Riemann Definition and 586 Threads

why does the einstein field tensor have the riemann tensor contracted? I am confused as to what purpose it serves. I have seen an explanation that it gets rid of extra information about spacetime or something like that. and also is the Ricci scalar added to einstein tensor so that the covariant...

What Lie groups are also Riemann manifolds? thanks

My crude understanding of GR in outline is that spacetime curvature is described by the way the components of the Riemann tensor vary from point to point in spacetime, that such variation is controlled by Einstein's field equations, and that the source of curvature is the energy-momentum tensor...

In the Riemann theory for a function f defined on all of R, we define its improper integral over R as the sum of two limits: \int_{-\infty}^{+\infty}f(x)dx = \lim_{a\rightarrow -\infty}\int_{a}^0f(x)dx+\lim_{b\rightarrow +\infty}\int_{0}^b f(x)dx and in general, this is not equal to...

Find the Riemann sum for this integral using the right-hand sums for n=4 Find the Riemann sum for this same integral, using the left-hand sums for n=4 Sorry the integral is attatched. I don't know how to get it onto here.

Hi I recently stumbled upon this: I know that the Riemann Integral is defined for every piecewise continouus curve. But now suppose you´re asked the following: you are given f(x,y)=\frac{xy^3}{(x^2+y^2)^2} with additional Definition f(0,0)=0. ( It´s a textbook problem :) ) Now surely...

Please HELP...Don't Understand Simple Concept on Riemann Sums Can someone please explain this to me... The number of subintervals in a partition approaches infinity as the norm of the partition approaches 0. That is, ||Triangle|| approaches 0 implies that n approaches infinity. I thought...

I often see people use the Riemann definition of the integral to solve a certain limit-series computation, but they usually just skip a step that I can follow one way but not the other. Given the integral, I can see the limit-series that comes from it, but when trying to find the integral from...

Hi guys, My gf is doing honours and is having some trouble with one question on her assignment for complex analysis. She is really stuck and I've only done this topic at an undergraduate level so I have no idea. Neither of us have done any subjects in Topology so we don't know what to do...

Could the maths of string theory or versions of it lead insight into the Riemann hypothesis as, for a start both are about mathematics in the complex plane. Anyone working on this connection at the moment?

In the course I'm taking, we are already done with Lebesgue integration on R, and while we have proven that for continuous fonctions, the Riemann integral and the Lebesgue integral give the same output, we have not investigated further the correspondance btw the two. So I have some questions...

What is the difference between the two? Does the Newton integral arise from the fundalmental theorem of calculus and the Riemann integral is the Newton integral but more rigorously defined?

Can somebody explain me about the trivial zeros? Why \zeta(-2) = \zeta(-4) = \zeta(-6) = 0 = \zeta(k) So \zeta(k) \sum_{n=1}^{ \infty} \frac{1}{n^k} = 0 ?

Homework Statement Let f:[a,b] -> R R being the set of real numbers If f^3 is Reimann-integrable, does that imply that f is? Homework Equations If f is Riemann-Integrable, then it has upper/lower step functions, such that the difference between the upper and lower sums is less...

Are there notes on the net or books that give a gentle introduction on Riemann surfaces ( say undergraduate math or math for physicists type level)? Always read of the importance and beauty of Riemann surfaces but can't find surveys or intros for outsiders. Same for elliptic curves...

http://www.arsmathematica.net/archives/2007/05/22/latest-on-latest-paper-on-arxiv/

Can someone elaborate on the relationship of the Riemann zeros and primes? How are the zeros harmonic to the primes? The quotes below mention the 'sum of its complex zeros' and 'other sums over prime numbers'. Can someone clarify this? From Answers.com: "The zeros of the Riemann...

Can someone please explain to what exactly the Riemann Hypothesis is? My friend said it is something to do with imaginary numbers and how they behave in a certain interval- just wondering.

Homework Statement The following sum \sqrt{9 - \left(\frac{3}{n}\right)^2} \cdot \frac{3}{n} + \sqrt{9 - \left(\frac{6}{n}\right)^2} \cdot \frac{3}{n} + \ldots + \sqrt{9 - \left(\frac{3 n}{n}\right)^2} \cdot \frac{3}{n} is a right Riemann sum for the definite integral. Solve as n->infinity...

