Riemann Definition and 586 Threads

Homework Statement I've been looking at how integrable functions behave under composition, and I know that if f and g are integrable, f(g(x)) is not necessarily integrable, but it -is- necessarily integrable if f is continuous, regardless of whether g is. So I was wondering, what about if g is...

The problem says: evaluate 4x^2+y by breaking into four congruent subrectangles and evaluating at the midpoints, 1=<x<=5 0=<y<=2 When i setup the rectangles these are my coordinates: (1,1/2),(1,3/2),(3,1/2),(3,3/2) and delta A = 2 My answer comes out to be 168...

Homework Statement Suppose f(x):[a,b]\rightarrow\Re is bounded, non-negative and f(x)=0. Prove that \int^{b}_{a}f=0. Homework Equations The Attempt at a Solution I am trying to use the idea that lower sums are zero, and show that the upper sums go to zero as the norm of the...

I read in Elie CARTAN book : "la Géométrie des espace de Riemann" that when R = cte, you can compare space-time to hydrostatic description of a liquid. Is it true ?

One of the axioms of Riemann's geometry holds that there are no parallel lines and that any two lines meet. Since Riemann's geometry fits for that of a sphere, any two great circles of the sphere should intersect. However, if we were to take 2 longitudinal lines, then it is possible that these...

Homework Statement My book presents the Riemann-Darboux integral. It has a small supplemental section on the Riemann integral. Then a later section on the Riemann-Stieljes integral. Then a later chapter on the Lebesgue integral. A supplementary text that I have has a section on...

Homework Statement Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown in the figure below where the upper line is defined by 6x + y = 12 and the other line is defined by y=x^2-4. The figure, which I can't get on here, is just the...

Homework Statement http://img4.imageshack.us/img4/898/integerqj5.jpg Homework Equations The Attempt at a Solution It does appear to be a Riemann sum, I figured the 1/n is probably the width of the intervals and the sum in brackets is related to the sums of the heights of the rectangles. But my...

The characteristic function of the RATIONALS is a well-known example of a bounded function that is not Riemann integrable. But is the characteristic function of the IRRATIONALS (that is, the function that is 1 at every irrational number and 0 at every rational number) Riemann integrable on an...

...But it may not exist yet. Has any mathematician thought about producing a formula or function which spits out all the prime numbers? i.e 1->2, 2->3, 3->3, 4->5, 5->7, 6->11 etc. Any attempts been made? What the closest that people have thought?

Homework Statement I am a first-year physics student learning calculus. my question is about the approximation of the area of a region bounded by y = 0. Homework Equations Use rectangles (four of them) to approximate the area of the region bounded by y = 5/x (already did this one), and y =...

Homework Statement I am given a left riemann sum program module in Mathematica and need to convert it into the right riemann sum. The program takes values for x and f/x and the partition and graphs on a certain interval provided. leftRiemannGraph[f_, a_, b_, n_] := Module[{expr}, expr[1]...

Homework Statement I need to find the following: \lim_{n\rightarrow\infty}\left(\frac{1^2+2^2+3^2+...+n^2}{n^3} \right) Homework Equations The Attempt at a Solution I know I could do the sum of the series to find the result but I would like to use Riemann sums. I think I have to start by...

Can anyone provide me with a website that has copies of the original works of Riemann, Taylor, famous mathematicians. I am looking for papers on proved theorems.

in the Riemann Zeta function, is it possible to have two complex zeros off the critical strip that both have the same imaginary part?

Homework Statement Let f be continuous on [a,b] and suppose that f(x) \geq 0 for all x Є [a,b] Prove that if there exists a point c Є [a,b] such that f(c) > 0 , then \int_{a}^{b} f > 0 Homework Equations The Attempt at a Solution Using my books notation, Suppose P =...

Need some urgent help with Riemann Sums. Homework Statement PART A: In all of this question, let I = \int ^{2}_{-2} f(x)dx where f(x) = -2x + 1 Evaluate I. PART B: Use the defintion of the definite integral to evaluate I. i.e Riemann Sum. Homework Equations The...

