Rigorous Definition and 154 Threads

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...

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...

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...

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...

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

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...

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...

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...

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...

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...

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...

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...

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...

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 =...

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

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...

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...

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")...

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...

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?

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...

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...

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...

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...

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...

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...

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...

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...

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

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

"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...

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...

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...

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...

Is math suppose to be rigorous? I love it and I hate it. Is this right? So far I'm in trig, I'm struggling a bit. Is math suppose to be tough, do you have to work at it everyday and study for 4 to 5 hrs or does it come easy to you? Please share. Thanks..

it is a well known result, but it is a rigorious result or not?

Looking for a little advice regarding proving things in mathematical way. I am a physics major currently taking a math methods course where we are asked to prove things, basically for the time in my schooling career. Sometimes I have trouble formulating a mathematically rigorous way of...

From a recent double-slit thread: Thank you for mentioning those papers! (Probably, I didn't notice the earlier mentions because I hardly ever read the endless "double-slit" threads. :-) The Fourier transform method occurred to me a while back as possibly a better way of deriving this...