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.

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  1. A

    MHB How Does Subtracting a Limit Value Affect Convergence?

    Prove that $\displaystyle \lim_{x\to a} f(x) = L \space \text{if and only if} \space \lim_{x\to a} [f(x) - L] = 0$ Provide a rigorous proof. I am not sure what he has given to us. Is $\displaystyle \lim_{x\to a} f(x) = L$ true? So, $|f(x) - L| < \epsilon$ for $|x - a| < \delta_1$ some...
  2. M

    Find a Rigorous Calculus of Multiple Variables Book for Advanced Learners

    Hi! I am looking for a very rigorous book on some of the topics covered in Calculus of Multiple Variables. My University uses the last part of Adams "Calculus: a complete course" and I found the presentation therein more fit for people needing to know enough to perform the calculations than for...
  3. kq6up

    Straightforward and Rigorous Rocket Thrust Derivation

    I have noticed many of the thrust derivations in textbooks I have seen do not do a straightforward derivation of rocket thrust. The all seem to use the same trick with infinitesimals in a sort of binomial form. For reference: Taylor, "Classical Mechanics" Pg. 85. I am working on a rigorous...
  4. T

    Starting my Self-Study of Rigorous Mathematics

    Hello everyone! I'm new here. I'm starting a self-study of more rigorous mathematics. My background is I have a B.S. in Mathematics(class of '14) and have had rigorous classes but they were, in my opinion, sub-par and not taught as rigorously as they should have been. Truth be told, I...
  5. W

    Somewhat Rigorous thermodynamics?

    Any thermodynamics book that's somewhat mathematical,but is still for beginners?
  6. W

    Calculus books, intuitive but rigorous?

    I've tried to learn calculus many times from many books,I've come to the conclusion that there is no ideal book on this subject. I've read Spivak's book,and greatly enjoyed its problems but I felt unstatisfied by the explanations and the illustration were very poor ,and the only chapter on...
  7. C

    Rigorous Definition of Infinitesimal Projection Operator?

    I've been reading Thomas Jordan's Linear Operators for Quantum Mechanics, and I am stalled out at the bottom of page 40. He has just defined the projection operator E(x) by E(x)(f(y)) = {f(y) if y≤x, or 0 if y>x.} Then he defines dE(x) as E(x)-E(x-ε) for ε>0 but smaller than the gap between...
  8. micromass

    MTW definition of differential froms made rigorous

    Hello guys! I've been trying to get some intuition for differential forms. I know the formal theory and I know how useful they are. But then I came across the following paper: https://dl.dropboxusercontent.com/u/828035/Mathematics/forms.pdf It describes an intuition for forms that is very...
  9. S

    Recommend a Rigorous Calculus Book

    I have an undergraduate degree in Mathematics from a Big 10 university. Right now I am looking for a Calculus text that would aid me in approaching philosophy and literature text. Mostly what I'm looking for is a combination of Rigor (strictness in definitions and such) and problems that...
  10. P

    Rigorous Definition of Kelvin?

    Hi all, I'm reading through Zwiebach's String Theory text on my own and am thinking about one of his very elementary exercises on "units." We are asked to define temperature in Kelvin with reference to the fundamental units of mass, length, and time. My thought is the following: We take...
  11. G

    What is the rigorous definition of set?

    Hi all, first math post here. I was just wondering- after having read from quite a few textbooks that intuitively, a set is a collection of objects- if there's a rigorous definition of the concept of set. It's just out of curiosity- I mean, is a rigorous definition even necessary? I guess I'm...
  12. B

    More Rigorous Text than Rudin?

    I was recommended Rudin's "Principles of Mathematical Analysis" as a text that assumes you know nothing and takes it from there. And make no mistake, I think it's an amazing book. I've learned techniques and overcome some hurdles all on my own that make me feel quite good about myself. So this...
  13. B

    Recommendations for Mathematically Rigorous Texts on Statistics and Probability

    Hi PF, looking for a recommendation on a mathematically rigorous text on statistics (and perhaps probability). I have a decent amount of knowledge on linear & abstract algebra and calculus and analysis, so I don't mind if it uses them. What I don't want is a text that fails to prove its theorems...
  14. 0

    How Does Error Propagation Affect Calculations in Physics?

