Curves Definition and 778 Threads

  1. cianfa72

    I Notion of congruent curve along a vector field

    Consider the following: suppose there is a smooth vector field ##X## defined on a manifold ##M##. Take a smooth curve ##\alpha(\tau)## between two different integral curves of ##X## where ##\tau## is a parameter along it. Let ##A## and ##B## the ##\alpha(\tau)## 's intersection points with the...
  2. K

    I Anomalous contribution to galactic rotation curves due to stochastic s

    Anomalous contribution to galactic rotation curves due to stochastic spacetime Jonathan Oppenheim, Andrea Russo Subjects: General Relativity and Quantum Cosmology (gr-qc); Astrophysics of Galaxies (astro-ph.GA); High Energy Physics - Theory (hep-th) another way to explain MOND in...
  3. S

    B Calculating shaded area: I'm getting discrepancies between methods

    **EDIT** Everything looked good in the preview, then I posted and saw that some stuff got cropped out along the right edge...give me some time and I'll fix it. Hello all, I"m trying to calculate shaded area, that is, the area bounded by the curves ##x=y^{2}-2, x=e^{y}, y=-1##, and ##y=1##...
  4. T

    Why are the paths of our cosmic explorations pretty?

    TL;DR Summary: Why are the paths of our cosmic explorations, pretty? OK, so I ask a lot of stupid questions. Here's another. Why is this picture, below, pretty? (They are the paths of all our cosmic explorations.) Now, I get the sine, cosine, circles, gravitational attraction, escape...
  5. berlinvic

    Prove orthogonality of these curves

    I am asked to prove orthogonality of these curves, however my attempts are wrong and there's something I fundamentally misunderstand as I am unable to properly find the graphs (I have only found for a, but I doubt the validity). Furthermore, I am familiar that to check for othogonality (based...
  6. F

    Timelike geodesic curves for two-dimensional metric

    Using EL equation, $$L=\left(\frac{t^2}{\alpha}\dot{x}^2-\frac{c^2t^2}{\alpha}\dot{t}^2\right)^{0.5} \Longrightarrow \mathrm{constant} =\left(\dot{x}^2 -c^2 \dot{t}^2\right)^{-0.5} \left(\frac{t^2}{\alpha}\right)^{0.5} \dot{x}$$. Get another equation from the metric: $$ds^2=-\frac{c^2t^2}\alpha...
  7. chwala

    I Determining directional Field for say ##\dfrac{dy}{dx}=y-x##

    I am looking at this now : My understanding is that in determining the directional fields for curves; establishing the turning points and/or inflection points if any is key...then one has to make use of limits and check behaviour of function as it approaches or moves away from these points thus...
  8. BloonAinte

    I Characteristic curves for ##u_t + (1-2u)u_x = -1/4, u(x,0) = f(x)##

    I woud like to find the characteristic curves for ##u_t + (1-2u)u_x = -1/4, u(x,0) = f(x)## where ##f(x) = \begin{cases} \frac{1}{4} & x > 0 \\ \frac{3}{4} & x < 0 \end{cases}##. By using the method of chacteristics, I obtain the Charpit-Lagrange system of ODEs: ##dt/ds = 1##, ##dx/ds = 1 -...
  9. Reuben_Leib

    I Help with Euler Lagrange equations: neighboring curves of the extremum

    I tried writing this out but I think there is a bug or something as its not always displaying the latex, so sorry for the image. I have gone through various sources and it seems that the reason for u being small varies. Sometimes it is needed because of the taylor expansion, this time (below) is...
  10. I

    A Finding Minimal Mean Distance Curves on the Unit Sphere

    **Problem:** Find parametric equations for a simple closed curve of length 4π on the unit sphere which minimizes the mean spherical distance from the curve to the sphere; the solution must include proof of minimization. Can you solve this problem with arbitrary L > 2π instead of 4π? There...
  11. SaintRodriguez

    I Worldlines & Curves in Spacetime: Exploring Possibilities

    Can there be worldlines that are neither timelike, nor null, nor spacelike? They can Are there curves in spacetime that are neither timelike, nor null, nor spacelike? Why?
  12. V9999

