In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's Elements: "The [curved] line is […] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width."This definition of a curve has been formalized in modern mathematics as: A curve is the image of an interval to a topological space by a continuous function. In some contexts, the function that defines the curve is called a parametrization, and the curve is a parametric curve. In this article, these curves are sometimes called topological curves to distinguish them from more constrained curves such as differentiable curves. This definition encompasses most curves that are studied in mathematics; notable exceptions are level curves (which are unions of curves and isolated points), and algebraic curves (see below). Level curves and algebraic curves are sometimes called implicit curves, since they are generally defined by implicit equations.
Nevertheless, the class of topological curves is very broad, and contains some curves that do not look as one may expect for a curve, or even cannot be drawn. This is the case of space-filling curves and fractal curves. For ensuring more regularity, the function that defines a curve is often supposed to be differentiable, and the curve is then said to be a differentiable curve.
A plane algebraic curve is the zero set of a polynomial in two indeterminates. More generally, an algebraic curve is the zero set of a finite set of polynomials, which satisfies the further condition of being an algebraic variety of dimension one. If the coefficients of the polynomials belong to a field k, the curve is said to be defined over k. In the common case of a real algebraic curve, where k is the field of real numbers, an algebraic curve is a finite union of topological curves. When complex zeros are considered, one has a complex algebraic curve, which, from the topological point of view, is not a curve, but a surface, and is often called a Riemann surface. Although not being curves in the common sense, algebraic curves defined over other fields have been widely studied. In particular, algebraic curves over a finite field are widely used in modern cryptography.
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...
**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##...
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...
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...
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...
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...
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 -...
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...
**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...
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?
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...
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...
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...
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...
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...
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...
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 :)
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...
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...
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...
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...
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...
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...
>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...
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’...
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...
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^{*}...
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...
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...
[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...
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...
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...
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...
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...
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...
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...
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...
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...
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.
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...
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...
i am new at relativity, it said mass can curve spacetime, does this mean spacetime will curve to a new 5th dimension (1-3 for space dimension, 4 for time dimension)?