How many degrees of freedom has Riemann Tensor in general D dimensions and how it can be calculated?

Use the riemann sums model to estimate the area under the curve f(x) = x^2, between x =2 and x = 10, using an infinite number of strips. Be sure to include appropriate diagriams and full explanation of the method of obtaining all numerical values, full working and justification. Does anybody...

Homework Statement let f:[0,oo) -> R be given by f(x) = sin(x) / x for x>0 and f(0) = c. Prove that f is improper riemann integrable without computing the integral explicitly The Attempt at a Solution I've attempted to find a upperbound for f(x) such that the integral does not diverge...

I was wondering if anyone could tell me more about the Riemann Zeta function, esp at negative values. Especially when \sum_{n=1}^{\infty}n= \frac{-1}{12} R where R is the Ramanujan Summation Operator. Could anyone post a proof?

Homework Statement Find the Riemann sum associated with f(x)=3 x^2 +3 ,\quad n=3 and the partition x_0=0,\quad x_1=3,\quad x_2=4,\quad x_3=6,\qquad \mbox{ of } [0,6] (a) when x_k^{*} is the right end-point of [x_{k-1},x_k]. . (b) when x_k^{*} is the mid-point of [x_{k-1},x_k]...

Is there ever a situation where it is more appropriate/advantageous to use Riemann summation as opposed to evaluating an integral, or is Riemann summation merely taught in order to help the student to understand what's going on?

Hello, I am doing a project on Riemann but I don't understand some of his contributions. For instance, I do not understand what Riemann zeta-function measures or what it does, what the Riemann hypothesis states, or what the difference is between Riemannian geometry and hilbert geometry. Any...

For what values of p>0 does the series Riemann Sum [n=1 to infinity] 1/ [n(ln n) (ln(ln n))^p] converge and for what values does it diverge? How do i do this question? Would somebody please kindly show me the steps? Do i use the intergral test?

The cauchy Riemann relations can be written: \frac{\partial f}{\partial \bar{z}}=0 Is there an 'easy to see reason' why a function should not depend on the independent variable [itex]\bar{z}[/tex] to be differentiable?

Hello, just going through some Riemann sum problems before I hit integrals and I am like 99% sure that this answer from my text is wrong but I want to make sure. It's not really an important question so if you have better things to do, help the next guy :) But checking this over would be...

"Riemann zeta function"...generalization.. Hello my question is if we define the "generalized" Riemann zeta function: \zeta(x,s,h)= \sum_{n=0}^{\infty}(x+nh)^{-s} which is equal to the usual "Riemann zeta function" if we set h=1, x=0 ,then my question is if we can extend the definition...

Can anyone tell me if Riemann's Prime Counting function can be solved by residue integration? Here it is: J(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{ln(\zeta(s))x^s}{s}ds which has the solution: J(x)=li(x)-\sum_{\rho}li(x^\rho)-ln(2)+ \int_x^{\infty}\frac{dt}{t(t^2-1)ln(t)} I...

In a book of mine, the author proceeds to the proof that a Riemann sum in a interval [a,b] must converge by proving that for S_m and S_n (m>n) where the span of the subdivisions is suffiencienly small, then |S_m - S_n)| < e(b-a) Where e can assume infinitly small values in dependence of...

Let H_{n}=\sum_{k=1}^{n}\frac{1}{k} be the nth harmonic number, then the Riemann hypothesis is equivalent to proving that for each n\geq 1, \sum_{d|n}d\leq H_{n}+\mbox{exp}(H_{n})\log H_{n} where equality holds iff n=1. The paper that this came from is here: An Elementary Problem...

Hi, maybe someone can help. When I think about it, I'm pretty sure that the following is true: Let c be a curve parametrized by t\in [a,b], let \sigma = \{t_0,...,t_N\} be a partition of [a,b] and \delta_{\sigma}=\max_{0\leq k \leq N-1}(t_{k+1}-t_k). Also define \Delta t_k=t_{k+1}-t_k Then...