    Book recommendations about singular points of algebraic curves

    I'm not quite sure if this is an appropriate question in this forum, but here is the situation. I have just finished my graduate studies. Now, I want to explore algebraic geometry. Precisely, I am interested in the following topics: Singular points of algebraic curves; General methods employed...
  13. B

    I Questions about algebraic curves and homogeneous polynomial equations

    It is generally well-known that a plane algebraic curve is a curve in ##\mathcal{CP}^{2}## given by a homogeneous polynomial equation ##f(x,y)= \sum^{N}_{i+j=0}a_{i\,j}x^{i}y^{j}=0##, where ##i## and ##j## are nonnegative integers and not all coefficients ##a_{ij}## are zero~[1]. In addition, if...
  14. K

    I Deur Gravitational self-interaction Doesn't Explain Galaxy Rotation Curves

    Deur Gravitational self-interaction Doesn't Explain Galaxy Rotation Curves this paper A. N. Lasenby, M. P. Hobson, W. E. V. Barker, "Gravitomagnetism and galaxy rotation curves: a cautionary tale" arXiv:2303.06115 (March 10, 2023). Directly comments on Deur's theory of self-interaction...
  15. bakerjay

    A Data on galaxy rotation curves vs visible matter

    I'm after some raw data for testing theories of dark matter in galaxies. Basically what I want is table showing visible mass vs total mass within different radii (or, observed rotational velocity vs expected rotational velocity without dark matter). Plus error percentages. And ideally, for...
  16. LCSphysicist

    I Can Curves Live Out of Embedding Diagrams?

    The embedding diagram is well known for its qualitative representation of how the stress energy tensor curve the spacetime. We can construct a map from a general spherical metric to a cylindrical metric if we want to construct such diagrams. Now, my confusion is if there exist curves out of the...
  17. cianfa72

    I Maps with the same image are actually different curves?

    Hi, I've a doubt about the definition of curve. A smooth curve in ##\mathbb R^2## is defined by the application ##\gamma : I \rightarrow \mathbb R^2##. Consider two maps ##\gamma## and ##\gamma'## that happen to have the same image (or trace) in ##\mathbb R^2##. At a given point on the...
  18. mncyapntsi

    Banked curves, coefficient of static friction

    The solution in my textbook says that for b, us = 0.234. However when I use the formula above I get 0.2364 which I feel like is too far off. Something must have gone wrong... Any help would be much appreciated! Thanks :)
  19. elcaro

    I Can magnetic fields explain anomalous galaxy rotation curves?

    Magnetic fields as an alternative explanation for the rotation curves of spiral galaxies ABSTRACT THE flat rotation curves of spiral galaxies are usually regarded as the most convincing evidence for dark matter. The assumption that gravity alone is responsible for the motion of gas beyond the...
  20. chwala

    Finding the relative extrema for a speed function using parametric curves

    I have no problem in following the literature on this, i find it pretty easy. My concern is on the derived function, i think the textbook is wrong, it ought to be, ##S^{'}(t)##=##\frac {4t} {\sqrt{1+4t^2}}=0## is this correct? if so then i guess i have to look for a different textbook to use...
  21. S

    Finding value of k so two curves are tangent

    I tried to equate the derivative of the two equations: $$\cos x=-ke^{-k}$$ Then how to continue? Is this question can be solved? Thanks
  22. Eclair_de_XII

    Integrating vector-valued functions along curves

    The following parametrizations assume a counter-clockwise orientation for the unit square; the bounds are ##0\leq t\leq 1##. Hypotenuse ##(C_1)## %%% ##r(t)=(1-t,1-t)## ##dr=(-1,-1)\,dt## ##f(r(t))=f(1-t,1-t)=(a(1-t)^2,b(1-t)^2)## ##f\cdot dr=-(a+b)(1-t^2)\,dt## \begin{align} \int_{C_1} f\cdot...
  23. T

    B Ways to abstract curves or surfaces

    Hello. So, we have curves and surfaces. We already know about generally manifolds and Riemannian manifolds but what i want to produce are ways to abstract curves or surfaces but i am not talking about manifolds. Do you have any ideas? Perhaps the feature of curvature would help? To make an...
  24. B

    Find the osculating plane and the curvature

    I know the osculating plane is normal to the binormal vector ##B(t)=(a,b,c)##. And since the point on which I am supposed to find the osculating plane is not given, I'm trying to find the osculating plane at an arbitrary point ##P(x_0,y_0,z_0)##. So, if ##R(x,y,z)## is a point on the plane, the...
  25. A

    I How do Limb-Darkening curves differ at two different wavelengths?

