Curvature Definition and 872 Threads

Hello guys! I'm stuck with this for a 4th day now.. I have a set of data and for every data point I want to calculate a curvature. In order to do that I use Catmullrom spline to interpolate points and get derivatives f' and f". Curvature is defined as y"/ (1+y'^2)^3/2. However, at some...

Gauss-Bonnet term extrinsic curvature calculations? In General Relativity if one wants to calculate the field equation with surface term, must use this equation: S=\frac{1}{16\pi G}\int\sqrt{-g} R d^{4} x+\frac{1}{8\pi G}\int\sqrt{-h} K d^{3} x The second term is so-called Gibbons-Hawking...

Homework Statement r(t)=cos(t)i+sin(t)j+sin(2t)k Find the curvature κ, the unit tangent vector T, the principal normal vector N and the binormal vector B at t=0. Find the tangential and normal components of the acceleration at t=∏/4 Homework Equations T(t)=r'(t)/|r'(t)| N(t)=T'(t)/|T't|...

Homework Statement Given the polar function r = 4cos(3θ) find the curvature. Homework Equations The Attempt at a Solution I know there is a formula for curvature of a polar function but I was never given that equation and was told to convert to parametric. and use ||v x a|| /...

In his article The Ricci and Weyl Tensors John Baez states that the tidal stretching and squashing caused by gravitational waves would not change the volume as there is 'only' Weyl- but no Ricci-curvature. No additional meaning is mentioned. But, beeing not an expert I still have no good...

Hi there: I've learned that there's no such thing as gravity, just the curvature of spacetime that makes objects that are close to each other act like it existed. Does Higgs Bossom discovery tell us that there is a gravity force after all?

Hi all, I am wondering if it is possible to calculate, using Einstein Tensor, the space time curvature around the Earth. As far as I understand, Einstein Field Equations tell us that the presence of a matter curves the space time. So space time curvature is gravity and gravity is space time...

Hi, I was wondering if anyone could clarify something for me. I have been reading about the curvature of Spacetime and have come across a few things in articles in conjunction with de Sitter and Anti de Sitter spaces "Negative curvature corresponds to an attractive force" and "Positive...

When I first started learning about GR, I understood that a vacuum solution is one where the Einstein tensor vanishes, for the simple reason that the stress-energy tensor, T, vanishes. I have since read many times that the Ricci tensor vanishes for a vacuum solution. I am confused because to...

A hydraulic steel tube of diameter 10mm and wall thickness 1.5mm has been bent to a radius of 100mm. Calculate the reverse radius of curvature that needs to be applied to straighten the tube. M/I= σ/Y = E/R I am really struggling on the above question. I think I need to read off the...

I've been trying to calculate the Riemann Curvature Tensor for a certain manifold in 3-dimensional Euclidean Space using Christoffel Symbols of the second kind, and so far everything has gone well however... It is extremely tedious and takes a very long time; there is also a high probability...

Need to find the Ricci scalar curvature of this metric: ds2 = e2a(z)(dx2 + dy2) + dz2 − e2b(z)dt2I tried to find the solution, but failed to pass the calculation of Riemann curvature tensor: <The Christoffel connection> Here a'(z) denotes the first derivative of a(z) respect to z...

Homework Statement Need to find the Ricci scalar curvature of this metric: ds2 = e2a(z)(dx2 + dy2) + dz2 − e2b(z)dt2 Homework Equations The Attempt at a Solution I tried to find the solution, but failed to pass the calculation of Riemann curvature tensor: <The Christoffel...

Radius of curvature and focal distance question? Hi everyone! Really struggling on these two questions and have spent way too many hours coming up with the wrong answers. Below are the questions and my attempts. Any help would be greatly appreciated! 1. Calculate the radius of curvature...

