Curvature Definition and 920 Threads

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.

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  1. D

    Why positive curvature implies finite universe?

    This post in influenced by 3 new threads in our cosmology forum. Recent observational data favors positive curvature of our Universe. The question I have, however, is why positive curvature implies spatially finite Universe? Yes, it might look quite obvious if we embed curved space into higher...
  2. N

    Integral of mean curvature function

    Hello everyone, I am self teaching some elementary notions of differential geometry. Rather, I should say I am concentrating on mean and gaussian curvature concepts related to a physics application I am interested in. I see one has to evaluate an integral that goes as...
  3. A

    Spaces of Constant Curvature and the Ricci Tensor

    Hi all, I was just interested in verification of a concept. If we are given the full Riemann tensor in the form which implies constant curvature (i.e. lambda multiplying metric components) does this imply that the Ricci tensor vanishes? The question stems from why the vacuum equations cannot be...
  4. G

    Double contraction of curvature tensor -> Ricci scalar times metric

    Double contraction of curvature tensor --> Ricci scalar times metric I'm trying to follow the derivation of the Einstein tensor through double contraction of the covariant derivative of the Bianchi identity. (Carroll presentation.) Only one step in this derivation still puzzles me. What I...
  5. F

    Understanding the Effects of Relative Curvature and Mass on Space and Observers

    Mass curves space. And speed near the the speed of light increases mass. So for someone traveling near c and is passing a partice at rest, the traveling observe feels like he's at rest and the other particle is moving. So if the other particle is moving wrt his rest frame, does he see an...
  6. F

    Conditions for curvature singularity

    Hi All, I was wondering if it is correct to say that a vanishing metric determinant is a necessary (but probably not sufficient) condition for a curvature singularity to exist at some point(s), or is one forced to construct the full Kretschmann scalar? Cheers! FD
  7. C

    Curvature of an orthogonal projection

    Homework Statement Let \vec{X(t)}: I \rightarrow ℝ3 be a parametrized curve, and let I \ni t be a fixed point where k(t) \neq 0. Define π: ℝ3 \rightarrow ℝ3 as the orthogonal projection of ℝ3 onto the osculating plane to \vec{X(t)} at t. Define γ=π\circ\vec{X(t)} as the orthogonal projection...
  8. G

    Radius of Curvature calculation in a magnetic field

    Homework Statement 1) The magnetic field everywhere is tangential to the magnetic field lines, \vec{B}=B\hat{e}t, where \hat{e}t is the tangential unit vector. We know \frac{d\hat{e}t}{ds}=(1/ρ)\hat{e}n , where ρ is the radius of curvature, s is the distance measured along a field line and...
  9. P

    Radius of Curvature, Lenses, Finding Di

    Homework Statement If you look at yourself in a shiny Christmas tree ball with a diameter of 8.1 cm when your face is 35. 0cm away from it, where is your image? Homework Equations 1/do + 1/di = 1/f The Attempt at a Solution 1/di = 1/4.05 cm - 1/35.0 cm di = 4.6 cm I...
  10. A

    How to calculate radius of curvature of beam?

    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...
  11. R

    What are the only possible surfaces with zero mean curvature?

    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 =...
  12. S

    Relationship between curvature of an arch and bending moment.

    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...
  13. E

    Determining the radius of curvature mass spectrometre question

    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...
  14. F

    Is it possible to have curvature without bending moment?

    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...
  15. M

    Principal curvature used in a contact problem

    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...
  16. Chris L T521

    MHB Manifold Theory: Computing Curvature and Verifying Geometries

    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: ----- Problem: (i) Let $\omega$ be a 1-form. Use the structure equations \[\begin{aligned}d\theta^1 &=...
  17. K

    Lawn Mower Curvature: Calculating & Applications

    "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...
  18. J

    Curvature of Time: Measuring Change in XYZ Axis

    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?
  19. G

    Curvature of Catmullrom spline

    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...
  20. S

    Gauss-Bonnet term extrinsic curvature

    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...
  21. N

    Problem involving tangent vector, normal vector, binormal vector and curvature

    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|...
  22. M

    Curvature of polar function r=4cos(3θ): Find Solution

    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|| /...
  23. T

    The meaning of Weyl curvature caused by gravitational waves

    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...
  24. R

    Does Higgs Boson rule out spacetime curvature?

    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?
  25. M

    Find curvature of spiral of archimedes

    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...
  26. G

    Space time Curvature around the Earth

    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...
  27. P

    Curvature of Spacetime related to mass and expansion

    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...
  28. m4r35n357

    Confused about curvature of vacuum solutions

    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...
  29. T

    Plastically Deformed Steel Tube - Radius of Curvature to Straighten

    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...
  30. H

    Calculating Riemann Curvature Tensor: Faster Methods?

    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...
  31. C

    Need to find the Ricci scalar curvature of this metric

    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...
  32. C

    Need to find the Ricci scalar curvature of this metric

    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...
  33. C

    Radius of curvature and focal distance question?

    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...
  34. J

    Is There an Easier Way to Find the Nth Derivative 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...
  35. C

    The relationship between the moment and curvature

    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
  36. C

    Need to find the riemann curvature for the following metric

    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...
  37. PerpStudent

    EFE: Why is there a curvature tensor and curvature scalar?

    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?
  38. P

    Zero gaussian curvature implies developability proof

    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...
  39. H

    Beam Curvature Measurement with a Capacitive Sensor

    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...
  40. R

    Calculating the Curvature of a Trigonometric Curve | Math Homework Help

    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...
  41. E

    Mechanics: acceleration & curvature relationship

    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...
  42. P

    Spacetime curvature observer and/or coordinate dependent?

    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...
  43. J

    Finding magnification, object distance from the mirror, and radius of curvature

    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...
  44. M

    Vector field curvature in the complex plane

    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...
  45. O

    I have some somewhat detailed questions about inflation, curvature, and entropy

    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...
  46. H

    What's dt in finding normal vector for curvature

    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...
  47. L

    The meaning of the curvature term

    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...
  48. M

    What Is the Radius of Curvature in Optical Applications?

    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...
  49. T

    Intersection of planes, curvature and osculating plane

    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...
  50. V

    Radius of Curvature at Soap Bubble Contact Point

    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>
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