Riemann Definition and 586 Threads

So, I know that a function is integrable on an interval [a,b] if ##U(f,P_n)-L(f,P_n)<\epsilon ## So I find ##U(f,P_n## and ##L(f,P_n## ##L(f,P_n)=5(3-\frac{1}{n}-0)+5(3+\frac{1}{n}-(3-\frac{1}{n}))+7(4-(3+\frac{1}{n}))=22-\frac{2}{n} ##...

Good day all. Since the gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. Then If we form the Gradient vector field...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.2 Existence Results ... ... I need some help in understanding the proof of Theorem 5.12 ...Theorem 5.12 and its...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.2 Existence Results ... ... I need some help in understanding the proof of Theorem 5.12 ...Theorem 5.12 and its...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.1 Riemann Sums ... ... I need some help in understanding the proof of Theorem 5.10 ...Theorem 5.10 and its proof...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.1 Riemann Sums ... ... I need some help in understanding the proof of Theorem 5.10 ...Theorem 5.10 and its proof...

I am reading John B. Conway's book: A First Course in Analysis and am focused on Chapter 3: Integration ... and in particular I am focused on Section 3.1: The Riemann Integral ... I need help with an aspect of the proof of Proposition 3.1.4 ...Proposition 3.1.4 and its proof read as follows...

ok basically t is 3 hours appart except between 7 and 12 of which I didn't know if we should intemperate. other wise it is just adding up the 4 $(t)\cdot(R(t))$s.

I am reading "Complex Analysis for Mathematics and Engineering" by John H. Mathews and Russel W. Howell (M&H) [Fifth Edition] ... ... I am focused on Section 3.2 The Cauchy Riemann Equations ... I need help in fully understanding the Proof of Theorem 3.4 ...The start of Theorem 3.4 and its...

I need a citeable source that gives the formula for the Gaussian curvature at a single point of an intrinsically defined Riemannian or Semi-Riemannian manifold given the intrinsic metric tensor and/or Riemann tensor. I've got sources for this already, but I'm not "allowed" to use them for this...

I derived this equation $$ A_{i,jk}-A_{i,kj}=R^r _{kij}A_r$$.But where do I use this $$A_{i,j}+A_{j,i}=0$$?

According to Schutz's second edition book, the equation for Riemann tensor to first order in ##h_{\mu\nu}## is: $$R_{\alpha\beta\mu\nu}= \frac{1}{2}(h_{\alpha \nu,\beta \mu}+h_{\beta\mu,\alpha\nu}-h_{\alpha\mu,\beta \nu}-h_{\beta \nu , \alpha \mu})$$ which (as stated in the book) can be derived...

Hi, I've coded Riemann tensor in python successfully. However, I recently stumbled onto another Riemann equation for the valence (0,4) as shown in the following link: Riemann (0,4) I'm having troubled coding the last part after the partial derivatives and the plus sign. Can anyone help me? Thanks

Hi, Although I'm using trigonometric form of Fourier transform, first I'd discuss both, exponential and trigonometric forms, for the sake of context. Now proceeding toward the main question and we would only be using trigonometric form. % file name...

Can someone please show how to write as ?

https://phys.org/news/2019-05-mathematicians-revive-abandoned-approach-riemann.html and more on the Reimann Hypothesis backstory from Numberphile:

The definition of the Riemann sums: https://en.wikipedia.org/wiki/Riemann_sum I'm stuck with a problem in my textbook involving upper and lower Riemann sums. The first question in the problem asks whether, given a function ##f## defined on ##[a,b]##, the upper and lower Riemann sums for ##f##...

Since in 2D the riemman curvature tensor has only one independent component, ## R = R_{ab} g^{ab} ## can be reversed to get the riemmann curvature tensor. Write ## R_{ab} = R g_{ab} ## Now ## R g_{ab} = R_{acbd} g^{cd}## Rewrite this as ## R_{acbd} = Rg_{ab} g_{cd} ## My issue is I'm not...

I was reading about this, and they seem the same. Of course, if they were the same, they wouldn't have different names.

Hello I have been going through the cosmology chapter in Choquet Bruhats GR and Einstein equations and in definition 3.1 of chapter 5 she defines the sectional curvature with a Riemann( X, Y;X, Y) (X and Y two vectors) I don't understand this notation, regarding the use of the semi colon, is it...