    A position of a particle in linear motion is given by: x = vt + 0.5at2 Calculate x with the error for: t = 25.3 ± 0.5s v = 10.1 ± 0.4m/s a = 2.5 ± 0.3m/s2 So for calculating vt: q = (10.1) (25.3) = 255.53 (exact) Δq = (10.1)(25.3) * √ (0.4/10.1)2 (0.5/25.3)2 = 11.31... Therefore, vt =...
  15. S

    What are the Best Books for College Physics with Tough Problems?

    Any suggestions for books with introductory college subjects(Classical Mechanics, Electromagnetism, Thermodynamics,etc) containing tough problems? Looking at IPhO level here.
  16. C

    Is this course rigorous enough?

    Well, to keep it short I'm thinking about going to a university near by. However after reading around on the internet it seems as though some think it's not so good at maths (which is what I'll be going to study.) I've looked online and found a sample of their past papers in the following...
  17. A

    Rigorous treatment of infinitesimal reparametrizations

    Hi! Reading some string theory books I always find that the introductory chapters discuss the relativistic free particle (see Lüst-Theisen, or Becker-Becker-Schwarz, page 21, exercise 2.3). Then they go on about showing that the action S=-m\int^{t_1}_{t_2} dx = -m...
  18. A

    Rigorous treatment for deforming functions in string theory

    Hi! Reading some string theory books I always find that the introductory chapters discuss the relativistic free particle. Then they go on about showing that the action S=-m\int^{t_1}_{t_2} dx = -m \int^{t_1}_{t_2}d\tau\sqrt{-\frac{dx^\mu}{d \tau} \frac{dx^\nu}{d \tau} \eta_{\mu\nu}} is...
  19. N

    Rigorous definitions in general relativity

    Hello (a) The universe U is a topological space whose elements are called events and as each event has a neighborhood homeomorphic to R^4. (b) A local coordinate system is a homeomorphism between an open subset of U and a bounded subset of R. (c) A world line segment is a continuous...
  20. P

    What's the difference between Rigorous Calculus and Analysis?

    I don't understand the difference between Rigorous Calculus books (Hubbard's "Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approch", Loomis and Sternberg's "Advanced Calculus", Spivak's "Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus")...
  21. N

    Rigorous definitions in quantum mechanics

    Hello It is question of specifying mathematical definitions which are cummunes in several theories. In classical physics, in special relativity, in quantum theories (wave functions and state vectors) and in general relativity, we can assert : (a) The universe U is a topological space...
  22. A

    Courses Is this course schedule too rigorous?

    Hi, I currently attend a community college and plan to take 3 courses next semester: Calculus 2, Intro to Stats, Chemistry 1. I've heard that taking two math courses at once is too much. Does my schedule sound too rigorous?
  23. L

    Physics with rigorous math - need advice

    Hi all, I have a doubt regarding the connection between physics and math. I'm studying physics at college but I'm a little confused. My main area of interest is general relativity, so what I really want to do is to work on that area, however I prefer (and I have more hability) to atack problems...
  24. R

    Rigorous dirrerential definition and physics

    My question is rather simple but it puzzles me for a long time actually. If we have a look at differential as physicists usually do we came up with a simple definition of "infinitesimal variable change". And this idea then preserves elsewhere like in the definition of entropy: \mathrm{d} S =...
  25. H

    Is Susskind's derivation Euler-La Grange rigorous?

    Beginning at 31:03 Dr. Susskind presents an intuitively very satisfying derivation of the Euler-La Grange equation(s). But, I'm not convinced it is rigorous. It seems his choice of variation is not the only possible choice for the neighborhood he selected. The reason this matters to me is...
  26. K

    Is this proof for convergence of 3^n/n rigorous enough?