So I am trying to find the nth derivative of the curvature function: \kappa(x) = \frac{f''(x)}{[1 + (f'(x))^2]^\frac{3}{2}} Now, I could go about using Faa Di Bruno's formula but when I did I realized that I also have to use the Leibniz formula as a substitution for a term in the Faa Di...

I want to know why the moment could be represented as the product of the bending stiffness and the curvature. I do not quite understand the function of the curvature in the formula. http://en.wikipedia.org/wiki/Curvature

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...

In the Einstein tensor equation for general relativity, why are there two terms for curvature: specifically the curvature tensor and the curvature scalar multiplied by the metric tensor?

Hi, I am trying to measure the center deflection of a simply supported beam with a capacitive sensor. The beam's surface that is the sensor's target is originally flat. After applying equal forces on the two edges of the beam, its center curves upwards (away from the capacitive sensor) thus...

Homework Statement I am trying to calculate the curvature of a curve given by the position function: \vec{r}(t)= sin(t) \vec{i} + 2 \ cos (t) \ vec{j} The correct answer must be: \kappa (t) = \frac{2}{(cos^2(t) + 4 \ sin^2 (t))^{3/2}} I tried several times but I can't arrive at this...

Is there a way to take a known path, all the existing forces/fields along it, and solve for a driving force function that results in an object moving along that path? but solving it without knowing the speed along that path, only the path itself, and maybe the total time taken from A to B along...

Spacetime curvature observer and/or coordinate dependent? In another topic several people suggested that spacetime curvature is not absolute, it apparently depends on the observer and/or coordinate system. Apparently if someone goes fast (whatever that might mean in relativity) curvature is...

Homework Statement You placed a 10cm high object in front of a mirror and got 5cm high virtual image at (-30cm). (Hint: watch the sign convention) a. Find the magnification b. Find the object distance from the mirror c. Find the radius of curvature of the mirror then I have to know if...

Hey all, I have a vector field described by a complex potential function (so I have potential lines and streamlines). I am looking for a way to express its curvature at every point, but I can't find such a formula in my books. I have searched in wikipedia and I read that the way to define it...

I posted these on Reddit but some questions weren't answered so I was wondering if people here could help: First Part I was informed that the universe did actually exponentially gain energy during inflation and perhaps other periods of its development. So how does this affect the entropy and...

A formula for finding the normal vector for curvature is: N=(dT/ds)/(||dT/ds||) Where dT=change in tangent vector ds=change in distance travelled Another fromula was: N=(dT/dt)/(||dT/dt||) What's dt ? Is it the same as ds? I don't think so cause the course notes said that...

I just wanted to make sure whether I've understood something correctly In the FRW equation: (\frac{ \dot a}{a})^2 = \frac{8 \pi G}{3} \rho - \frac{k}{a^2} ...there is this curvature term. I'm confused about the meaning of this k. Sometimes they say it can ONLY be -1 , 0 or +1...

What exactly is the radius of curvature of an object? And how would this be applied to a question such as the following: A glass porthole of a submerged craft has parallel curved sides, both of radius of curvature R. What would R be in order that an object in the water 2m away from the...

Homework Statement The equations sin(xyz) = 0 and x + xy + z^3 = 0 define planes in R^3. Find the osculating plane and the curvature of the intersection of the curves at (1, 0, -1)Homework Equations Osculating plane of a curve = {f + s*f' + t*f'' : s, r are reals} Curvature = ||T'|| where T is...

Two soap bubbles of radius r and R are in touch find the radius of curvature of their point of contact? (Both bubbles are touching each other with their external surfaces) I have no idea about this question. can you please try to help>

Hi all, I'm looking for an equation which will give me the focal length of a biconvex lens given that we know both Radii of curvature, the thickness of the lens and the refractive index inside and outside. An equation is given on wikipedia here http://en.wikipedia.org/wiki/Focal_length as I...

Hi all, I'm trying to write a Matlab simulation that determines the velocity of a car of known mass and moment of inertia which travels on a track whose curvature is also known. To say the least, I'm at a loss as to what approach I should take to create my simulation. I'm finding it...

