What is Riemann: Definition and 613 Discussions

Georg Friedrich Bernhard Riemann (German: [ˈɡeːɔʁk ˈfʁiːdʁɪç ˈbɛʁnhaʁt ˈʁiːman] (listen); 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry.
In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis.
His famous 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as one of the most influential papers in analytic number theory.
Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity. He is considered by many to be one of the greatest mathematicians of all time.

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  1. Math Amateur

    MHB The Riemann Integral .... Conway, Proposition 3.1.4 ....

    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...
  2. karush

    MHB 2.7.3 AP calculus Exam Riemann sum

    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.
  3. Math Amateur

    MHB Understanding Theorem 3.4: Mean Value Theorem & Cauchy Riemann Equations

    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...
  4. The Bill

    Geometry Source for a coordinate-based formula for Gauss from Riemann

    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...
  5. E

    How can I use the covariant derivative to derive the Riemann curvature tensor?

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

    I Derivation of Riemann tensor's first order-equation

    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...
  7. Vick

    Coding Riemann (0,4) Equation in Python - Help Needed!

    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
  8. PainterGuy

    MATLAB Approximating the inverse FT of a unit pulse using a Riemann sum

    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...
  9. berlinspeed

    A How Do You Derive the Riemann Component?

    Can someone please show how to write as ?
  10. jedishrfu

    B Mathematicians Revive Abandoned Approach to the Riemann Hypothesis

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

    Riemann sums for discontinuous functions

    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##...
  12. W

    Riemann Curvature Tensor in 2D

    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...
  13. S

    I What's the difference between the Riemann & Darboux Integrals?

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

    A Riemann Tensor Notation Explained | Choquet-Bruhat GR

    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...
  15. nomadreid

    I Hausdorff dimension of Riemann zeta function assuming RH

    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...
  16. V

    I Riemann Tensor: Questions & Geometric Interpretation

    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...
  17. Chromatic_Universe

    Specific proof of the Riemann tensor for FRW metric

    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 =...
  18. M

    I Riemann Integral: Does g(x)=f(x) Almost Everywhere?

    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.
  19. P

    Calculus 2 for Engineers: Riemann sums

    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...
  20. T

    MHB Definite integral with Riemann sums

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  21. Spinnor

    B Get up to speed with the Riemann Hypothesis

    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...
  22. N

    A Has the Riemann hypothesis been proven?

    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...
  23. A

    I Learning the theory of the n-dimensional Riemann integral

    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...
  24. A

    MHB Reference request for self-studying multiple Riemann integrals

    As the title says, I would like to self-study multivariable real analysis (integration, specifically; the Riemann integral) and I need some recommendations (resources, books, videos, ...). I'm from Croatia and got my hands on some Croatian notes about multivariable real analysis so if some of...
  25. Ken Gallock

    I Gaussian normal coordinates and Riemann normal coordinates

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

    How to calculate the Riemann curvature at r=2GM?

    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...
  27. C

    I Riemann Tensor knowing Christoffel symbols (check my result)

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  28. M

    MHB Volume Problem - Riemann Sum

    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'
  29. W

    Proving the symmetry property of Riemann curvature tensor

    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...
  30. t_r_theta_phi

    I Intuitive explanation for Riemann tensor definition

    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...
  31. Z

    I Components of Riemann Tensor: 4 Indexes, 16x16 Matrix

    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} &...
  32. T

    A simple case of translation invariance of Riemann integrals

    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...
  33. C

    I Riemann curvature tensor derivation

    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...
  34. MountEvariste

    MHB Sum of powers limit via Riemann sums?

    One of the many excellent problems by lfdahl in the challenge questions and puzzles subforum was recently: https://mathhelpboards.com/challenge-questions-puzzles-28/prove-limit-23480.html My first idea was Riemann sums! I didn't succeed. So I ask, can this limit be calculated via Riemann...
  35. nightingale123

    I N-th dimensional Riemann integral

    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...
  36. Z

    MHB Solution to the Riemann Hypothesis in plain English

    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...
  37. S

    Finding Riemann Integrability for f(x) on [0,1]

    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...
  38. P

    I Non-zero components of Riemann curvature tensor with Schwarzschild metric

    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...
  39. Math Amateur

    MHB Cauchy Riemann Equations - Proof of Sufficiency .... Conway Theorem 2.29 .... Proof ....

    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 ...
  40. Math Amateur

    MHB Riemann Integration - The Additivity Theorem .... B&S Theorem 7.2.9 ....

    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...
  41. D

    Calculating the Limit Using Riemann Sum with Starred Part?

    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
  42. D

    Find the limit using Riemann sum

    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...
  43. P

    MHB Dale's questions via Facebook about Riemann Sums

    (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...
  44. binbagsss

    Zeros of Riemann zeta function, functional equation and Euler product

    Homework Statement Question Use the functional equation to show that for : a) ##k \in Z^+ ## that ## \zeta (-2k)=0## b) Use the functional equation and the euler product to show that these are the only zeros of ##\zeta(s) ## for ##Re(s)<0## . And conclude that the other zeros are all located...
  45. S

    Mathematica How to compute the Riemann tensor using RGTC mathematica?

    I am trying to learn RGTC. It gives value of particular Riemann tensor but not all of it. What should I do?
  46. M

    B Isn't the Riemann Hypothesis just a convention?

    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?
  47. FallArk

    MHB How to calculate the lower and upper riemann sum

    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}) =...
  48. jedishrfu

    I Quantum Mechanics does Riemann Zeta Function

    "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/
  49. D

    A Paper About the Riemann Zeta Function

    What do you think of the following paper about the Riemann Zeta Function? http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.130201
  50. S

    I Number of independent components of the Riemann tensor

    I've thought of a new way (at least I never read it anywhere) of counting the independent components of the Riemann tensor, but I am not sure whether my arguments are valid, so I would like to ask whether my argument is sound or total bonkers. The Riemann tensor gives the deviation of a vector A...
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