    Hi, I am trying to self study analysis and was practicing some problems. I wasn't sure if this solution to one of the problems I came across was rigorous enough. Basically, by writing down the first few terms of 3^n and n!, I figured I can say 3^n < 3*(n-1)! for all n>=13...without...
  27. M

    Need help articulating an idea in rigorous language

    So I'm working on a personal project, and I have most of the details worked out, but I'm having difficulty expressing an idea in precise terms. Premise: We start with a set which has only itself as an element, which results in the following infinite regress: {{{...}}} Now what I...
  28. P

    Rigorous definition of a limit

    First off I want to apologize for bombarding this subforum with my gazillion questions. If my continuous barrage of questions poses a problem just let me know and I'll stop. Homework Statement For each value of ε, find a positive value of δ such that the graph of the function leaves the window...
  29. E

    Most Rigorous General Chemistry Textbook

    I am seeking the most challenging and rigorous gen. chem. textbook to learn from. I have it pretty well narrowed to Oxtoby vs. Atkins. Caltech appears to use Oxtoby. Furthermore, the description for the Oxtoby book touts itself as "the most modern, rigorous, and chemically and mathematically...
  30. C

    Rigorous math and physics textbooks

    Hello, physics forums. As an introduction to the community, I'm 15 years old and live in northwestern Ontario. I've recently became very interested in physics, but I've always excelled in math. I've looked into some textbooks, particularly Apostol's I and II, along with Spivak to bridge the two...
  31. J

    Fun and Rigorous math subjects

    Hey everyone! Ok, this summer I am self-studying Alg II so I can test out of it in August for my high school. I was also planning on learning a lot of pre-calc/calc from a friend, but I'm somewhat skeptical about it because he seems to not have that much time on his hands and I don't want to...
  32. danjordan

    Rigorous Quantum Mechanics text or online

    Hi everybody, I'm taking Quantum Mechanics 1 this semester. Thing is many of my classmates and I feel the professor is horrible and not a good teacher. He is often disorganized and unprepared for class. Problem is there's no way to change him so we're stuck with him. He apparently wants to...
  33. D

    Good rigorous intro combinatorics textbook?

    Hey guys, I'm thinking of self studying some math this summer, and combinatorics is one field I never really got into but want to learn a bit of. I rigorous proof experience from reading other books/taking rigorous classes, so I wanted to start off with something relatively rigorous. Thanks
  34. K

    Is there a more rigorous way to prove this?

    Homework Statement Show that d(v^2)/dt = 2 . (d^2r/dt^2) . (dr/dt) HINT: v^2 = ||dr/dt||^2 = dr/dt . dr/dt Homework Equations The Attempt at a Solution I did it another method: d(v^2)/dt = d/dv(v^2) . dv/dt --------------chain rule = 2v . dv/dt since v=dr/dt and dv/dt = d^2r/dt^2...
  35. B

    Vector Space R^3 and rigorous proof

    Homework Statement 1) Consider the 3 norms in vector space R^3, ##\| \|_i## where i=1,2 and infinity. Given x = (2, -5,3) and y = ( -3, 2,0). Calculate ##\|x\|_1, \|x+y\|_2, \|x-2y\|_\infty## 2)Prove Rigorously that ##\displaystyle \lim_{n \to \infty}=\frac{4n^2+1}{2n^2-1}=2##Homework...
  36. D

    Testing Rigorous Exam in Physics and Mathematics.

    I am very much interested in Physics and Mathematics. I want some rigorous hard or tough competitions exam. I live in India so I want international exams and can be recognize everywhere in the world. The Exam should be International and I am doing by graduation currently. I want to...
  37. M

    Learn Rigorous Mathematics - Time Management Tips

    I've taken a liking to studying mathematics, though I'm a physics major I've always tried to learn things as rigorous as possible whether it's mathematics or physics. Now, I haven't quite gotten to the level where I just breeze through proofs or at least when I study the theorems it still takes...
  38. M

    Rigorous classical education in the arts?

