What is Curvature: Definition and 910 Discussions

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. S

    What Does Oil/Water Curvature Indicate about Emulsions and Microemulsions?

    When it comes to emulsions and microemulsions; what does oil/water curvature mean? What does it mean that microemulsions have high oil/water curvatures, while emulsions have small oil/water curvatures?
  2. H

    Gaussian Curvature: Use Gauss Map to Find Sphere/Cylinder Radii

    I really need help answering these questions: use the gauss map to find the gaussian curvature of a sphere of radius r at any point. also, use the gauss map to find the gassian curvature of a cylinder of radius r at any point.
  3. O

    Degrees of freedom in the curvature tensor

    The Einstein field equations (EFE) in 4 dimensions have 10 degrees of freedom; The Riemannian curvature tensor in 4 dimensions has 20. If I understood this correctly, one can split up the curvature tensor and describe the remaining degrees of freedom by its traceless part, which is called the...
  4. F

    Radius of Curvature Derivation for Bending of Beam w/Overhangs

    Homework Statement Derive an expression geometrically for the radius of curvature of the following beam. This is part of a lab assignment for the bending of a simply supported beam with overhangs. ** I did this crappy diagram with AutoCAD, so I couldn't ( or didn't know how to ) include...
  5. F

    What Does 'To Lowest Order' Mean in the Derivation of Riemann Curvature Tensor?

    http://www.mth.uct.ac.za/omei/gr/chap6/frame6.html" is a derivation of the components of the riemann curvature tensor. the problem is that i can't understand the transition between eq97 and eq89 . what does "To lowest order " mean ?
  6. R

    Curvature form of Cartan connections

    Why is the definition of the curvature form for a Cartan connection the correct definition, and what does it actually tell you? I've read that it "measures the failure of the structural equation" but I suppose I don't really understand the structural equation (of a Maurer-Cartan form) anyway...
  7. S

    Radius of curvature atom path

    A doubly charged helium atom (mass = 6.68 x 10-27 kg) is accelerated through a potential difference of 4.00x 103 V. What will be the radius of curvature of the path of the atom if it is in a uniform 0.450 T magnetic field? the equation i was using was r = mV/|q|B m = 6.68 x 10^-27 V =...
  8. E

    Curvature using exterior differential forms

    Hello, I have a question related to the calculation of curvature using exterior differential forms (Misner, pp. 354-363). In all the examples given in the book (i.e. Friedmann, Schwarzschild, pulsating star metrics), the "guess and check" method used to find the connection forms (Eq. (14.31))...
  9. D

    Calculating Radius of Curvature for Upright Image Mirror

    A production line inspector wants a mirror that produces an upright image within magnification of 7.9 when it is located 10.0 mm from a machine part. What is its radius of curvature? I used: r/2 = f 1/f=(1/do) + or - (1/di) m=di/do first i did 7.9=10/do then i found that...
  10. N

    Radii of Curvature for Separating CO2 with 12C and 14C

    Homework Statement Use the same device to separate singly charged CO2 having 12C and 14C. What are the radii of curvature? This is a follow up question based on a HW problem I answered last week. The original problem was this: A doubly charged helium atom is accelerated by a voltage 2700V...
  11. N

    Circle of Curvature: Find Equation for Space Curve

    Homework Statement I am given a space curve r(t)= t i + sin(t) j and point (pi/2,1). They ask me to find an equation for the circle of curvature. Homework Equations Kappa, T, N, not sure The Attempt at a Solution So I have found the radius of curvature which is row= 1/kappa=...
  12. B

    Find the curvature of x = e^(t)

    Homework Statement Find the curvature of x = e^(t) y = e^(-t) z = t t = 0 Homework Equations I've used the equation of k(t) = |r'(t) x r''(t) |/ |r'(t)|^3 The Attempt at a Solution k(t) = |r'(t) x r''(t) |/ |r'(t)|^3 = |e^t i + -e^(-t)j + 1k| x |e^t i + e^(-t)j +...
  13. J

    Curvature in terms of the tangent vector

    My teacher wrote an alternative equation on the board for curvature, and I am wondering how it is true: k = | dT/dt / |dR/dt| | where T is the unit tangent vector. I know k = |R' x R''| / |R'|^3 = |dT/ds| but I am not sure about the formula in question. How is it true/derived?
  14. K

    How does Curvature Relate to 'Gravitation'?

