Riemann Definition and 618 Threads

Homework Statement The notation for a Riemann sum - Ʃ f(x*i)Δx - is very similar to the notation for the integral (the Ʃ becomes ∫, the f(x*i) becomes f(x) and the Δx becomes dx). \int f(x)dx = \lim_{n \to \infty}\sum_{k=0}^{n} f(x_i) Δx Is there a way to explicitly define the values on the...

Hi, I was wondering if someone wouldn't mind breaking down the geometrical differences between the Riemann, Ricci, and Weyl tensor. My current understanding is that the Ricci tensor describes the change in volume of a n-dimensional object in curved space from flat Euclidean space and that if we...

Homework Statement If R = [0,4]x[-1,2], use a Riemann sum with m=2, n=3 to estimate the value of ∫∫(1-xy^2)dA. Take the sample points to be the lower right corners. Homework Equations NoneThe Attempt at a Solution 2*1[f(2,-1) + f(2,0) + f(2,1) + f(4,-1) + f(4,0) + f(4,1)] = some value Just...

Hello, it is well-known that with stereographic projection we can obtain a 1-1 correspondence between the points of the 2d Cartesian plane (plus the point at infinity), and the points on the Riemann sphere. What is the geometrical construction that corresponds to a 1-1 mapping between the...

Homework Statement The reimann function (if x is rational then f(x)=1/q, if x is irrational then f(X)=0)is not a step function,then, for any essilope, construct a step function k:[0,1] ~>R s.t. ||f-k||=sup{|f(t)-k(t)|:t is in[0,1]}<essilope,can someone help me to construct such step function...

Homework Statement Do the following: (a) find an interval I and functions f, g: I → R such that f and g are both Riemann integrable, but f g is not Riemann integrable. (b) find an interval I and functions f, g: I → R such that f and g are both Riemann integrable, but g ◦ f is not...

Homework Statement I'm trying to compute the sum of the following series: S=1+\frac{1}{4}-\frac{1}{16}-\frac{1}{64}+\frac{1}{256} Homework Equations The Attempt at a Solution I'm not really sure how to begin this one. I know it probably involves Riemann's Rearrangement Theorem...

The Riemann hypothesis is arguably the most difficult and perplexing unsolved theorems in all of mathematics. There is currently a $1,000,000 prize for it's solution. It's been 153 years since it's inception in 1859 and no mathematician has ever been able to solve it, not even Bernhard Riemann...

I'm trying to find if a number is odd or not, basically if X % 2 = 1. Can this be expressed through some function? Like the sum of 1 + 2 .. +n is n(n+1) /2 Or as a Riemann sum? I'm trying to add only the odd numbers from a random set of N integers to a sum.

Homework Statement Show that sin(z) satisfies the condition. (Stated in the title) Homework Equations The Attempt at a Solution f(z) = sin (z) = sin (x + iy) = sin x cosh y + i cos x sinh y thus, u(x,y)=sin x cosh y ... v(x,y)= cos x sinh y du/dx = cos x...

this is a riemann sum question and i need help with part 2 let Sn denote the finite sum 1+2^ 3/2 +...+n^ 3/2 i) use suitable upper and lower riemann sums for the function f(x)=x^3/2 on the interval [0,100] to prove that S99<J<100 ummm i did this and found 40000<J<41000 II) hence, or...

*SOLVED*Riemann Sum Question *SOLVED* My question is quite simple. I probably just missed something somewhere. I've looked for hours and cannot find the mistake. Homework Statement Find the area under the curve using the definition of an integral and Gauss summation equations: f(x) = 3 -...

Can somebody explain in layman's terms the connection between local flat space (tangent planes) on a manifold and the Riemann tensor.

I have learned that integral is the Riemann sum of infinite rectangle, that: Ʃ^{n=1}_{∞}f(xi)Δxi = ∫^{b}_{a}f(x)dx However, I think that (a,b) is the continuous interval, so the number of rectangle should be c instead of \aleph0 (cardinality of natural number N). So I wonder whether there are...

Homework Statement let f(x)=x^2 Calculate upper sum and lower sum on the interval [-2,2] when n=2The Attempt at a Solution since n=2 I divide the interval into [-2,0]\cup[0,2] then on the interval [-2,0] the function f(x)=x^2 has the highest valute at x=-2, f(-2)=4=M_{0} and the lowest value...

Homework Statement \int_0^2 x^2 \, dx using true definition involving Riemann Sums (w/o fundamental theorem). Homework Equations I don't know what the relevant equations may be, perhaps some type of lim\sum f(x)(x_{j}-x_{j-1}) The Attempt at a Solution No attempt. Just seeking the...

