Riemann Definition and 619 Threads

For you math people who like to express objects in a coordinate-free way, how would you denote the Riemann and the Ricci curvature tensors? They are both usually denoted R but with different indices to show which one is which. Is there a standard way to write them without the coordinate indices?

Greetings everyone. First, it's great that the site is back again and I hope it can be merged soon enough. :D Here's the question: Let \( f \) be a bounded function on \( [a,b] \). Suppose there exist sequences \( (U_n) \) and \( (L_n) \) of upper and lower Darboux sums such that \( \lim (U_n -...

Homework Statement Show the Thomae's function f : [0,1] → ℝ which is defined by f(x) = \begin{cases} \frac{1}{n}, & \text{if $x = \frac{m}{n}$, where $m, n \in \mathbb{N}$ and are relatively prime} \\ 0, & \text{otherwise} \end{cases} is Riemann integrable. Homework Equations Thm: If fn...

Homework Statement Find the Riemann function for uxy + xyux = 0, in x + y > 0 u = x, uy = 0, on x+y = 0 Homework Equations The Attempt at a Solution I think the Riemann function, R(x,y;s,n), must satisfy: 0 = Rxy - (xyR)x Rx = 0 on y =n Ry = xyR on x = s R = 1 at (x,y) = (s,n) But I...

I don't know anything of complex analysis or analytic number theory or analytic continuation. But i read about zeta function and riemann hypothesis over wikipedia, clay institute's website and few other sources. I started with original zeta function...

Is the imaginary axis considered a closed curve on the Riemann Sphere?

Let p_n be number of Non-Isomorphic Abelian Groups by order n. For R(s)>1 with \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s} we define Riemann zeta function. Fundamental theorem of arithmetic is biconditional with fact that \zeta(s)=\prod_{p} (1-p^{-s})^{-1} for R(s)>1. Proove that for R(s)>1 is...

$\displaystyle \int_{0}^{1} \ln \ x \ dx $ is not a proper Riemann integral since $\ln \ x $ is not bounded on $[0,1]$. Yet $ \displaystyle \int_{0}^{1} \ln \ x \ dx = \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \ln \left(\frac{k}{n} \right)$. Is this because $\ln \ x$ is monotone on $(0,1]$?

Hi guys I'm trying to understand Riemann Zeta functions particularly for s=1/3 I know \zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s} and converges for Re(s)>1 Ok, but what about for s=1/3, then \zeta(1/3)=\sum_{n=1}^\infty\frac{1}{n^{1/3}}= \sum_{n=1}^\infty\frac{1}{\sqrt[3]{n}}...

Homework Statement For which integral, is the below example, a Riemann sum approximation.? The example is: 1/30( sqrt(1/30) + sqrt(2/30) + sqrt(3/30)+...+sqrt(30/30)) A. Integral 0 to 1 sqrt(x/30) B. Integral 0 to 1 sqrt(x) C. (1/30) Integral 0 to 30 sqrt(x) D. (1/30) Integral 0 to 1...

In general 4d space time, the Riemann tensor has 20 independent components. However, in a more symmetric metric, does the number of independent components reduce? Specifically, for the Schwartzchild metric, how many IC does the corresponding Riemann tensor have? (I think it is 4, but I...

Homework Statement Use the form of the definition of the integral to evaluate the following: lim (n \rightarrow ∞) \sum^{n}_{i=1} x_{i}\cdotln(x_{i}^{2} + 1)Δx on the interval [2, 6] Homework Equations x_{i} = 2 + \frac{4}{n}i Δx = \frac{4}{n} Ʃ^{n}_{i=1}i^{2} =...

Homework Statement Is there any painless way of calculating the Riemann tensor? I have the metric, and finding the Christoffel symbols isn't that hard, especially if I'm given a diagonal metric. Out of 40 components, most will be zero. But how do I know how to pick the indices of...

Homework Statement I have the metric of a three sphere: g_{\mu \nu} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & r^2 & 0 \\ 0 & 0 & r^2\sin^2\theta \end{pmatrix} Find Riemann tensor, Ricci tensor and Ricci scalar for the given metric. Homework Equations I have all the formulas I need, and I...

Let Pn denote the partition of the given interval [a,b] into n sub intervals of equal length Δxi = (b-a)/n Evaluate L(f,Pn) and U(f,Pn) for the given functions f and the given values of n. f(x)=x on [0,2], with n=8 2.My solution x0 = 0, x1 = 1/4, x2 = 1/2, x3 = 3/4, x4 = 1, x5 = 5/4...