My assignment: Solve for pi using a Riemann Sum with n= 40,000,000. The function is the antiderivate of 4/(1+x^2) dx. The bounds are from 0 to 1. Solving this gives you pi. Anyone know how to do this? Preferably with fortran77?

Homework Statement Express the integral as a limit of Riemann sums. Do not evaluate the limit. Homework Equations \int_0^{2\pi} x^{2}sin(x)\,dx The Attempt at a Solution I really don't know where to start...any help getting me started would be highly appreciated!

What is an example where it's Riemann integrable int(f(t),t,a,x) but no derivative exists at certain pts?

What's Example of Lebesgue Integrable function which is not Riemann Integrable?

Here is the classic Dirichlet function: Let, for x ∈ [0, 1], f (x) =1 /q if x = p /q, p,q in Z or 0 if x is irrational. Show that f (x) is Riemann integrable and give the value of the integral. Is this actually true?

Homework Statement Find the limit, as n -> infinity, of \sum_{k=1}^nk3/n4 Homework Equations Riemann sum: S(f, \pi, \sigma) = \sum_{k=1}^nf(\xi)(xk - xk-1) The Attempt at a Solution My guess is that I should try to put this sum in terms of a Riemann sum, and then taking n -> infinity will...

\zeta (s)= \frac{1}{(1-2^{1-s})} \sum_{n=0}^{\infty} \frac {1}{(2^{n+1})} \sum_{k=0}^{n}(-1)^k{n \choose k}(k+1)^{-s} Is the main problem with trying to prove the hypothesis algebraically boil down to the fact that s is an exponent to a "k" term? Would a derivation of the function that had...

I have two questions: Why hasn't the hypothesis been proved yet? Is it because we don't know why re(s) has to be 1/2 and thus can't prove it, or is it because we know why re(s) has to be 1/2 but we just don't know how to prove it. Why exactly does re(s) have to be 1/2? \zeta...

This is driving me nuts. (I originally posted this to the coursework section, but in thinking about this, I felt that it might not be the right place (this is for a term paper, not really any ongoing coursework, so there). Hope I'm not imposing ... I feel quite embarrassed on this one, since it...

I still don't understand a few things. Let's say we had a non-trivial zero counting function, Z_n(n), for the riemann zeta function. Couldn't we fairly easily prove the riemann hypothesis by evaluating \zeta (\sigma+iZ_n), solving for \sigma , then proving it for all n using induction...

Homework Statement Show that when f(z)=x^3+i(1-y)^3, it is legitimate to write: f'(z)=u_x+iv_x=3x^2 only when z=i Homework Equations Cauchy riemann equations: u_x=v_y , u_y=-v_x f'(z)=u_x+i*v_y The Attempt at a Solution u=x^3 v=(1-y)^3 u_x=3*x^2 v_y=-3*(1-y)^2...

Alright, I started doing Riemann sums and I am ripping my hair out in frustration. I just can't wrap my head around how I'm supposed to do it, and my woefully vague textbook isn't helping either. I'm wondering how I'm supposed to solve a Riemann sum question with sigma notation (no limits), and...

Can someone show me the steps to evaluating \zeta(c + xi), where 0 \leq c<1?

It is a standard fact that at any point p in a Riemannian space one can find coordinates such that \left.g_{\mu\nu}\right|_p = \eta_{\mu\nu} and \left.\partial_\lambda g_{\mu\nu}\right|_p. Consider the Taylor expansion of g_{\mu\nu} about p in these coordinates: g_{\mu\nu} = \eta_{\mu\nu}...

Hello folks, this is going to be a bit longish, but please bear with me, I'm going nuts over this. For a term paper I am working through a paper on higher dimensional spacetimes by Andrew, Bolen and Middleton. You can http://arxiv.org/abs/0708.0373" . My problem/confusion is in...

Homework Statement (My first post on this forum) Background: I am teaching myself General Relativity using Dirac's (very thin) 'General Theory of Relativity' (Princeton, 1996). Chapter 11 introduces the (Riemann) curvature tensor (page 20 in my edition). Problem: Dirac lists several...

