Riemann Definition and 586 Threads

Just looking at the summand, I can see that the function is ln(pi/4 + x^2) as the (i pi/2n) term is the 'x' term. How do I determine the limits of the integral, however? I was thinking about using the lower bound of the summation --> this given the (pi / 2n)^2 term, implying that nothing was...

I understand mathematically several ways to test whether an infinite series converges or diverges. However, I came across one particular equation that is stumping me, ## \sum_{n=1}^{\infty} 1/n ##. I understand how to mathematically apply series tests to show it diverges. But intuitively, I...

Is it correct to define Riemann integrability as follows: 'For any ϵ>0, there exists a δ>0 such that if the maximum interval length of the partition is less than δ, then the difference between the upper and lower Riemann sums is less than or equal to ϵ'? I wanted to define Riemann integrability...

There is a very interesting topic in mathematics called the Hopf fibration. In 1931, Heinz Hopf published his work on the construction he discovered in topology, which in history was called "Hopf fibration". The essence of this design was based on the geometric designs of William Kingdon...

Another GR question... in the thick of revision season. I would appreciate a sketch of how to approach the problem. You basically are given a metric, involving a positive function ##A(z)##, $$g = A(z)^2(-dt^2 + dx^2 + dy^2) + dz^2$$The game is to figure out somehow that the null-energy...

Was reading the Reimann integrals chapter of Understanding Analysis by Stephen Abbott and got stuck on exercise 7.2.5. In the solutions they went from having |f-f_n|<epsilon/3(a-b) to having |M_k-N_k|<epsilon/3(a-b), but I’m confused how did they do this. We know that fn uniformly converges to...

I was reading Bernhardt Riemann's old foundational text on abelian functions, and I found a part that really confused me. What he is trying to do is set up an invariant to classify 2d surfaces as simply connected, multiply connected, etc via some kind of "connectivity number". From the text, I...

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What is the zeta function and the Riemann hypothesis.

In the book general relativity by Hobson the gravitational wave of a binary merger is computed in the frame of the binary merger as well as the TT-gauge. I considered what components of the Riemann tensor along the x-axis in both gauges. The equation for the metric in the source and TT-gauge are...

Hi! I am having trouble finalizing this problem. The interval is given so we know that a = 1 and b = 2. From there you can figure out that ∆x = 1/n, xiR = 1 + i/n. Using logarithmic properties, I rearranged the expression and wrote (1 + i/n)(1/n)ln[(n + i)/n]. I can guess that the function is...

Hi! I understand that this is an expanded Riemann sum but I'm having trouble determining its original form. I don't actually have any ideas as to how to find it, but I know that once I determine the original form of the Riemann sum, I will be able to figure out the values for a, b, and f. If...

This is an Argand diagram showing the first 40,000 terms of the series form of the Riemann Zeta function, for the argument ##\sigma + i t = 1/2 + 62854.13 \thinspace i## The blue lines are the first 100 (or so) terms, and the rest of the terms are in red. The plot shows a kind of approximate...

I'm reading Carroll's GR notes and I'm having trouble deciphering a particular expression for the Riemann curvature tensor. The coordinate-free definition is (eq. 3.71 in the notes): $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ An index-based expression is also given in (eq...

Given a function ##f##, interval ##[a,b]##, and its tagged partition ##\dot P##. The Riemann Sum is defined over ##\dot P## is as follows: $$ S (f, \dot P) = \sum f(t_i) (x_k - x_{k-1})$$ A function is integrable on ##[a,b]##, if for every ##\varepsilon \gt 0##, there exists a...

Partition each closed interval ##[a_i,b_i]## in the Cartesian product, ##A##. Denote the partition for the i-th closed interval as ##\{x_i^1,\ldots,x_i^{k_i}\}##. The Cartesian product of the partitions forms a partition of ##A## (think: a lattice of points that coincide with the points of each...

How to calculate the contraction of second and fourth indices of Riemann tensor?I can only deal with other indices.Thank you！

Hello PF! If ##\Re (s)## is the real part of ##s## and ##\Im (s)## is the imaginary part, then t is very easy to prove that $$\zeta (s) = \zeta ( \Re (s) ) \zeta ( \Im (s)i) - \displaystyle\sum_{n=1}^\infty \frac{1}{n^{\Re (s)}} [\displaystyle\sum_{k \in S, \mathbb{Z} \S = n}...

