Riemann Definition and 586 Threads

I'm not sure if this is the right place to post this in, but I'm trying to recreate the "Deformation of water by a magnetic field" experiment by Chen et al. The PDF version of the paper can be accessed via Google (for some reason it won't let me provide a direct link). On the 2nd page of the...

Homework Statement Here is a link to the problem which I am currently working on: http://math.umn.edu/~jara0025/Math4603/Math4603H9Answers.pdf Homework EquationsThe Attempt at a Solution [/B] The specific problem I am working on is found on page 3, and is the first problem on that page. The...

Hi brand new to the site. I keep on having a syntax error when I run the code below on my casio fx-cg10. Btw I also put a display triangle on the last m as well

Hi It's just that last step I'm not getting, so you have: [1 / Kz] - [1 / (2K)z] = [ (2K)z - Kz ] / [(2K)z * Kz] = [ (2)z - 1 ] / [(2K)z*] Then what? Thanks

The question provides a table of values for time and velocity. Part c of the question asks to use a Riemann sum to approximate (not specifying which one). Part d asks what the answer to part c represents and to explain my reasoning. The answer that I got for the sum is 58.5 feet, but I do not...

Hi I was wondering if there any observations that have only been described using the Riemann Zeta function? What would it mean in physics to assign a divergent series a finite value? Thank you Edit Sorry I overlooked a thread just posted that asked about this so this might need to be deleted.

Hi, I am currently teaching myself complex analysis (using Stein and Shakarchi) and wondered if someone can guide me with this: Find all the complex numbers z∈ C such that f(z)=z cos (z ̅). [z ̅ is z-bar, the complex conjugate). Thanks!

I am reading Manfred Stoll's book: Introduction to Real Analysis. I need help with Exercise 2(a) from Stoll's Exercises 6.2 on page 229 ... Exercise 2 reads as follows: I was somewhat puzzled about how to do this exercise ... BUT ... even more puzzled when I read Stoll's hint for solving the...

I am reading Manfred Stoll's book: Introduction to Real Analysis. I need help with Stoll's proof of Riemann's Criterion for Integrability - Stoll: Theorem 6.17 Stoll's statement of this theorem and its proof reads as follows: https://www.physicsforums.com/attachments/3941In the above proof we...

Homework Statement Prove or give a counter example of the following statement: If f: [a,b] \to [c,d] is linear and g:[c,d] \to \mathbb{R} is Riemann integrable then g \circ f is Riemann integrable Homework EquationsThe Attempt at a Solution I'm going to attempt to prove the statement is...

We have ##R^{1}_{212}## as the single independent Riemann tensor component, and I'm after ##R##. From symmetry properties and contracting we can attain the other non-zero components. The solution then states that ##R_{11}=R^{1}_{111} + R^{2}_{121}=R^{2}_{121}## . I thought it would have been...

Hello everyone, I'm studying Weinberg's 'Gravitation and Cosmology'. In particular, in the 'Curvature' chapter it says that the Riemann tensor cannot depend on ##g_{\mu\nu}## and its first derivatives only since: What I don't understand is how introducing the second derivatives should change...

I understand how to calculate values of positive values ζ(s), it's pretty straightforward convergence. But when you expand s into the complex plane, like ζ(δ+bi), how do you assign a value with i as an exponent? Take for example ζ(1/2 + i) This is the sequence 1/1^(1/2+i) + 1/2^(1/2+i) +...

Homework Statement (Self study.) Several sources give the following for the Riemann Curvature Tensor: The above is from Wikipedia. My question is what is \nabla_{[u,v]} ? Homework Equations [A,B] as general purpose commutator: AB-BA (where A & B are, possibly, non-commutative operators)...

How do we explains space-time through Riemann Calculus?

I have come about few mathematical problems related to Riemann Tensor analysis while learning General Relativity. Should I ask these questions in this section or in the homework section. They are pretty hard!

Hello, This question is purely inspired by: http://mathhelpboards.com/calculus-10/evaluating-infinite-sum-e-x-using-integrals-12838.html My other question. Anyhow, How do you find the integral for a given specific Riemann sum. Suppose the same one given in the link; $= \displaystyle...

Does anyone remember/know what the lowest co-efficient is of the imaginary part of the exponent for infinite Riemann zeta sums? I think it's (9/2)*pi, but I'm not sure.