    The above is from another thread and I was rather intrigued as to what the posted spoke about. My schooling focused more on "how should my answers be for me to get maximum marks?" and as such, the phrase "rigorous classical education in the arts" seems quite vague to me. Would you guys be so...
  39. K

    Bosons and Fermions in a rigorous QFT

    I'm wondering, is there still a sharp distinction between Bosons and Fermions in a rigorous QFT, if exsits? My question is motivated by the following, consider one of the equations of motion of QED: \partial_\nu F^{\nu \mu} = e \bar{\psi} \gamma^\mu \psi In our familiar perturbative QED (Here...
  40. T

    Rigorous Lie Group and Lie Algebra Textbooks for Physicists

    Hi everyone, I was just wondering if anyone had any suggestions of more-mathematically-rigorous textbooks on Lie groups and Lie algebras for (high-energy) physicists than, say, Howard Georgi's book. I have been eying books such as "Symmetries, Lie Algebras And Representations: A Graduate...
  41. C

    Rigorous definition of derivative

    Hi, I have a question about the definition of derivative. As far as I know, for a real valued function f defined on a subset of R, the derivative of f at a is (f(x+h)-f(x))/h as h → 0. And if it exists it's said f is differentiable at x. What if I define f : Q → R as follows...
  42. Hepth

    Mathematically rigorous way of expanding propagators?

    Such propagators as found in HQET \frac{i}{2 v \cdot k} come about from expanding the full propagator. I'm wondering what the method is to properly Taylor expand denominators that contain 4-dimensional dot products. Lets start with something like : \frac{1}{2 v \cdot k + k^2} If we...
  43. N

    Rigorous introduction to electromagnetism

    I'm looking for a good introductory text on electromagnetism for someone who has very little experience with physics but is comfortable with multivariate and vector calculus and ODE's. I'd prefer something that starts from the basics and works up in a rigorous fashion. Note that I am proficient...
  44. K

    Rigorous definition of continuity on an open vs closed interval

    Let I be an open interval and f : I → ℝ is a function. How do you define "f is continuous on I" ? would the following be sufficient? : f is continuous on the open interval I=(a,b) if \stackrel{lim}{x\rightarrow}c \frac{f(x)-f(c)}{x-c} exists \forall c\in (a, b) is this correct? Also, what...
  45. N

    Rigorous Feynman pathintegral derivation

    Hey, I'm trying to do exercise I.2.1. from Zee's QFT in a nutshell but I ran into a problem. The exercise is to derive the QM path integral with a Hamiltonian of the form 1/2 m p^2 + V(q). In the textbook he shows the proof for a free hamiltonian. He gets to a point where he has (I left out...
  46. Q

    Rigorous delta-epsilon limit proof?

    I'm trying to prove the limit as x approaches 3 of sqrt(3x-5)=2. Call delta D and epsilon E So we have 0<abs(x-3)<D and must prove abs(sqrt(3x-5)-2)<E. abs(sqrt(3x-5)-2)=abs(sqrt (3(x-3)+4)-2)<E if we choose delta to be ((E+2)^2-4)/3, which simplifies to (E^2+4E)/3 My questions arise...
  47. M

    Virtual particles in rigorous quantum field theory

    "virtual particles" in rigorous quantum field theory If I am not mistaken "virtual particles" are just a name someone put to some integrals that we use to calculate different things, and those integrals depends on the perturbation scheme and on the gauge selected, and they don't even exist in...
  48. Q

    Rigorous proof about the nature of rolling motion

    In Resnick/Halliday, they describe how rolling can be described as the sum of a rotational force centered at the center of mass (for a wheel, say) and translational motion. The next part involves them saying that the motion can also be described as a completely rotational motion centered at the...
  49. H

    What Are the Best Textbooks for Challenging Physics Problems?

    Hello everyone I've recently begun studying "Introduction to Classical Mechanics" by David Morin, and the problems are AMAZING. Do you know of any other textbooks/collections of problems where problems like these can be found? (With solutions) This applies to any undergraduate physics...
  50. bcrowell

    Book stacking - how rigorous is the standard proof?

    book stacking -- how rigorous is the standard proof? There is a classic problem in mechanics, which is that you have n identical books, and you want to place them in a stack at the edge of a table so that they stick out as far as possible. Here is a typical, fairly careful statement of the...
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