    Does the limb-darkening curve fall off faster at shorter wavelengths or at longer wavelengths?
  26. A

    I Finding intersection of two algebraic curves

    Given two algebraic curves: ##f_1(z,w)=a_0(z)+a_1(z)w+\cdots+a_n(z)w^n=0## ##f_2(z,w)=b_0(z)+b_1(z)w+\cdots+b_k(z)w^k=0## Is there a general, numeric approach to finding where the first curve ##w_1(z)## intersects the second curve ##w_2(z)##? I know for low degree like quadratic or cubics...
  27. yucheng

    Simmons 7.10 & 7.11: Find Curves Intersecting at Angle pi/4

    >10. Let a family of curves be integral curves of a differential equation ##y^{\prime}=f(x, y) .## Let a second family have the property that at each point ##P=(x, y)## the angle from the curve of the first family through ##P## to the curve of the second family through ##P## is ##\alpha .## Show...
  28. S

    MHB Determine the area of a region between two curves defined by algebraic functions

    R is the region bounded by the functions f(x)=3√x−4 and g(x)=3x/5−8/5. Find the area A of R. Enter answer using exact values.
  29. B

    Automotive Two IC Engines: Comparing Torque Curves

    Hello there , hopefully someone can shed some light on this for me . let's say you have two IC engines . all variables are the same . Engine A has a completely flat torque curve at 500’ lbs to 550’ lbs . Engine B has a torque curve that starts out at 450’lbs but gradually makes its way to 650’...
  30. B

    A Levi-Civita Connection & Length of Curves in GR

    I am studying GR and I have these two following unresolved questions up until now: The first one concerns the Levi-Civita connection. There are two conditions which determine the affine connections. The first one is that the connection is torsion-free (which is true for space-time and comes...
  31. LCSphysicist

    I About groups and continuous curves

    Define $$\phi(A)$$ a transformation which, acting on a vector x, returns $$AxA^{*}$$, in such way that if A belongs to the group $$SL(2,C)$$, $$||\phi(A)x||^2 = ||x||^2$$, so it conserves the metric and so is a Lorentz transformation. $$\phi(AB)x = (AB)x(AB)^{*} = ABxB^{*}A^{*} = A(BxB^{*})A^{*}...
  32. M

    Physics C: Mechanics - Negative Energy and Potential Energy Curves

    I'm currently taking a course where we are working to teach older physics concepts and combine them with calculus. I was assigned to work on teaching a unit about energy; for the most part, it stays relatively consistent and can be solved algebraically. Another topic in this unit is Potential...
  33. K

    I How do I compute the cumulative probabilites of multiple bell curves?

    I took statistics in university about two years ago, but I'm rusty. I was trying to write a zero player game - except sometimes, the player can control one of the characters, and I needed to be able to compute these probabilities. That said, I almost put this in homework help, but it is not...
  34. S

    B Solving for integral curves- how to account for changing charts?

    [Ref. 'Core Principles of Special and General Relativity by Luscombe] Let ##\gamma:\mathbb{R}\supset I\to M## be a curve that we'll parameterize using ##t##, i.e. ##\gamma(t)\in M##. It's stated that: Immediately after there's an example: if ##X=x\partial_x+y\partial_y##, then ##dx/dt=x## and...
  35. karush

    MHB -b.2.2.33 - Homogeneous first order ODEs, direction fields and integral curves

    $\dfrac{dy}{dx}=\dfrac{4y-3x}{2x-y}$ OK I assume u subst so we can separate $$\dfrac{dy}{dx}= \dfrac{y/x-3}{2-y/x} $$
  36. Astrowolf_13

    Find the area delimited by two polar curves

    I attempted to solve this problem by finding the angles of an intersection point by equalling both ##r=sin(\theta)## and ##r=\sqrt 3*cos(\theta)##. The angle of the first intersection point is pi/3. The second intersection point is, obviously, at the pole point (if theta=0 for the sine curve and...
  37. bigfooted

    Geometry Is Walker's Textbook the Best Resource for Algebraic Curves?