    I have searche many general relativity texts and have not found an answer to the following question: How does curvature translate into the Newtonian idea of gravitation? For example, how is Newton's law of gravitation, where all matter attracts all matter, an approximation to the idea of...
  15. F

    What's the curvature of spacetime describe?

    Does the curvature of spacetime describe gravity or tidal force?
  16. 3

    Radii of curvature for pseudosphere

    This is a problem I'm having reading Visual Complex Analysis, page 295. If you look up "pseudosphere circles of curvature" on Google, it should be the first thing listed. On a point of a psuedosphere, there are 2 "circles of curvature", one with its center on the normal pointing out and the...
  17. J

    Does a particle at max speed have zero acceleration?

    Homework Statement Show that this statement is false: when a moving particle along a curve reaches its max. speed at t=3, its acceleration is 0. Homework Equations a = d^2 R / dt^2 = d|v|/dt * T + k |v|^2 N where k = |dT/ds|, T = v/|v|, N = 1/k dT/ds The Attempt at a Solution...
  18. N

    Radius of Curvature of Ion in Magnetic Field

    Homework Statement A doubly ionized atom (charge = +2e) whose mass is 5.15E-26 kg is accelerated by a voltage of 3450 V and enters a region where a uniform magnetic field B = 0.100 T acts perpendicular to its motion. What is the radius of curvature of the path of the ion in the B-field...
  19. N

    Gravity: force or space-time curvature?

    Please note that I do NOT want to discuss whether gravity is a force or the effect of space-time curvature here. If you want to discuss this, please post a separate topic about it. What I wanted to ask is what Einstein's own beliefs on this were. Up till now I had always believed he had...
  20. jinksys

    Find the curvature of the generating curve.

    Homework Statement A surface is formed by revolving y=x^(8/5) / 4 between [0,5] about the y-axis. Find the curvature of the generating curve. Graph the function of the generating curve. This is a question from a mathematica project. The book we are using is Stewart Multivariable Calculus...
  21. Y

    Guiding Centre, curvature drift (astro phys)

    Take a 10 keV ion in the van Allen belts 1000km above Earth's surface in a dipole magnetic field of 100 Gauss. Estimate the grad-B and curvature drift if the particle is a proton and compare this drift with the gravitational drift. 3. I know all the formulas needed, but do not know what the...
  22. stevebd1

    Kretschmann Scalar for Rotating Black Holes: Curvature Invariant

    According to wiki under http://en.wikipedia.org/wiki/Kretschmann_scalar" - While Riemann curvature tensor is proportional to tidal forces (\Delta g=2Gm/r^3), in some models of rotating (and charged) black holes, K is considered to diverge at the Cauchy horizon while tidal forces remain finite-...
  23. A

    Electric field at the center of curvature of a solid hemisphere

    Homework Statement Hi, I'm wondering if this is the proper way to approach this problem. The question says to: a)find the electric field at the center of curvature of the hemisphere (center of the flat bottom). Homework Equations Gauss's law: integral E*da = Qencl/epsilon...
  24. B

    Curvature and Osculating circle

    Homework Statement Find a formula for the curvature of the curve: x=(e^t + e^(-t))/2 y=(e^t - e^(-t))/2 Write an equation of the osculating circle when t=0. Homework Equations curvature=|x'y'' - x''y'|/(x'^2 + y'^2)^(3/2) The Attempt at a Solution First, wouldn't the formula for...
  25. M

    Exploring Intrinsic Curvature in Different Dimensions

    Hello all. I understand that a two dimensional surface can have curvature without it being referred to a higher dimension. So that a surface such as that of a sphere does not need to refer to a third dimension to determine its own intrinsic curvature and so on for higher dimensions. Can a...
  26. R

    Curvature of Spacetime: an inacurate Description?