Use the Riemann sum and a limit to evaluate the exact area under the graph of y = 2x^2 + 4 on [0, 1] I know how to do this normally but now they ask to do it w/ a limit and I'm not sure how. (LaTex corrected by HallsofIvy.)

Given a function f: [0,1] \to \mathbb{R}. Suppose f(x) = 0 if x is irrational and f(x) = 1/q if x = p/q, where p and q are relatively prime. Prove that f is Riemann integrable.

A doubt..why einstein Chose Riemann Tensor for GR?..i know its covariant derivative is zero and all that..but Why Riemann tensor?...was not other tensor avaliable or simpler than that?..i studied that and found that for Geodesic deviation ( i didn,t understand that concept..sorry) the Riemann...

so i know what it is (i think lol) ... but what are its applications?

I was hoping someone could check my answer? Use the limit of a Riemann sum to find the area of the region bounded by the graphs of y=2x^3+1, y=0, x=0, x=2. Area=2

let \zeta(z)=\sum_{n \in \mathbb{N}} n^{-z} ~ {{a+ib}}>1 then, \zeta(z)=0 iff z=-2n where n is a natural number. pi(x)=\int_0^\infty\frac{dx}{\xS[x+1]} gamma(x+) where S[x+1]= \sum_{n \in \mathbb{N}} n^-{x+1} I have discovered that pi(x)=\int_a^b\frac{dx}/logx = 1/log b+ 2/log b...

They claim that the trivial 0s (zeta(z)=0) occur when z is a negative odd integer (with no imaginary component). But it seems obviously wrong. Take z=-2 zeta(-2)=1+1/(2^-2)+1/(3^-2)... =1+4+9... Obviously this series will not equal 0. Where have I gone wrong? Have I misunderstood...

hi, is it possible to find the riemann sum of (cos1)^x? it looks divergent to me can someome please help me... even if it is convergent, i don't know how to find the sum of a trigonometric function

Is it possible to show by induction that f:[a,b]->R, a bounded function, is Riemann integrable if f has a countable number of discontinuities? I'm told this is usually done with Lebesgue integrals, but I don't see why an inductive proof of this using Riemann/Darboux integrals can't work.

"Let (r_n) be any list of rational numbers in [0,1]. Let (a_n) be a sequence such that 1>a_1>a_2>...>a_n>... converging to 0. Let f be defined such that f=0 if x is irrational =a_n if x is equal to r_x Prove that f is Riemann integrable." We are doing integrals from Darboux' approach, so no...

Let f:[a,b]->R be bounded. Further, let it be continuous on [a,b] except at points a1, a2, ...,an,... such that a1>a2>a3>...>an>...> a where an converges to a. Prove that f is Riemann integrable on [a,b]. It suffices to prove that f is integrable on [a,a1) (I've worked out that part). And...

Problem states: (A) Use mathematical induction to prove that for x\geq0 and any positive integer n. e^x\geq1+x+\frac{x^2}{2!}+...+\frac{x^n}{n!} (B) Use part (A) to show that e>2.7. (C) Use part (A) to show that \lim_{x\rightarrow\infty} \frac{e^x}{x^k} =...

This problem is in the section of my book called "The Natural Logarithmic Function" so I am guessing that I would have to use a natural log somewhere...but anyways...the problem says: For what values of m do the line y=mx and the curve y=\frac{x}{(x^2+1)} enclose a region? Find the area of the...

I always wonder how the definitions of curvatures of curves and surfaces be unified by the Riemann Tensor symbols. For surfaces, I know R_{1,2,1,2} corresponds to the Gaussian curvature of a surface. How come R_{1,1,1,1}=0 and not corresponds to the curvature of a curve in \RE^2 or in \Re^3...

Just wondering if any of the posters seriously tried solving Riemann Hypothesis. And if yes, then what kinds of problems did you run into?

How do i find the number of independent components of the Riemann curvature tensor in D space-time dimensions. One is given that the Riemann tensor is an (2,2) irreducible rep of GL(4, \mathbb{R}) and obeys Bianchi I R_{[\mu\nu|\rho]\lambda}=0 Been trying this problem for 3 days and...