. Let C be a smooth curve with arc length L, and suppose that f(x, y) = P(x, y)i +Q(x, y)j is a vector field such that $|| f|(x,y) || \leq M $ for all (x,y) on C. Show that $\left\vert\displaystyle\int_C f \cdot dr \right\vert \leq ML $ Hint: Recall that $\left\vert\displaystyle\int_a^b g(x)...

I got something like this, but I'm not sure it is correct, because if it has the same order of convergence as trapezoidal rule which is 2, it should yield the same result as trapezoidal rule but mine doesn't (?). For example sin(x) for [0,1], n with trapezoidal rule = 0.420735... With my own...

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I just need to compare my results of the Ricci and Riemann Tensors of FRW metric, but only considering the spatial coordinates.

i am facing problems in the definition of dual oF some objects which has pair of anti symmetric indices e.g. Weyl curvature tensor. Double dual is there in the literature but given that how to find the anti self dual part of that. the problem is written in attached the file.

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According to Wikipedia, the definition of the Riemann Tensor can be taken as ##R^{\rho}_{\sigma \mu \nu} = dx^{\rho}[\nabla_{\mu},\nabla_{\nu}]\partial_{\sigma}##. Note that I dropped the Lie Bracket term and used the commutator since I'm looking at calculating this w.r.t. the basis. I...

I am reading J. J. Duistermaat and J. A. C. Kolk: Multidimensional Analysis Vol.II Chapter 6: Integration ... I need help with the proof of Theorem 6.2.8 Part (iii) ...The Definition of Riemann integrable functions with compact support and Theorem 6.2.8 and a brief indication of its proof...

I am not sure of the overall purpose of the concepts developed below regarding Riemann integrable functions with compact support ... nor am I sure of the details ... so I am sketching out the meaning as I understand it in 2 dimensions and depicting the relevant entities in diagrams ... I am...

I'm working on a problem involving some hypothetical spacetimes (i.e. no tables/data-sheets available) and need to calculate a bunch of ##R_{\mu \nu \rho \sigma}## and ##R_{\mu \nu}## values, as well as ##R##. The metrics contain some arbitrary functions ##f(x^i)## of the spatial co-ordinates...

We show that there is a partition s.t. the upper sum and the lower sum of ##f## w.r.t. this partition converge onto one another. Let ##\epsilon>0##. Define a sequence of functions ##g_n:[a,b]\setminus(\{a_n\}_{n\in\mathbb{N}}\cup\{y_0\})## s.t. ##g_n(x)=|f(x)-f(a_n)|##. Suppose there is a...

Hi, I have to find the real and imaginary parts and then using Cauchy Riemann calculate ##\frac{df}{dz}## First, ##\frac{df}{dz} = \frac{1}{(1+z)^2}## Then, ##f(z)= \frac{1}{1+z} = \frac{1}{1+ x +iy} => \frac{1+x}{(1+x)^2 +y^2} - \frac{-iy}{(1+x^2) + y^2}## thus, ##\frac{df}{dz} =...

Hello everyone, in equation 3.86 of this online version of Carroll´s lecture notes on general relativity (https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll3.html) the covariant derviative of the Riemann tensor is simply given by the partial derivative, the terms carrying the...

Everyone who is currently studying GR must be familiar with this picture. We find Riemann curvature by paraller transport a "test vector" around and see whether the vector changes its direction. My question. How does it work with one dimensional Ring? A geomteric ring is intuitively curved but...

To calculate the Riemann coefficient for a metric ##g##, one can employ the second Cartan's structure equation: $$\frac{1}{2} \Omega_{ab} (\theta^a \wedge \theta^b) = -\frac{1}{4} R_{ijkl} (dx^i \wedge dx^j)(dx^k \wedge dx^l)$$ and using the tetrad formalism to compute the coefficients of the...