In several places, for example https://xxx.lanl.gov/pdf/chao-dyn/9406003v1, it is claimed that the Riemann zeta function is a fractal under the assumption of a positive result for the Riemann Hypothesis, because (1) the Voronin Universality Theorem, and (2) if the RH is true, then the zeta...

Tensor of Riemann. Geometric interpretation.The Riemann tensor gives the variation of a vector displaced parallel in a closed loop, say a small rectangle formed by geodesic sides, (δa) and δb) first, starting from a vertex A and going to another vertex in the diagonal, B; then starting from the...

Homework Statement Prove Rijkl= k/R2 * (gik gjl-gil gjk) where gik is the 3 metric for FRW universe and K =0,+1,-1, and i,j=1,2,3, that is, spatial coordinates. . Homework Equations The Christoffel symbol definition: Γμνρ = ½gμσ(∂ρgνσ+∂νgρσ-∂σgνρ) and the Riemann tensor definition: Rμνσρ =...

Let f(x) be a bounded continuous function on [0,1]. Let g(x)=f(x) on all rational points in [0,1]. Let g(x) be Riemann integrable on [0,1]. Does g(x)=f(x) almost everywhere in the interval? If so - proof? If not -counterexample.

Homework Statement a. Write down a Riemann sum for the integral ∫x3dx from 0 to 1. b. Given the following identity 13+23+33...+N3=(N(N+1)/2)2, show that the Riemann sums for ∫x3dx from 0 to 1 converge to 1/4. The Attempt at a Solution I believe I have gotten part a. I got ∑i^3/N^4 from i=0 to...

Heya, So, I know this is a pretty simple problem, but I seem stuck on it nevertheless. Here's the question Calculate the upper and lower sums , on a regular partition of the intervals, for the following integrals \begin{align*} \int_{1}^{3}(1-7x)dx \end{align*} Please correct me if I'm doing...

I found the following blog post from a Ph.D., "The Riemann Hypothesis, explained", while trying to get up to speed with the Riemann Hypothesis. It is at, https://medium.com/@JorgenVeisdal/the-riemann-hypothesis-explained-fa01c1f75d3f It seemed well written as I could understand it. Delete or...

I am very baffled. I have heard through the grapevine that the Riemann hypothesis has been proven. My first reaction was of course to dismiss it as yet another failed attempt by someone who was not careful or by a crackpot, or some type of April's fool joke made a few months late.But what I...

I would like to learn (self-study) the theory behind the n-dimensional Riemann integral (multiple Riemann integrals, not Lebesgue integral). I am from Croatia and found lecture notes which Croatian students use but they are not suitable for self-study. The notes seem to be based on the book: J...

Hi. I was wondering what is the relationship between Gaussian normal coordinates and Riemann normal coordinates. Thanks.

As the coordinate singularity at r=2GM doesn't mean a physical singularity as Riemann curvature tensor is smooth although [g][/rr] metric behaves oddly in the case of Schwarzschild solution. Do somebody tell me an authentic reference how is this value is calculated? I have references writing...

I need to find all the non-zero components of the Riemann Tensor in a two-dimensional geometry knowing that the only two non-zero components of the Christoffel symbols are: \Gamma^x_{xx}=\frac{1}{x} and \Gamma^y_{yy}=\frac{2}{y} knowing that: R^\alpha_{\beta\gamma\delta}=\partial_\gamma...

THE QUESTION By using Riemann’s sum, synthesise a mathematical model for finding the exact volume of any ‘tepee’ tent of side s and height h. HERE'S WHAT I HAVE Am currently stuck on writing a side length for the hexagon at any height 'x'

Homework Statement Hi everyone! Just wondering if there's a way to prove the symmetry property of the Riemann curvature tensor $$ R_{abcd} = R_{cdab}$$ without using the anti-symmetry property $$ R_{abcd} = -R_{bacd} = -R_{abdc} $$? I'm only able to prove it with the anti-symmetry property and...

Many sources give explanations of the Riemann tensor that involve parallel transporting a vector around a loop and finding its deviation when it returns. They then show that this same tensor can be derived by taking the commutator of second covariant derivatives. Is there a way to understand why...

