In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number.
For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or manifold. This leads to the concepts of maximal curvature, minimal curvature, and mean curvature.
For Riemannian manifolds (of dimension at least two) that are not necessarily embedded in a Euclidean space, one can define the curvature intrinsically, that is without referring to an external space. See Curvature of Riemannian manifolds for the definition, which is done in terms of lengths of curves traced on the manifold, and expressed, using linear algebra, by the Riemann curvature tensor.
If I start out with a flat beam of length a and then I fix one side and then bend the other side up to form an arc with height h, is that enough information to determine the radius of curvature of the bent beam? If so, how would I do it?
Thanks...
Homework Statement
Prove that the only surfaces with zero mean curvature are either planes or hyperbolic curves with the equation: y = \frac{\cosh (ax+b)}{a} rotating alone the x axis.Homework Equations
The Attempt at a Solution
I made an attempt by devoting the equation of the surface as r =...
Homework Statement
A bridge has two arches, with one abutment in the center labeled A connecting the two arches, and a point b on the right.
I have calculated that the maximum bending moment in the structure would be at point A, right on the abutment that is connecting the two arches. I am...
Homework Statement
A mass spectrometer is an important tool in the study of air pollution. However, one of the difficulties faced by scientists is that carbon monoxide molecules (CO), which are major contributors to air pollution, have very nearly the same mass as harmless nitrogen molecules...
The simple question is whether it is possible to have curvature in a beam with no bending moment (similar to how there can be strain without stress)?The main example I have to discuss what lead me to this question is a beam which has been prestressed concentrically and so is undergoing only...
Hi,
I am trying to understand a theoretical problem involving the contact between two surfaces. I have uploaded a screen shot of the mathematical formulations of the solution.
I understand most of the solution, except the principal curvatures. I have tried to look up principal curvature...
I've posted a bunch of analysis questions as of late. I'm going to change things up a little bit and ask something that involves manifold theory. Here's this week's problem:
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Problem: (i) Let $\omega$ be a 1-form. Use the structure equations
\[\begin{aligned}d\theta^1 &=...
"Lawn Mower Curvature"
Given a curve C in the plane, if you pick a consistent perpendicular direction, you can construct a new curve by moving out a fixed distance ε from the curve in that direction. For small values of ε, the area between these two curves will be approximately equal to ε times...
So in GR spacetime is curved in the x, y, z, and t axis but suppose you are given a surface in x, y, z could you find the curvature of time by simply measuring the rate in change in curvature of the surface in the x, y, z axis?
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?
Homework Statement
Find the curvature of the spiral of Archimedes r = 2θ
Homework Equations
||v x a || / ||v||^3
The Attempt at a Solution
I tried to convert the polar equation into parametric and got
x = 2θsinθ
y = 2θsinθ
z = 0
I think took the derivative of x y and z...
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?
Homework Statement
Hey guys, I have a small question on a proof in Struik's Differential Geometry book. It concerns the proof that all surfaces with K = 0 (gaussian curvature zero) are developable, i.e. all surfaces with K = 0 are ruled and the tangent planes along the rulings are parallel...
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?