Task in real analysis: P is a uniform partition on [0, π] and is divided into 6 equal subintervals. Show that the lower and upper riemann sums of sin (x) over P is lesser than 1.5 and larger than 2.4 respectively. My attempt at the solution: The greates value and the least value of sin x over...

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...

I am looking for a introductiory book on Riemann surfaces in context of bosonic String theory, or heterotic String theory for that matter. Prices should be affordable but should not matter, of I nead guide books this also does not matter...Help is seriusly apreciated.

I plotted the real and imaginary parts of a complex function \(z^{1/3}\). The two plots are similar to the Riemann surface is that correct?

Homework Statement Let ##f:[a,b] \rightarrow R## be a differentiable function. Show that if ##P = \{ x_0 , x_1 , ... , x_n \}## is a partition of ##[a,b]## then $$L(P,f')=\sum_{j=1}^n m_j \Delta x_j \leq f(b) - f(a)$$ where ##m_j=inf \{ f'(t) : t \in [x_{j-1} , x_j ] \}## and ##\Delta x_j =...

Homework Statement Integrate √(25 - x^2) dx from 0 to 5 using an infinite Riemann Sum Homework Equations lim n→∞ Σ_(i=1)^n i = n(n+1)/2 lim n→∞ Σ_(i=1)^n i^2 = n(n+1)(2n+1)/6 The Attempt at a Solution Δx = (b - a)/n Δx = (5 - 0)/n Δx = 5/n f(x_i) = √(25 - [a + iΔx]^2) f(x_i) = √(25 - [0 +...

Hi All, I was reading through Kreyzeig's Advanced Engineering Mathematics and came across two theorems in Complex Analysis. Theorem 1: Let f(z) = u(x,y) + iv(x,y) be defined and continuous in some neighborhood of a point z = x+iy and differentiable at z itself. Then, at that point, the...

I am trying to expand $$\varepsilon^{{abcd}} R_{{abcd}}$$ by using four identities of the Riemann curvature tensor: Symmetry $$R_{{abcd}} = R_{{cdab}}$$ Antisymmetry first pair of indicies $$R_{{abcd}} = - R_{{bacd}}$$ Antisymmetry last pair of indicies $$R_{{abcd}} = - R_{{abdc}}$$...

Hello Normally in order to change the order of limit and integration in rimann integration, you need uniform convergence. But let's say that you are not able to prove uniform convergence, but only pointwise convergence. And let's say you are able to prove that the functions are also...

Homework Statement I found the function V, which is the conjugate harmonic function for U(x,y)=sin(x)cosh(y). I am attaching my work. It turns out to be a two-term function with trig factors. I am then to write F(Z) in terms of Z, but is plugging in x, and y, in terms of Z into my trig...

We know how objects such as the metric tensor and the Cristoffel symbol have symmetry to them (which is why g12 = g21 or \Gamma112 = \Gamma121) Well I was wondering if the Riemann tensor Rabmv had any such symmetry. Are there any two or more particular indices that I could interchange and...

Draft re Ricci vs Riemann tensors This one is really just the beginning of a musing. I can't even remember if I came to any conclusion or just forgot about it. I started a thread in Jan 2014, a couple of months after this blog post, on the related issue of what the physical significance of...

Throughout this note, I'll give a brief, explanatory, informal introduction to the Riemann Hypothesis (RH) explaining the statement of the conjecture, the difficulties of approaching it as well as some notable consequences of RH in the field of number theory. I hope the readers will enjoy the...

The Riemann curvature of a unit sphere is sine-squared theta, where theta is the usual azimuthal angle in spherical co-ordinates, and this is shown in many textbooks. But since a sphere is completely specified by its radius, then as far as I can see its curvature should be a function of its...

Hey guys, I'd appreciate some help for this problem set I'm working on currently The u-substitution for the first one is somewhat tricky. I ended up getting 1/40(u)^5/2 - 2 (u) ^3/2 +C, which I'm not too sure about. I took u from radical 3+2x^4. For the second question, I split the integral...

Dear All, I am trying to calculate the Riemann tensor for a 4D sphere. In D'inverno's book, I have this equation R^{a}_{bcd}=\partial_{c}\Gamma^{a}_{bd}-\partial_{d}\Gamma^{a}_{bc}+\Gamma^{e}_{bd}\Gamma^{a}_{ec}-\Gamma^{e}_{bc}\Gamma^{a}_{ed} But the exercise asks me to calculate R_{abcd}. Do...