In the proof of the the Cauchy-Riemann's conditions we have and equality between differentials of the same function (f(z)) by x(real part) and by iy(imaginary part?). Why do we "say" that both differentials should be equal when it's normally possible to have different differentials according...

Riemann "sphere" in infinite dimensional space? I was just reading about the Riemann sphere, in 3-space and find it very interesting. With it you assign a point on the sphere to every location in the xy-plane. Then I thought, in 4-space you would be able to assign every point in 3-space to a...

I got trouble to understand the cyclic sum identity (the first Bianchi identity) of the Riemann curvature tensor: {R^\alpha}_{[ \beta \gamma \delta ]}=0 or equivalently, {R^\alpha}_{\beta \gamma \delta}+{R^\alpha}_{\gamma \delta \beta}+{R^\alpha}_{\delta \beta \gamma}=0. I can understand the...

Homework Statement f:[0,1]→R where f(x)= 0 if x=0 and f(x)=1/n when 1/(1+n) < x ≤ 1/n, n \in N. is f Riemann integrable Homework Equations R integrable only when L(f) =U(f) L(f) = largest element of the set of lower sums for n partitions U(f) = least element of the set of upper...

Homework Statement Homework Equations The Attempt at a Solution This is what I got so far. I haven't seen any similar problems like this, and this is my first attempt here. I wonder if I did it right. Can anyone check this for me? If you need additional explains or...

Hi all, I encounter a technical problem about tensor calculation when studying general relativity. I think it should be proper to post it here. Riemann curvature tensor has Bianchi identity: R^\alpha_{[\beta\gamma\delta;\epsilon]}=0 Now given double (Hodge)dual of Riemann tensor: G = *R*, in...

Hi all, I'm now reading Chap 11 of Gravitation by Wheeler, etc. In exercise 11.7, by introducing Jacobi curvature tensor, which contains exactly the same information content as Riemann curvature tensor, we are asked to show that we can actually measure ALL components of Jacobi curvature tensor...

I am currently reading about riemann sums and several different sources uses these abbreviations, inf and sup, and I am not certain what they mean. Could someone explain them to me?

I once had a math professor who said that although they might not admit it, most mathematicians try to solve the Riemann hypothesis in their spare time. How true is this? Who is trying to solve the Riemann hypothesis? I don't just mean "are you?" (although feel free to speak up if you are)...

So Euler derived the analytic expression for the even integers of the Riemann Zeta Function. I was wondering if there is a link to his derivation somewhere? Also, is there anyone else who used a different method to get the same answer as Euler? Thank you

I am confused on what is meant by the modulus of a Riemann surface. I have done some very quick searches through my books but haven't found anything. For context I will paraphrase a portion of Penrose's road to reality that I'm reading. In the complex plane construct a parallelogram by...

I can't understand how to recover a Riemann surface from the branch data. In particular, given a group acting on the Riemann sphere with some points removed, how can I construct a Riemann surface?

The plot below is a horn torus. Is that a valid genus-1 normal Riemann surface? I believe it is but I'm just a novice. I'm unsure about the single point in the center and if it "technically" still has a hole in it. Maybe I need to review that.

Given the function: w=\sqrt[3]{(z-5)(z+5)} which is fully-ramified at both the finite singular points and at infinity, how does one create the normal Riemann surface for this function? It's a torus but I do not understand how to map a triple covering onto the torus so that it's fully-ramifed...

Hi all, I am working through Visser's notes http://msor.victoria.ac.nz/twiki/pub/Courses/MATH465_2012T1/WebHome/notes-464-2011.pdf section 3.5 onward. I am trying to differentiate between the torsion and the Riemann curvature tensor in a heuristic manner. It appears from "Geometric...

Homework Statement We know \sum_{n=1}^{\infty}\frac{1}{n^x} is uniformly convergent on the interval x\in(1,\infty) and that its sum is called \zeta(x). Proof that \zeta(x) \rightarrow \infty as x \rightarrow 1^+. Homework Equations We cannot find the formula that \zeta is given...

Homework Statement Calculate the Riemann curvature for the metric: ds2 = -(1+gx)2dt2+dx2+dy2+dz2 showing spacetime is flat Homework Equations Riemann curvature eqn: \Gammaαβγδ=(∂\Gammaαβδ)/∂xγ)-(∂\Gammaαβγ)/∂xδ)+(\Gammaαγε)(Rεβδ)-(\Gammaαδε)(\Gammaεβγ) The Attempt at a Solution...

