Curvature Definition and 872 Threads

Hi guys, The terms above (asperity density and asperity radius of curvature) have confused me for quite a while. I've no clue what they are. Could anyone give me a hand? And is there any relation between them and the summit radius & area per summit? Thanks! CC

I am looking for some intuition into a way of looking at the Gauss curvature on a surface that describes it as the divergence of a potential function - at least locally. I am not sure exactly what the intuition is - but this way of looking at things seems suggestive. Any insight is welcome. In...

Hi there. This is my first posting to this forum, and in fact to any forum in many years, so please excuse me if I have not followed the rules correctly or chosen the correct forum. I have been watching a lot of Neil DeGrasse Tyson-related videos and it has lead me to a line of thinking...

A year ago I learned in multi-var calc about curvature, and since then I've wondered something. It came up again today when my dad tried to talk to me about curvature like it was the second derivative. :P Is there a way, or at least any attempt or resource at all, about parameterizing a...

My professor asks, "Double check a formula that specifies how Riemann tensor is proportional to a curvature scalar." in our homework. The closet thing I can find is the relation between the ricci tensor and the curvature scalar in einstein's field equation for empty space.

Hi all! Let's start from the begin to see where I get lost. Extrinsic curvature defines the way an object relates to the radius of curvature of circles that touch the object (a couple of further nicer definitions come from physics, for the moments I am not mentioning them), and intrinsic...

Hi I know that the sphere has positive curvature everywhere, and the torus has positive an negative curvature. Is there a space homeomorphic to the sphere such that this space has negative curvature everywhere? And, is there a space homeomorphic to the torus such that the curvature is always...

Let us say there is a curved region of spacetime whose curvature is \kappa(s). How does one find the coordinates of the unit vector normal to a certain point on the region of spacetime? I tried searching Hamilton's principle and the general theory of relativity but I could not find any equation...

Hi everybody! what happens if an electron passes by with a speed of, say, 99.999999999...% of the speed of light (relative to me). Its mass will then be enormous. Will this electron cause a relevant curvature of spacetime? Can it be so fast that it acts like a black hole? I guess not. But why?

Let x(u,v) be a coordinate patch. Define a new patch by y(u, v) = c x (u, v) where c is a constant. Show that K_y = \frac {1}{c^2}K_x where K_x is the gaussian curvature calculated using x(u, v) and K_y is the gaussian curvature calculated using y(u, v). my book expects us to use the...

Let the elliptical orbit be bisected through the semi-minor axis... Ellipses are symetrical and both sides here are mirror images of each other, true? If so, then they share matching curvature at opposite ends of the semi-major axis (and opposite ends of the semi-minor axis, and reflections...

Homework Statement Find the curvature of r(t)= <t^2, lnt, t lnt> at the point P(1,0,0) Homework Equations K(t) = |r'(t) x r''(t)|/(|r'(t)|^3) The Attempt at a Solution r'(t) = <2t, t^-1, lnt+1> r''(t) = <2, -t^-2, t^-1> |r'(t) x r''(t)| = sqrt[t^-4(4 + 4 lnt + ln^2t) + (4...

I posted earlier asking if my science fair idea—to use many mirrors to reflect light onto one point—would work. After getting a reply that it would, I now have another question. It's clear that I have to use a curved board to mount the mirrors onto, or I would just get reflection of straight...

Homework Statement Parallel light in air enters a transparent medium of refractive index 1.33 and is focused 35 mm behind the surface. Calculate the radius of curvature of the surface of the medium Homework Equations f = \frac{R}{2} \frac{1}{f}=(n-1) \left(...

I'm not sure if this belongs here or in the physics section. The mathematical definition of curvature is the derivative of the unit tangent vector normalized to the arc length: \kappa = \frac{dT}{ds}. If we apply this to a parabola with equation y = x^{2} we get \frac{2}{(1+4x^{2})^{3/2}}...

Curvature form with respect to principal connection Hi all, I have a question. Let us suppose that P is a principal bundle with G standard group, \omega a principal connection (as a split of tangent space in direct sum of vertical and horizontal vectors, at every point in a differential way)...