    I recently became interested in algebraic curves, specifically topics like parametrization and its links to differential equations. I read a number of papers but I'm looking for a good (introduction) textbook on (planar) algebraic curves that gives a solid background, not pure theoretical but...
  38. A

    Differential Equations and Damper Curves

    Good evening, I have been wrestling with the following and thought I would ask for help. I am trying to come up with the equations of motion and energy stored in individual suspension components when a wheel is fired towards the car but, there is a twist! I am assuming a quarter car type...
  39. J

    Exploring Anodic & Cathodic Tafel Curves: Uneven Electrodeposition?

    The shapes of the anodic and cathodic Tafel curves are different. What does it mean? Does it mean that the electrodeposition of the copper onto a surface of an electrode is uneven? If yes, I am also thinking that this has something to do with the macrothrowing power? Since it was done in an...
  40. S

    Asymptote of x^3 - x^5 / ( x^2 + 1) and similar curves

    Playing with some numerical simulations, I plotted this in Wolfram Cloud / Mathematica: ##x^3-\frac{x^5}{x^2+2}## I had naively expected it to approach ##x^3−x^3=0##, but that isn't the case. It approaches 2x. I can now vaguely understand that the two terms need not cancel at infinity, but I'd...
  41. N

    Transistor Load line doesn't intersect characteristic curves....

    Hello, I've carried out an experiment to plot the characteristic curves (Ic vs Vce) for a BC108 transistor and then attempted to find where the load-line intersects those curves. Below are my results: ...as you can see, the load-line doesn't intersect the characteristic curves at all...
  42. K

    I Question about Galactic Rotation curves in the Milky Way galaxy

    The graph in Wikipedia, article Milky Way, section Galactic Rotation, shows the actual rotation speeds in blue and the calculated speeds due to observed mass in red. (The graph is to the right of the article.) At about 3 kpc the actual speed is about 205 km/s. To account for the decrease in...
  43. R

    Engineering How to calculate the total central angle of railroad curves

    0.3 19 5.7 tangent track miles 0.55 19 10.45 6 degree curves 0.15 19 2.85 10 degree curves Length of Curve = I ? Dc
  44. S

    I Applying the spacetime interval to regular vectors instead of curves

    I have some questions. Let us assume for these questions that I am using the (- + + +) sign convention. Firstly, we know that if you have a parameterized curve ξ(s), then you can find the proper time between two events at points s1 and s2 by using this formula (assuming that the curve is...
  45. Z

    Do floating objects clump together on curves in rivers?

    I recall reading somewhere about when anything, (eg. cars on the road, balls in a flowing stream) tend to clump together on curves. I don't remember where I read it but I seem to think there was some principal involved. Has anyone ever heard of this before, or am I mistaken. Thanks.
  46. S

    I Calculus- Area between two curves (minimize it)

    Hi, This is my first question here, so please apologise me if something is amiss. I have two curves such that Wa = f(k,Ea,dxa) and Wb = f(k,Eb,dxb). I need to minimize the area between these two curves in terms of Eb in the bounded limit of k=0 and k=pi/dx. So to say, all the variables can...
  47. igorrn

    Area between two curves (x = cos(y) and y = cos (x))

    I tried this: X = cos(y) → y = arccos(x) for x E(-1,1) and y E (0,2) Then: There's a point I(Xi,Yi) in which: Cos(Xi) =Arccos(Xi) Then I said area1 (file: A1) A1 = ∫cosx dx definite in 0, Xi And A2 (file:A2): A2 = ∫cosy dy definite in 0, Yi And the overlapping area as A3 (file: A3): A3 = ∫Yi dx...
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