    I think about general relativity often, specifically about the curvature of spacetime in the presence of matter (gravity). For a while, I understood much of this concept, but certain things escaped me: when objects are moving, it is easy to see how curved space causes matter to move in the way...
  27. L

    Radius of curvature experiment

    Hi there In an experiment I have calculated the radius of curvature for a +0.25D lens using Newtons rings and have obtained a value for 892mm, but the radius of curvature for a +0.25D lens is just over 2 metres. why is the value i got so much lower? do i have to double the value i got for the...
  28. C

    Maximum space-time curvature?

    First I thought If nothing can move faster than c, then nothing can accelerate faster than c/sec, right? Well, that means that the maximum amount space-time can curve is up to the ol' 45 degree slope, not like straight down as some black-hole pictures are made. Right? And then I thought...
  29. M

    What is the Surface of Revolution with Constant Curvature +1?

    Homework Statement I'm trying to find a surface of revolution with Gauss curvature K of +1 at all points, which doesn't lie in a sphere. Homework Equations The surface is parametrized as \psi (t, \theta ) = ( x(t), y(t) cos \theta , y(t) sin \theta ) I have the equation K =...
  30. M

    Curvature Proof Problem

    Homework Statement If c is given in terms of some other parameter t and c'(t) is never zero, show that k = ||c'(t) x c"(t)||/||c'(t)||3 The first two parts of this problem involved a path parametrized by arc length, but this part says nothing about that, so I assume that this path is not...
  31. S

    Find Curvature of Ellipse: x=3*cos(t), y=4*sin(t)

    Find Curvature of Ellipse given x=3*cos(t) and y=4*sin(t) at the points (3,0) and (0,4) Relevant equations: curvature at r(s) is k(s)=||dT/ds|| when r(s) is arc length parametrization and T is the unit tangent vector I usually use the formula k(t)= (||r'(t) x r''(t)||)/||r'(t)||^3 So...
  32. N

    Electromagnetic spacetime curvature effects

    In a nearby thread: I believe it was correctly concluded "yes" because an electromagnetic (EM) field has energy and pressure so it curves spacetime; it does have a gravitational effect. Although it was not discussed, this effect is much, much smaller than the electromagnetic effects because...
  33. P

    Killing Vector and Ricci curvature scalar

    Homework Statement I'm currently self-studying Carroll's GR book and get stuck by proving the following identity: K^\lambda \nabla _\lambda R = 0 where K is Killing vector and R is the Ricci ScalarHomework Equations Mr.Carroll said that it is suffice to show this by knowing: \nabla _\mu...
  34. S

    Questions about the Kretschmann curvature scalar

    The Kretschmann curvature scalar is defined to be K = RabcdRabcd where Rabcd is the Riemann tensor. I believe I heard in class that this scalar can be used to demonstrate the existence of a curvature singularity. Can somebody tell me why this is so? Also, I heard that it is better (easier I...
  35. S

    Radius of curvature of the cornea

    Homework Statement The cornea, a boundary between the air and the aqueous humor, has a 3.0cm focal length when acting alone. What is its radius of curvature? Homework Equations n(1)/s + n(2)/s' = (n(2)-n(1))/R where, n is the index of refraction (unitless), s' is the img dist (cm)...
  36. S

    Relative kinematics and radius of curvature

    Homework Statement (Only need help for b) I think but I'll post the whole problem) (All values are SI units) (O; i, j, k) orthonormal basis. A particle moves following this law: r = 8t2 j A disk with radius 2 in the plane XOY rotates around Z with constant angular speed: ω = 3 k. At the time...
  37. JasonJo

    How Does Constant Sectional Curvature Relate to Locally Symmetric Spaces?

    Curvature Questions, Please Help! Homework Statement 1) Prove that if M is locally symmetric (i.e. the Riemann tensor is constant), connected and 2 dimensional, then M has constant sectional curvature. 2) Prove that if M has constant (sectional) curvature, then M is a locally symmetric...
  38. C

    Curvature of curve with arbitrary parametrization

    Hi, I'm preparing a little exposition of curvature and torsion for my Calculus class and so I need to include some simple proofs for the things I'll use to define curvature. So I'm looking for a proof of the formula for the curvature of an arbitrarily parametrized curve that doens't use the...
  39. C

    Motion in space due to space-time curvature

    Why do things orbiting, i.e. free-falling, around Earth float away from each other? Why don't they both free fall toward Earth together? I remeber hearing once that if you let go of 2 objects while 'floating' in space they both go away from you and away from each other. Is this due to curving...
  40. RyanSchw

    Curvature, More of a cross product question really.