Homework Statement Question: "Find the unit tangent, normal and binormal vectors T, N, B, and the curvature of the curve x = 4t, y = -3t^2, z = -4t^3 at t = 1." Answer: T = 0.285714285714286 i - 0.428571428571429 j - 0.857142857142857 k N = -0.75644794981871 i + 0.448265451744421 -...

Please, anyone tell me how to proof this equation: {R^\rho}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma} Given a definition...

Now, I know that tittle is messy, so I'm going to explain it as clearly I can. One of the proofs to the fact that time is relative is, as I've heard. Putting one clock on the ground, and another a few feets above it. When these clocks measure time, the one above the ground will do it faster...

Hi, I know that the mean curvature at an extremum point where the function vanishes must be nonpositive.can this say something about the sign of the mean curvature at the farthest point on a close surface from the origin? Thank's Hedi

Homework Statement A line segment extends horizontally to the left from (-1,-1) and another line extends horizontally to the right from the point (1,1) find a curve of the form y=ax^5+bx^3+cx that connects the two endpoints so that the curvature and slope are zero at the endpoints...

Here is the question: At what point does the curve y=e^(32x) have maximum curvature? I have tried this method: http://www.math.washington.edu/~conroy/m126-general/exams/mt2SolMath126Win2006.pdf Problem 4. Adapting for 32x rather than x it seems to get a bit lengthier with 32x than...

Can the twin paradox provide us with insight into time curvature? If my twin boards a ship that can travel near the speed of light, special relativity says that on arrival back on Earth, my twin should be younger that I am. Has my twin experienced a time curvature?

Since it's been observed that mass causes the curvature of the spacetime continuum. I'm wondering how it curves the spacetime continuum at a distance. For example, a planet will curve the spacetime around it millions of miles away, yet all around the planet is the almost perfect vacuum of...

hi, I am trying to understand the meaning of the following equation in the simplest way possible thanks in advance

hi, I am trying to understand the meaning of the following equation in the simplest way possible https://public.blu.livefilestore.com/y1pWwu86vlTmLHRY35RBhm3I55eYrMWCtWPmdVjAM807ltH2EfInsaFIBk6nCFhnIdwno9Mz4Oa4qWC8Zv9xND3KA/tensorp.png?psid=1 thanks in advance

I just want to ask, because it is easy to imagine on small scale (eg draw triangle on small sphere and than make sum of its angles). But how it is really done at universe scale, when we cannot leave our position to see the big picture?

the radii of the curvature of the spherical surfaces which is a lens of required focal length are not same. it forms image of an object. the surfaces of the lens facing the object and the image are interhanged. will the position of the image change?

I am wondering how space geographers would measure curvature of space around a large isolated star. i am thinking of the set up where there are two nearby spheres surrounding the star whose circumferences are already known. The remaining step is to measure the length of a radial geodesic segment...

Dave made a comment about curvature being a variable when determining strength of a material. I was wondering how curvature does affect strength in structures/materials. I know that "the more circular an object is the stronger it will be" is common knowledge. I am looking for something more...

I wanted to say I love the posts on this Cosmology forum - very good reads. My question is whether there is enough to large-scale curvature alone to cause the expansion of the universe as we measure it? Why can't galaxies accelerating away from us be moving in curved space? I agree with...

Hi, are there two non isomorphic surfaces with the same gaussian curvature? thank's Hedi

Hi, I was doing some exercises from the book on curvature by Lee to buff up my differential geometry. I came a cross the following question and it seems to me the question isn't completely correct, but I'm not so good at differential geometry that I am confident. Maybe someone else is...

Hello there. I'm new to the forums obviously. And I'm also quite new to physics. I've read many existing answers about how this works but I can't really understand how it would logically work out. From what I've understood. Gravity is not a force. Objects move in a straight line unaltered by...