Hello, Riemann tensor ##R^i_{jkl}## 4 indexes, and it should be matrix 16x16 in spacetime if we have time coirdinate - 0 and space coordinates -1,2,3. But how should I write the components to matrix? For example ##\begin{pmatrix}R^0_{000} & R^1_{000} & R^2_{000} ... \\ R^0_{100} & R^1_{100} &...

Homework Statement Show that \int_{A} 1 = \int_{T(A)} 1 given A is an arbitrary region in R^n (not necessarily a rectangle) and T is a translation in R^n. Homework Equations Normally we find Riemann integrals by creating a rectangle R that includes A and set the function to be zero when x...

Riemann tensor is defined mathematically like this: ##∇_k∇_jv_i-∇_j∇_kv_i={R^l}_{ijk}v_l## Using covariant derivative formula for covariant tensors and covariant vectors. which are ##∇_av_b=∂_av_b-{Γ^c}_{ab}v_c## ##∇_aT_{bc}=∂_av_{bc}-{Γ^d}_{ac}v_{db}-{Γ^d}_{ab}v_{dc} ##, I got these...

Hello I have a question regarding something we wrote in class today. Let ##A## be a bounded subset of ##R^n##, let ##f,g:A\to \mathbb{R}## be integrable functions on A. ##a)## if ## A## has a volume and ##\forall x \in A :m\leq f(x) \leq M## then ##mV(A)\leq \int_{A}f(x)\leq MV(A)## this...

The number line at x=1/2 is mediated by a concurrent incentive field whose shape can be extrapolated through the placement of prime numbers. Each prime number is a turning point in the n-dimensional movement of the imaginary number line, whose degree and angle can be determined through all the...

Homework Statement Find a such that f is Riemann integrable on [0,1], where: ##f = x^acos(1/x)##, x>0 and f(0) = 0 Homework EquationsThe Attempt at a Solution I found at previous points a such that f is continuous, bounded and derivable, but I am not sure how to use that (as all these...

I was working out the components of the Riemann curvature tensor using the Schwarzschild metric a while back just as an exercise (I’m not a student, and Mathematica is expensive, so I don’t have access to any computing programs that can do it for me, and now that I’m thinking about it, does...

I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ... I am currently focussed on Chapter III Elementary Properties and Examples of Analytic Functions ... Section 2: Analytic Functions ... ... I need help in fully understanding aspects of Theorem 2.29 ...

I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 7: The Riemann Integral ... I need help in fully understanding an aspect of the proof of Theorem 7.2.9 ...Theorem 7.2.9 and its proof ... ... read as follows...

Homework Statement http://i66.tinypic.com/aesd1u.png can someone explain to me how can i get the limit using riemann sum especially the starred part? i was so confused thanks! Homework Equations The Attempt at a Solution attempt at a solution in the picture

Homework Statement i want to find limit value using riemann sum \lim_{n\to\infty}\sum_{i = 1}^{2n} f(a+\frac{(b-a)k}{n})\cdot\frac{(b-a)}{n}= \int_a^b f(x)dx<br> question : <br> \lim_{h \to \infty} =\frac{1}{2n+1}+\frac{1}{2n+3}+...+\frac{1}{2n+(2n-1)}<br> Homework EquationsThe Attempt at a...

(a) Since a = 2, that means $\displaystyle \begin{align*} \Delta x = \frac{2}{5} \end{align*}$ (b) $\displaystyle \begin{align*} f \left( \frac{7\,\Delta x}{2} \right) &= f \left( \frac{7}{5} \right) \\ &= \left( \frac{7}{5} \right) ^2 \\ &= \frac{49}{25} \end{align*}$ (c) $\displaystyle...

If the zeta function intersects the critical line when the real part is 1/2, then it will intersect some other line when some other real part is used. Isn't the Riemann Hypothesis just based on a particular convention for the critical line?

I ran into some issues when trying to calculate the lower Riemann sum of $$f\left(x\right)={x}^{3}$$, $$x\in[0,1]$$ I am asked to use the standard partition $${P}_{n}$$ of $$[0,1]$$ with n equal subintervals and evaluate $$L(f,{P}_{n})$$ and $$U(f,{P}_{n})$$ What I did: $$L(f,{P}_{n}) =...

"Physicists are attempting to map the distribution of the prime numbers to the energy levels of a particular quantum system." https://www.quantamagazine.org/20170404-quantum-physicists-attack-the-riemann-hypothesis/