Hi, given the algebraic function: f(z,w)=a_n(z)w^n+a_{n-1}(z)w^{n-1}+\cdots+a_0(z)=0 how can I determine the geometry of it's underlying Riemann surfaces? For example, here's a contrived example: f(z,w)=(w-1)(w-2)^2(w-3)^3-z=0 That one has a single sheet manifold, a double-sheet...

Homework Statement Prove that: lim n->inf1/n*Ʃn-1k=0ekx/n = (ex-1)/x x>0 Homework Equations That was all the information provided. Any progress i have made is below. I didn't want to add any of that to this section because this is all speculation on my part so far. The Attempt at a...

My professor asks, "Double check a formula that specifies how Riemann tensor is proportional to a curvature scalar." in our homework. The closet thing I can find is the relation between the ricci tensor and the curvature scalar in einstein's field equation for empty space.

This isn't a homework question. My adviser has me studying basic analysis and has lately pushed me towards the following question: "Let f be any continuous function. Can we prove that there exists a SEQUENCE of step functions that converges UNIFORMLY to f?" I have noticed this idea is...

Hi, Currently, I need to read some reference about Integrable System, but I am stuck in Riemann Surface, genus, divisors, and Riemann Theta Functions. This makes me anxious. Is there introduction or pedagogical reference on this topic? I think I can spend some time read it during winter...

Suppose we have: f(x)= 1 if 0\leq x \leq 1 AND 2 if 1\leq x \leq 2 Using the definition, show that f is Riemann integrable on [0, 2] and find its value? I have a general idea of how to complete this question using partitions and the L(f,P) U(f,P) definition, but am not quite receiving the...

Let f:[0,1]→ℝ be an increasing function. Show that for all x in (0,1], \frac{1}{x}\int_{0}^{x}f (t) \,dt \le \int_{0}^{1}f (t) \,dt So by working backwards I got to trying to show that (1-x)\int_{0}^{1}f (t) \,dt \le \int_{x}^{1}f (t) \,dt . While I know both sides are equal at x=1, the...

I know that the improper integral \int_2^\infty \left(\frac{1}{x\log^2x}\right)^p \, dx converges for p=1, but does it diverge for p>1? How do you show this?

I learned something about genus in Topology. The concept Genus in Topology is intuitive and lucid. Now I am confronted with the Genus in Riemann Surface. I do not know what is Genus on Riemann Surface. Is it relevant to "singularity"? Anyone can help me make it a bit clear? Thanks.

I'm reviewing my Calc 1 material for better understanding. So, I was reading about the area under a curve and approximating it using Riemann sums. Now, I understand the method, but I was a little confused by finding xi*. I know there is a formula for it xi*=a+Δx(i). What does the "i" stand for...

Hi, this is fairly fundamental and basic, but I cannot seem to make sense of it I know z = x + iy and hence a function of this variable would be in the form h = f(z). BUT I do not understand why f(z) = u(x,y) + iv(x,y) why so? in z = x + iy, x is the real part and iy is the imaginary...

I've been having some trouble with a maths problem and I hoped someone might be able to help. We don't seem to have been taught most of what we need to do this, I understand Riemann integrals but what we've been taught and what they're asking for is just different. I could do with a...

The simple pole at on is due to that its value of course is not closed due to it is an infinite value. My question is: is this value of infinity, positive or negative. or both??

Hey guys, I saw these just showed up on arXiv, published by some unknown who claims to have invented his own number system and is not affiliated with any academic institutions. What do you make of this? http://arxiv.org/abs/1110.3465 http://arxiv.org/abs/1110.2952

Homework Statement Establish the theorem that any 2-dimensional Riemann manifold is conformally flat in the case of a metric of signature 0. Hint: Use null curves as coordinate curves, that is, change to new coordinate curves \lambda = \lambda(x0, x1), \nu = \nu(x0, x1) satisfying...

let be the function \sum_{\rho} (\rho )^{-1} =Z and let be the sum S= \sum_{\gamma}\frac{1}{1/4+ \gamma ^{2}} here 'gamma' runs over the imaginary part of the Riemann Zeros then is the Riemann Hypothesis equivalent to the assertion that S=2Z ??

Hello, I have read in many articles that the trivial zeros of the Riemann zeta function are only the negative even integers (-2, -4, -6, -8, -10, ...). The reason why these are the only ones is that when substituting them in the functional equation, the function is 0 because...

