What is Riemman: Definition and 15 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. 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...
  2. JorgeM

    I Does the Integral of Riemman Zeta Function have a meaning?

    I have been trying to use numerical methods with this function but now I realize that I if I could suggest a Polynomial in theory, I could get some value for the Integral at least in any interval. In general, does the Integral of the Riemman dseta function has a meaning by itself?
  3. GaussianSurface

    How can I find this displacement?

    Homework Statement Your task is to estimate how far an object traveled during the time interval 0≤t≤8, but you only have the following data about the velocity of the object. *First image You decide to use a left endpoint Riemann sum to estimate the total displacement. So, you pick up a blue...
  4. T

    Find the Riemannian curvature tensor component

    Given the metric of the gravitational field of a central gravitational body: ds2 = -ev(r)dt2 + eμ(r)dr2 + r2 (dθ2 + sin2θdΦ2) And the Chritofell connection components: Find the Riemannian curvature tensor component R0110 (which is non-zero). I believe the answer uses the Ricci tensor...
  5. P

    B Gaussian Curvature and Riemmanian Geometry

    Please bear with me because I'm only in Pre-calculus and am taking basic high school physics. This is completely outside of my realm but curiosity has taken the better of me. I just learned last week about the difference between Euclidean Geometry and Riemmanian Geometry (from another thread...
  6. P

    Riemann on Deductive vs Creative Reasoning

    I understand that Riemann was very shy, so he didn't talk much. Something that he said was: 'If only I had the theorems! Then I should find the proofs easily enough' . What do you think meant by that? I suspect he was comparing deductive reasoning (proofs) with imagination and the 'seeing over...
  7. O

    Does an explicit list of 20 independent compenents of Riemman exist?

    Hello all, I have been given a problem where I am asked to calculate "all" the components of the Riemann tensor in a gross non-diagonalized metric. I know there exists at most 20 independent components of Riemann, but I want to actually compose a list of these combinations. It is easy...
  8. T

    Using a Riemman Sum to find the area under a curve (n intervals, left endpoint)

    Hello, I'm having a bit of trouble calculating the area under the curve of x^2 on the interval x=-3 to x=1. The question says that there I have to use n subintervals and left endpoints. Relevant Equations Δx=b-a/n xi=a+(Δx)(i-1) -its i-1 because we're using a left endpoint, otherwise...
  9. A

    Riemann Sums: Understanding Expression of Angular Coordinate Theta

    there is correct the expresion \int^{-\pi+\epsilon}_{\pi-\epsilon} d\theta...where \theta is a angular coordinate between (-\pi,\pi)...¿what means this?... i believe that this mean that the angular coordinate theta runs from \pi-\epsilon to -\pi+\epsilon in the sense anti clock (figure)
  10. V

    Does the Riemman tensor and the covariant derivate commute?

    A random question - Does the Riemman tensor and the covariant derivate commute? a yes/no answer would suffice, but any explanation would be welcome:) From the equations, it looks as though they do for flat metrics - but if we have other manifolds, it seems to me that the Christoffel...
  11. A

    Riemman Integral: Path of \theta for \int^{\pi-\epsilon}_{-\pi+\epsilon}d\theta

    If the integral is \int^{\pi-\epsilon}_{-\pi+\epsilon}d\theta. where \theta is a angular coordinate. In the riemman integral , i don't understand if tetha follows the path grenn in figure 1, or \theta follows the path red in figure 2.
  12. T

    Riemman-Stieltjes Integral: Proving Supremum Property

    Let α>0, J:=[-a,a] and f:J→ℝ a bounded function. Let α an increases monotonically on J and P* the set of all partitions P of J containing 0 and such that are symmetric,i.e, x in P iff -x in P. Prove that ∫fdα= sup L(P,f,α) with P in P*
  13. L

    Understanding Riemann Sums and Limits | Homework #16 Question

    Homework Statement Question regarding #16 Homework Equations Riemman Sum The Attempt at a Solution I know that the limit of the Riemman Sum is basically the integral. However, I do not know where to go from there. Do I need to use the Summation formulas? Thanks
  14. M

    Solving Riemann Sums: a=3,b=8 & a=5,b=10, What is f(x) & g(x)?

    The following sum Sqrt(5+5/n) * (5/n) + Sqrt(5 +10/n) * (5/n)... is a right Riemann sum for the definite integral a=3 and b= 8 what does f(x) equal? I got a and b but could not find f(x) It is also a Riemann sum for the definite integral Sqrt(5+5/n) * (5/n) (same as above)...
  15. M

    Solving Riemann Sum: Velocity Function v(t) = t^2 -5t + 6

    The velocity function is v(t)=t^2 -5t + 6 for a particle moving along a line. Find the displacement traveled by the particle during the time interval [0,5]. What is the displacement? What is the distance traveled? I think that the information should look like this: (1)(1^2 -5(1)...
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