If a certain space-time region has a constant curvature (caused by, say, an even distribution of energy over the region) how would radiation be effected by the curvature? Would it create a red-shift / blue-shift as the radiation moved through the region or would it be un-effected? Has anyone...

Homework Statement show that the curvature of a plane curve is \kappa=|\frac{d\phi}{ds}| where phi is the angle between T and i; that is, phi is the inclination of the tangent line.Homework Equations The Attempt at a Solution I'm not sure how to start this one out. Any ideas?

Find the curvature of the plane curve given by r(t) = (3cost)i + (3sint)j at the point (√(2), √(7) ). I know that κ=|r'(t) x r"(t)| / |r'(t)|^3 However, I believe that you are not allowed to do cross product unless there is an x, y, and z component and this question only has an x and y...

I'm trying to follow the math in Wald's General Relativity where he starts out with the equation for covariant derivative: \nablab\omegac = \partialb\omegac - \Gammadbc\omegad He uses that to derive the equation for a double covariant derivative: \nablaa\nablab\omegac =...

How does Ricci curvature represent "volume deficit"? Hi all, I've been reading some general relativity in my spare time (using Hartle). I'm a bit confused about something. I understand that Riemann curvature is defined in terms of geodesic deviation; the equation of geodesic deviation is...

Homework Statement http://www.mathhelpforum.com/math-help/attachments/f6/22423d1317129472-curvature-torsion-untitled.png The Attempt at a Solution What I did was I calculated the unit vector for dN/ds={.21i+0.91j-0.42k}/ sqrt(.21^2+.91^2+0.42^2)=.205i+0.889j-.4102k then, I...

Homework Statement A civil engineer is asked to design a curved section of roadway that meets the following conditions: With ice on the road, when the coefficient of static friction between the road and rubber is 0.1, a car at rest must not slide into the ditch and a car traveling less than 80...

The title of an old paper... It mentions that in order to use the full information of a hessian in 2nd order optimization that you should make a part of your iterative step to include v (eigenvector corresponding to smallest eigenvalue, assuming that the eigenvalue is negative). By doing the...

A formula I know for the number of functionally independent components of the curvature tensor is: (n^2)(n^2 -1)/12. It gives 1 for n=2, 6 for n=2, 20 for n=4. However, for a metric space (with symmetric metric), the curvature tensor is completely specified by the metric tensor. For n=4...

Riemann Curvature? i was watching this documentary that mentioned that riemann came up with a method to deduce whether we were on a curved surface, or on a flat surface, without leaving the surface to make the deduction. for example, for a curved 2d surface, we know it is as such as we can...

Spacetime curvature and the force pulling an object "down" the curvature ok , I have a question about the current model of gravity. If mass bends spacetime I understand that it accounts for things like how long light takes to travel through its geodesics but "why" does this curvature make...

I could understand gravity involve in object revolving around a big mass as the path of the object is circular. But couldn't understand how gravity is explained when an object thrown up returns back.

I'm assuming that the observer witnesses all other particles enter a horizon, be it an event horizon or the particle horizon. Thus the observer's observable universe would contain himself only. Is this understanding correct and can it happen in finite time? Also does such a universe have time...

How Does Reflection Behave In Arbitrary Surfaces Hi I am interested to know how reflection would behave in a mirror on a surface of negative [gaussian] curvature. I tried googleing it and found nothing useful Thanks Edit: Reflection in a sphere behaves like inversion in a sphere...

Curvature of spacetime tells us how the body is moving when it moves inertially. But if the body is not moving inertially does it causes backreaction by affecting spacetime curvature? Say if we compare body that is in free fall toward planet with body that is at rest on the surface of planet.

Homework Statement A dentist uses a spherical mirror to examine a tooth. The tooth is 1.13 cm in front of the mirror, and the image is formed 10.8 cm behind the mirror. Determine the mirror's radius of curvature. Homework Equations 1/p+1/q=1/f f=R/2 The Attempt at a Solution...