After my studies of metric tensors and Cristoffel symbols, I decided to move on to the Riemann tensor and the Ricci curvature tensor. Now I noticed that the Einstein Field Equations contain the Ricci curvature tensor (R\mu\nu). Some sources say that you can derive this tensor by simply...

How could I find the lim as n-> infinity of the expression I attached? The only way I could find was to express it in terms of a definite integral. Integral of xe^(-2x) from 0 to 1. What is the other way?

On the spacetime manifold in general relativity, one chooses a basis at a point and express it by the partial derivatives with respect to the four coordinates in the coordinate system. And then the basis vectors in the dual space will be the differentials of the coordinates. Why do one do that...

Let f:[0,1]-> R be bounded on [0,1] and continuous on (0,1). Prove that f is Riemann integrable on [0,1]. Hint: Show that for any epsilon > 0 there exists a partition P of [0,1] such that U(f,P)-L(f,P) < epsilon. So Let P = {0 = t0 < t1 < ... < tn = 1} Since its bounded on [0,1] (|f(x)| <= S)...

Hello again! (Blush) Let $f:[a,b] \to \mathbb{R}$ bounded and $c \in (a,b)$.Then $f$ is integrable at $[a,b]$ iff $f$ is integrable at $[a,c]$ and $[c,b]$.In this case,we have $\int_a^b f = \int_a^c f + \int_c^b f$. The proof for the direction $\Rightarrow$ is like that: Suppose that $f$ is...

Hello! (Wave) I am looking at the proof that if $f$ is integrable and $k \in \mathbb{R}$,then $kf$ is also integrable and $\int_a^b{(kf)}=k \int_a^b{f}$. The following identity is used at my textbook: $$U(kf,P)=\left\{\begin{matrix} k \cdot U(f,P), \text{ if } k>0\\ k \cdot L(f,P), \text{ if...

What's the difference between Euclidean and Riemann space? As far as I know ##\mathbb{R}^n## is Euclidean space.

So my textbook asks to show \int^{3}_{1} x^{2}dx = \frac{26}{3}. They let the partition P = {x_{0},...,x_{n}}, and define the upper Riemann sum as U(P) = \sum^{i=1}_{n} x_{i}Δx_{i} and lower sum as L(P) = \sum^{i=1}_{n} x_{i-1}Δx_{i} I understand this part, but the next part is where I'm...

What do all possible combinations of the pochhammer contour over the normal Riemann surface for the function ##w=z^{1/2}(1-z)^{1/3}## look like? I imagine like a pumpkin with six ridges longitudinally from north pole to south, one for each joining along the cut between zero and one, the contour...

So the question is Evaluate (x-2)dx as the integral goes from -2 to 2 using the definition of a definite integral, choosing your sample points to be the right endpoints of the subintervals… Ok, so i understand how to do this problem if it gave me an actual number of interval like n=6 but it...

The Riemann hypothesis states, whenever the Riemann zeta function hits 0, the real part of the input must be 0.5. Does any input with real part being 0.5 make the function hit 0? Also, assuming the hypothesis is true, would it suffice to prove that if the input's real part is not 0.5, then the...

The Euler-Maclaurin summation formula and the Riemann zeta function The Euler-Maclaurin summation formula states that if $f(x)$ has $(2p+1)$ continuous derivatives on the interval $[m,n]$ (where $m$ and $n$ are natural numbers), then $$ \sum_{k=m}^{n-1} f(k) = \int_{m}^{n} f(x) \ dx -...

Here is the question: I have posted a link there to this topic so the OP can see my work.

Homework Statement https://scontent-b-mia.xx.fbcdn.net/hphotos-prn2/v/1456973_10201043975243279_1765184125_n.jpg?oh=05b39611ad70d28d837ed219e1b0f2aa&oe=52838593 Homework Equations The area can be approximated by using the sum of the areas of the rectangles. Area of rectangle = change...

Homework Statement I wish to prove that for s>1 $$ \sum\limits_{n=1}^{\infty}\frac{\mu(n)}{n^s}=\prod_{p}(1-p^{-s})=\frac{1}{\zeta(s)}. $$ The Attempt at a Solution (1) I first showed that $$ \prod_{p}(1-p^{-s})=\frac{1}{\zeta(s)}. $$ It was a given theorem in the text that $$...

Recently some interesting material about the Riemann Zeta Function appeared on MHB and I also contributed in the post... http://mathhelpboards.com/challenge-questions-puzzles-28/simplifying-quotient-7235.html#post33008 ... where has been obtained the expression... $\displaystyle \zeta (s) =...