Alright, I cannot seem to get this subroutine to return the correct sums for the trapezoidal rule... Where do I need to fix? SUBROUTINE atrap(i) USE space_data IMPLICIT NONE INTEGER :: i, j REAL :: f_b1, f_b2, f_x1, f_x2, trap_area REAL :: delta_x trap_area = 0 f_b1 = lower f_b2...

Can someone give me an example of a bounded function f defined on a closed interval [a,b] such that f does not attain its sup (or inf) on this interval? Obviously, f cannot be continuous, but for whatever reason (stupidity? lack of imagination?) I can't think of an example of a discontinuous...

(1) Let s be a complex number like s = a + b i, then we define \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} Our aim: to compute ζ(\frac{1}{2}+14.1347 i) with the help of the programming language Aribas (2) Web Links Aribas...

Homework Statement At my old university, Calculus was taught much differently than it is where I am now. My old school focused on numerical things, which this school focuses much more on pictures, abstract, etc. and it's very difficult for me. At my old school, we were given a shape...

I have started with the space-time metric in a weak gravitational field (with the assumption of low velocity): ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2) Where \phi<<1 is the gravitational potential. Using the standard form for the Christoffel symbols have found: \Gamma^0_{00}=\phi_{,0}...

I have started with the space-time metric in a weak gravitational field (with the assumption of low velocity): ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2) Where \phi<<1 is the gravitational potential. Using the standard form for the Christoffel symbols have found...

Riemann Curvature? i was watching this documentary that mentioned that riemann came up with a method to deduce whether we were on a curved surface, or on a flat surface, without leaving the surface to make the deduction. for example, for a curved 2d surface, we know it is as such as we can...

Homework Statement Consider the integral, \int _3 ^7 (\frac{3}{x} + 2) dx a) Find the Riemann Sum for this integral using right endpoints and n=4. b) Find the Riemann Sum for this integral using left endpoints and n=4. Homework Equations The sum, \sum^{n = 4} (\frac{3}{x} + 2) The graph...

I am reading a chapter on Complex Functions, Laplace Transforms & Cauchy Riemann (as part of Control theory) And I don't understand how they arrive at a particular part. [ I tried to type it out in tex, but it takes way too much time so uploaded a screenshot to flickr]...

Hello. I have to solve some integrals using both the standard theorem of calculus and infinite Riemann sums. \int_{1}^{7} (x^2-4x+2) dx = \lim_{n \to \infty } \sum f(x_i)\Delta x_i = \lim_{n \to \infty } \sum (x_i^2 - 4x_i + 2)6/n Evaluating the definite integral results in an answer of 30...

I was just going over Riemann integrability and how to prove it, and was just wondering is it possible to have a function f that is not Riemann integrable but |f| is Riemann integrable? Say on an interval [0,1] for example. (as that is what most examples I have done are on so easiest for me to...

what would Rabcd;e look like in terms of it's christoffels? or Rab;c

Given that \zeta (2n)=\frac{{\pi}^{2n}}{m} Then how do you find m with respect to n where n is a natural number. For n=1, m=6 n=2, m=90 n=3, m=945 n=4, m=9450 n=5, m=93555 n=6, m=\frac{638512875}{691} n=7, m=\frac{18243225}{2} n=8, m=\frac{325641566250}{3617} n=9...

Hi! While studying a text " A First Course in Real Analysis" by protter, I've been asked to prove a property of riemann stieltjes integral. The propery is as follows ; Suppose a<c<b. Assume that not both f and g are discontinuous at c. If \intfdg from a to c and \intfdg ffrom c to b exist...

Hello, There is a definition of simple-connectedness for a region R of the complex plane C that states that a region R is simply-connected in C if the complement of the region in the Riemann Sphere is connected. I don't know if I'm missing something; I guess we are actually consider...

Homework Statement Hi! I'm studying line integral in vector calculus and I've faced a difficulty related with the additivity of line integral. I really hope to get an answer for my question through this site. (2.17) Theorem : If \intfd\phi from a to b exists and a<c<b, then \intfd\phi...

Homework Statement 1. Express as a sum of riemann and write the integral to express the area of the trapezoid with vertex (0,0) , (1,3) , (3,3) , (5,0). 2. find the intersection points limited by these equations y = xsquare -3x and y = -2x +3 = 0 3. the trapezoid with vertex...

Homework Statement a.) Use definition 2 to find an expression for the area under the curve y=x^3 from 0 to 1 as a limit. b.)Evaluate the (above) limit using the sum of the cubes of the n integers. Homework Equations (\frac{n(n+1)}{2})^{2} The Attempt at a Solution For part a.) I wrote my...