We know in Lorentzian signature spacetime, in the case of timelike or spacelike hypersurfaces ##\Sigma## with \begin{align} n^\alpha n_\alpha=\epsilon=\pm1 \end{align} where ##\epsilon=1## for timelike and ##-1## for spacelike. We can define a tensor ## h_{\alpha\beta}## on ##\Sigma## by...

(a) I imagine there are several rectangles to represent the area under graph of p vs t then I try to make equation for the total area. Since the question asks about time when the container holds 22 fewer liters than it does at time t = 9, I think the total area of rectangles starting from t = b...

Hello! (Wave) I am looking at the Riemann integral and I have two questions. Theorem: Let $f: [a,b] \to \mathbb{R}$ bounded and $c \in (a,b)$. Then $f$ is integrable in $[a,b]$ iff it is integrable in $[a,c]$ and in $[c,b]$. In this case we have $\int_a^b f=\int_a^c f + \int_c^b f$. At the...

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So Riemann integrals on ℚ can be <> 0 but Lebesgue integrals on ℚ all have measure zero?

In terms of diff geo it seems like an obvious fact, that a manifold can be equipped with quite a variety of different Riemann metrics. But when it comes to physics (relativity theory in particular) it seems there is a very specific metric singled out. Now i do not entirely understand the...

Can someone please explain why the formula for midpoint approximation looks like the equation above instead of something like $$M_n=(f(\frac{x_0+x_1}2)+f(\frac{x_1+x_2}2)+\cdots+f(\frac{x_{n-1}+x_n}2))\frac{b-a}n$$? Thanks in advance!

Hello everyone, in a solution to my measure theory assignment, I have seen the equation $$ \int_{\mathbb{R}}^{} \frac {1}{|x|}\, d\lambda(x)=\infty $$ with ##\lambda## as the 1⁻dim Lebesgue measure. I was wondering how that integral was evaluated as we had never proven any theorem that states...

I have worked out (and then verified against some sources) that ##R^\theta_{\phi\theta\phi} = sin^2(\theta)##. The rest of the components are either zero or the same as ##R^\theta_{\phi\theta\phi} ## some with the sign flipped. I was surprised at this, because it implies that the curvature...

YouTube has been suggesting videos about category theory of late, and I have spent some time skimming through them, without really understanding where it's all going. A question came to mind, namely: It seems reasonably conceivable that group theory could perhaps supply a vital key to the...

This is the Riemann Zeta function ##Z(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}##. It equals 0 only at the negative integers on the real axis and numbers of form ##1/2+x i##. The series can be expanded to this: $$\sum_{n=1}^{\infty} \frac{1}{n^s} = \sum_{n=1}^{\infty} \frac{1}{n^{1/2 + xi}} =...

Hello everyone, I have to find an interval of this Riemann integral. Does anybody know the easiest way how to do it? I think we need to do something with denominator, enlarge it somehow. My another guess is the integral is always larger than 0 (A=0) because the whole function is still larger...

I saw briefly that the Riemann tensor can be obtained via Stoke's theorem and parallel transport along a closed curve. If one does add winding number then it can give several results, does it imply that this tensor is multivalued ?

I tested the first function with the Cauchy Riemann equations and it seemed to fail that test, so I don't believe that function is analytic. However, I'm really not sure how to show that it is or is not analytic using the definition of the complex derivative.

Could someone please write out or post a link to the Riemann Tensor written out solely in terms of the metric and its first and second derivatives--i.e. with the Christoffel symbol gammas and their first derivatives not explicitly appearing in the formula. Thanks.

I want to compute the Riemann Tensor of the following metric $$ds^2 = dr^2+(r^2+b^2)d \theta^2 +(r^2+b^2)\sin^2 \theta d \phi^2 -dt^2$$ Before going through it I'd like to try to predict how many non-trivial components we'd expect to get, based on the Riemann tensor basic rule: It is...

import numpy as np def num_int(f,a,b,n): dx=(b-a)/n x=np.arange(a,b,step=dx) y=f(x) return y.sum()*dx def rational_func(x): return 1/(1+x**2) print(num_int(rational_func,2,5,10)) Here is my code for the left endpoint, I know this code works because I compared it to an...