Like many of these forum dwellers, I've been reading the Elegant Universe and I've hit a fit of confusion. So I've got a couple of questions. In the book, it is explained that accelerated motion results in the warping of space and time (I'm thinking specifically of his example of the rigidly...

Hey Guys/Girls and thanks in advance Not quite sure this is in the correct forum since its not a homework question, more private study and curiosity lolz! Im trying to evaluate the curvature along the streamline within a hydrodynamic potential field (fluid flow). I have no issue calculating...

1) Is it always a given that the spacetime curvature will be flat in a region in which there is no mass? 2) Therefore is the curvature directly dependent on the mass in a particular region? 3) Also, what exactly is included in the term "mass"? 4) If there are no matter fields to curve...

I'm familiar with space and time together being 4 dimensions and that mass causes a curvature in this spacetime. When I consider a line that is curved, I can view the curvature because the line is drawn on a 2D surface (plane). So, it seems an additional dimension is required for a...

If one day gravitons are discovered, would their action be complementary to the gravitational attraction due to curved spacetime? Can gravity arise from both curved spacetime and exchange of gravitons? IH

In the Schwarzschild spacetime setting we have a vacuum solution of the Einstein field equations, that is an idealized universe without any matter at the geodesics that are solutions of the equations. This spacetime has however a curvature in both the temporal and spatial component that comes...

Could it be possible that space-time curvature is not caused by matter but is an inherent characteristic of space-time? Wouldn't this explain dark matter?

While watching a video on youtube about space/time, it explained space/time like a fabric with a ball on it. Rolling another ball past this first ball caused the second ball to curve. I get that part. then they said that Arthur Eddington went to test general relativity by photographing a...

Homework Statement I need to calculate the radius of curvature of a bimetallic strip when the two strips are subjected to different temperatures. in the problem, the two metals themselves are in different temperatures. One at 180°C, other at 160°C. Anyone with good solid mechanics knowledge...

[FONT="Georgia"][FONT="Tahoma"]Homework Statement Assume that we have a metric like: ds^{2}=f dr^{2}+ g d\theta^{2}+ h d\varphi^{2} where r,\theta , \varphi are spherical coordinates. f,g and h are some functions of r and theta but not phi. Homework Equations How can I calculate...

Hey! If a (pseudo) Riemannian manifold has an orthonormal basis, does it mean that Riemann curvature tensor vanishes? Orthonormal basis means that the metric tensor is of the form (g_{\alpha\beta}) = \text{diag}(-1,+1,+1,+1) what causes Christoffel symbols to vanish and puts Riemann...

I am thinking about train paradox here. The only difference between the passenger and the person waiting at the station is that the passenger is experiencing a force. They have exactly the same relative velocity and relative acceleration. The difference of this paradox from twin paradox is that...

When explaining the concept of spacetime curvature many popular science books (but I've also seen it mentioned in some textbooks) recur to the difference between intrinsic and extrinsic curvature, and how in a 2-D world with ants or bugs that are two dimensional, they would not be able to detect...

Given is a surface embedded in Euclidean 3 space whose Gauss curvature is everywhere positive. Its metric is <Xu,Xu> = E <Xu,Xv> = F <Xv,Xv> = G for an arbitrary coordinate neighborhood on the surface. The principal curvatures determine a new metric. In principal coordinates this new...

hi, we have a spline which consists of two splines,is it possible the two splines have the same curvature but different slopes?

hi all =D i have a question about evaporation rate depending on curvature. say a droplet of water is placed on a curved surface- one concave, one convex but with the same radius of curvature. will the evaporation rate be different?

Alright, my first thread with this title got locked down. Let's see how long this one lasts ;-) Actually, this time I have a specific queston. It is often stated there is no test that can determine if space time curvature is truly real. What about LIGO? As I understand it, with the...

Hello everybody, How do you prove that,given an n-dimensional manifold with constant curvature , i.e. the constant K is given by : K= R/n(n-1) (R denotes the scalar curvature)? I tried to contract the Riemann tensor in the expression above to obtain on the left side the scalar...