The work I have been following has me very confused... and I am almost sure I am making a mistake somewhere! After working up to this equation: \delta V = dX^{\mu}\delta X^{\nu} [\nabla_{\mu} \nabla_{\nu}]V I am asked to calculate the curvature tensor. Now the way I did it, turned out...

Homework Statement Consider a 2D spacetime where space is a circle of radius R and time has the usual description as a line. Thus spacetime can be pictured as a cylinder of radius R with time running vertically. Take the metric of this spacetime to be ds^{2}=-dt^{2}+R^{2}d\phi^{2} in the...

1. Homework Statement Write an m.file that will integrate a function f(x, y) on any given rectangle (a,b)\times(c,d) and returns the value of the integral from a to b and c to d of the function f(x,y) . Include error-catching code in the case that the integral diverges. The program...

Homework Statement χ is the Riemann Surface defined by P(w, z) = 0, where P is a complex polynomial of two variables of degree 2 in w and of degree 4 in z, with no mixed products. Find the fundamental group of χ.Homework Equations A variation of the Riemann-Hurwitz Formula states that if χ is...

Homework Statement Let f:[a, b]\rightarrow[m, M] be a Riemann integrable function and let \phi:[m, M]\rightarrowR be a continuously differentable function such that \phi'(t) \geq0 \forallt (i.e. \phi is monotone increasing). Using only Reimann lemma, show that the composition \phi\circf...

Question: A solid has a rectangular base that lies in the first quadrant and is bounded by the x and y-axes and the lines x=2, y=1. The height of the solid above point (x,y) is 1+3x. Find the Riemann approximation of the solid. Solution: I already know that the solution is \sum_{i=1}^{n}...

Hi, Can someone here help me understand how to illustrate maps of analytically-continuous paths over algebraic functions onto their normal Riemann surfaces? For example, consider w=\sqrt{(z-5)(z+5)} and it's normal Riemann surfaces which is a double covering of the complex plane onto a single...

Prove the following: If f is Riemann integrable on an interval [a,b], show that ∀ε>0, there are a pair of step functions L(x)≤f(x)≤U(x) s.t. ∫_a^b▒(U(x)-L(x))dx<ε My proof: Since f is Riemann integrable on [a,b] then, by Theorem 8.16, ∀ε>0, there is at least one partition π of the interval...

Homework Statement Find limn->∞ (1/n)(Ʃk=1 to n ln(2n/(n+k))) Homework Equations The Attempt at a Solution I'm not sure if this is even a riemann sum at all, but I don't see what else it could be. I wanted to find the riemann portion first to get rid of the sigma notation then find the...

I am reading a recent (2003) paper, "Fatou and Julia Sets of Quadratic Polynomials" by Jerelyn T. Watanabe. A superattracting fixed point is a fixed point where the derivative is zero. The polynomial P(z) = z2 has fixed points P(0) = 0 and P(∞) = ∞ (note we are working in \hat{\mathbb{C}} =...

Homework Statement Suppose f:[a,b] → ℜ is bounded and for each ε > 0, ∃ a partition P such that for any refinements Q1 and Q2 of P, regardless of how marked ⎟S(Q1,f) – S(Q2,f)⎟ < ε. Prove that f is integrable on [a,b]. Homework Equations If P and Q1 and Q2 are partitions of [a, b], with...

Happy April fool's day! (Rofl)

Homework Statement I've seen two methods that prove the integral test for convergence, but I fear they contradict each other. Each method uses an improper integral where the function f(x) is positive, decreasing, and continuous and f(x) = an. What confuses me is one method starts off the...

The problem states: Decide if the following function is integrable on [-1, 1] f(x)=\left\{{sin(\frac1{x^2})\;\text{if}\;x\in[-1,\;0)\cup(0,\;1]\atop a\;\text{if}\;x=0} where a is the grade, from 1 to 10, you want to give the lecturer in this course What I don't understand is how to find L(f...

In Felix's Klein's pamphlet, "On Riemann's Theory of Algebraic Functions and their Integrals" he describes ways to construct divergence free irrotational flows on a compact Riemann surfaces such as the torus. One method is simply to cover the surface with a conducting material and place two...

Homework Statement to prove that a function bounded on [a,b] with finite discontinuities is Riemann integrable on [a,b] Homework Equations if f is R-integrable on [a,b], then \forall \epsilon > 0 \exists a partition P of [a,b] such that U(P,f)-L(P,f)<\epsilon The Attempt at a...

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?