    I've been using various methods of finding curvature, and using the forumla K=\frac{||\vec{r}'X \vec{r}''||}{||\vec{r}'||^3} I took the cross product of my two vectors and came up with (just assume this is correct, if my question isn't answered i'll post all related work)...
  41. T

    Radius of curvature of 1 MeV electron in 1 Tesla B field

    i need to calculate radius of curvature of 1 MeV KE electron in 1 Tesla magnetic field. r = mv / eB what is the (relativistic) speed, v, of the electron? (ans: 0.941 c ??) then i think you use p(rho) = mo. V / sqrt (1 - v^2 /c^2) = mv (ans: 7.59E-22 kg m/s) e = 1.6E-19 C B = 1...
  42. A

    Physical interpretation of spacetime curvature?

    Folks, I'm in the process of trying to understand spacetime curvature in general relativity. My question might sound odd, but I'm wondering how to best conceptualize spacetime distortions due to a moving mass. If there is a large mass, e.g. a planet, moving through spacetime, the curvature...
  43. C

    How do we argue that gravity is not a force due to curvature of space-time?

    Homework Statement How do we argue that gravity is not a force due to curvature of space-time? Homework Equations I'm new.. I don't even understand the eqn of tensor calculus. The Attempt at a Solution No force is needed for as massive objects follows the curvature of spacetime...
  44. S

    Yau's result for the Ricci curvature on Kahler manifold

    Hi, I've been reading through Yau's proof of the Calabi conjecture (1) and I was quite intrigued by the relation R_{i\bar{j}} = - \frac{\partial^2}{\partial z^i \partial \bar{z}^j } [\log \det (g_{s\bar{t}}) ] derived therein. g_{s \bar{t} } is a Kahler metric on a Kahler manifold (I'm...
  45. O

    Measuring curvature with parallel transport

    Parallel transport, as one means of quantifying the curvature of a coordinate space, enables changes in a vector's components, when it is carried around variously oriented loops in that space, to be properly measured, i.e. by comparisons made at the same location. Those changes which are...
  46. K

    How much curvature are we talking about?

    Homework Statement I've attached a picture which include the question -- I found it on the net, and it's exactly the same as my homework. Homework Equations I'm not sure which formula to use. It's due pretty soon, so please help! :cry:
  47. L

    What is the Curvature of a Helix Given by a Parametric Equation?

    Homework Statement Find the curvature of a helix given by the parametric equation r(t)=<acost, asint, bt> where a and b are real numbers Homework Equations I know k=|T'(t)/r'(t)| The Attempt at a Solution and I believe the answer to be k=b/(a2+b2)1/2, I just don't know how to get there
  48. L

    Finding Curvature of Helix Given Parametric Equation

    Homework Statement Find the curvature of a helix given by the parametric equation r(t)=<acost, asint, bt> where a and b are real numbers Homework Equations I know k=|T'(t)/r'(t)| The Attempt at a Solution and I believe the answer to be k=b/(a2+b2)1/2, I just don't know how to get there
  49. T

    Proof: Curvature Zero -> Motion along a line

    Homework Statement Proof that, if a particle moves along a space curve with curvature 0, then its motion is a along a line.Homework Equations K=\frac{||r'(t)\times r''(t)||}{(||r'(t)||)^3} (curvature of a space curve)The Attempt at a Solution Assume the curve is smooth, so r'(t) cannot be the...
  50. A

    Acceleration and Curvature of space-time

    I'm confused, but when objects travel along the straight lines in curved space-time, do they undergo acceleration? We know that following geodesics is equivalent to inertial motion (one example is free-fall), but when these inertially moving objects travel in curved spacetime, they accelerate...
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