Hi, Given a large number of test particles N, it should be possible to determine the Riemann curvature tensor by tracking their motion as they undergo geodesic deviation. Is there a minimum number N that will achieve this in any situation, or does it vary from problem to problem? How...

Hi everyone, hope this is the right place to put this :) I have just finished "Theory of Functions" Vol. 1 & 2 by Konrad Knopp. I'd like to continue with a book that picks up where the second volume it left off. (Especially would be nice is a more "modern" book) The second volume is about...

Hi There. Was working on these and I think I managed to get most of them but still have a few niggling parts. I've managed to do questions 2,3,3Part2 and I've shown my working out so I'd be greatful if you could verify whether they are correct. Please could you also guide me on Q1 & 4...

I have a question concerning the Riemann Hypothesis, a conjecture about the distribution of zeros of the Riemann-zeta function. the trivial zeros (s=-2, s= -4, s=-6) arent much of a concern as the NON-trivial zeros, where any real part of the non-trivial zero is = 1/2. What i am having...

Lagarias’ equivalence to the Riemann hypothesis should be discussed, i.e., if hn := n-th harmonic number := 1/1 + 1/2 + · · · + 1/n, and σn := divisor function of n := sum of positive divisors of n, then if n > 1, hn + ehn ln hn > σn. There is a $1,000,000 prize for the proof of this at...

http://arxiv.org/abs/0806.0892"

i am a high school student i want to prove the riemann hypothesis but i do not how to start:confused:

Hello I plan on applying to the university of waterloo next year and due to the fact that many of my marks are not that great (failed gr 10 math) I decided to start a site to showcase my ability in math and programing. For those of you who are interested I wrote a program to graph regions of...

f(x) = x , if x is rational = 0 , if x is irrational on the interval [0,1] i just wanted to check if my reasoning is right. take the equipartition of n equal subintervals with choices of t_r's as r/n for each subinterval. calculating the integral as limit of this sum (and...

Can anyone tell we how this: http://mathworld.wolfram.com/RiemannSeriesTheorem.html can be proved? The book that I read it in said that it was "beyond the scope of the book". It one of the coolest theorems I've read about. For example, it means that for any number (pi, phi, ...) there's...

[SOLVED] Riemann Sum with Fortran 90 My assignment: Use Reimann Sums to estimate pi to 6 decimal places (ie: you can stop when successive iterations yield a change of less than 0.000001. For the Reimann Sums solution, an iteration equals 2X the number of segments as the trial before. Print out...

Suppose you have a spacetime with an observer at rest at the origin, and the surface at t = 0 going through the origin, and passing through the surface there are geodesics along increasing time. Then as you get a small ways away from the surface, the geodesics start to deviate from each other...

[SOLVED] Riemann sum Important stuff: \sum i^2 = \frac{n(n+1)(2n+1)}{6} \sum i = \frac{n(n+1)}{2} And the solution: (Where I write "lim" I mean limit as n-->infinity. Where I write the summation sign I mean from i=1 to n.) lim \sum t^2 + 6t - 4 \Delta t \Delta t = \frac{5 -...

[SOLVED] Summation - Riemann Intergral - URGENT Homework Statement Im working on the upper and lower riemann sums of f(x) = exp(-x) where Pn donates the partition of [0,1] into n subintervals of equal length (so that Pn = {0,1/n,2/n,...,1}) Homework Equations The Attempt at...

Homework Statement Prove that the function specified below is Riemann integrable and that its integral is equal to zero. Homework Equations f(x)=1 for x=1/n (n is a natural number) and 0 elsewhere on the interval [0,1]. The Attempt at a Solution I have divided the partition into...

Could anybody please give advice for the study of complex analysis, Riemann surfaces & complex mappings. These subjects form the content of chapters 7 & 8 of Roger Penrose's "The Road to Reality". Any advice will do: maybe suggestions on books to supplement the learning, or books to further my...

Homework Statement Let f, g : [a, b] \rightarrow R be integrable on [a, b]. Then, prove that h(x) = max{f(x), g(x)} for x \in [a, b] is integrable. 1 Homework Equations Definition of integrability: for each epsilon greater than zero there exists a partition P so that...