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.
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...
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?
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...
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...
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...
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...
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...
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...
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)
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...
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.
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*
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
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)...
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)...