatment of the trigonometric functions from their definitions through to basic relationships and inequalities through to their differentiation and integration ... and perhaps further ... Can someone please suggest (i) an online...
50. ### Rigorous Precalculus and Calculus Textbooks + Intro to Linear Algebra

I am currently studying A level Further Pure Mathematics, and I will be joining university next year. I want to brush up on precalculus (algebra, trigonometry, geometry) and calculus (differential, integral [single variable]), except that this time I need a more rigorous treatment of those...