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.
I am trying to develop an equation or algorithm to determine the roulette curve that results from rolling a curve over another fixed curve.
A general method to determine the roulette using any two curves, rolling and fixed, seems to be presented here...
Homework Statement
Use Desmos to graph the spiral ##r=\theta## on the interval ##0\leq\theta\leq4\pi##, and then determine the exact length of the curve and a four decimal approximation.
Hint: ##\int \sec^3(x)dx=\frac{1}{2}\sec(x)tan(x)+\frac{1}{2}\ln\left|\sec(x)+\tan(x)\right|+C##
Homework...
COD stands for co-ordinate.
As the title says, you have two co-ordinates of a point, x and y, on a unit square.
What's the formula for converting these two co-ordinates into a single Hilbert curve co-ordinate?
Which represents the percentile along the length of the Hilbert Curve that point is on.
<Moderator's note: Member has been informed to separate different question into different threads in the future.>
1. Homework Statement
Please see picture attached...
The diagram shows the curve with equation y = f (x)
The coordinates of the minimum point of the curve are (–2, –1)
(a) Write...
Homework Statement
Homework EquationsThe Attempt at a Solution
I don’t have a clue on how to do the calculations. I know the equivalency point is at the steepest part of the derivative graph. Please don’t say just figure it out, I truly need help. I haven’t learned titrations in lecture yet...
Recently I have studied that from the rotation curve of spiral galaxies, the nearly constt. behaviour of velocity of the stars situated far away from the central core suggests mass(r) ~ r ,rather than 1/√r as expected.
Are there any other theory which proves the existence of dark energy ??
In 2-D Cartesian coordinate system let's there exist a scaler field Φ(x1,x2) ,now we want to find how Φ changes with a curve which is described by the parameter(arc length) s
dΦ/ds=(∂Φ/∂xi)dxi/ds
Can we say for Cartesian coordinate system that along the curve at any s dxi always points in the...
Homework Statement
I have to find length of the curve: y2 = (x-1)3 from (1,0) to (2,1)
Homework Equations
s = ∫ √(1 + (f '(x) )2 ) dx where we have integral from a to b
The Attempt at a Solution
I'm bit confused:
I'm thinking of writing function regarding x, f(x)...
I'd like to plot a Bragg curve for an alpha particle in air.
I'd like to make one plot for 239Pu (5.156 MeV alpha) and one plot for 238U (4.267 MeV).
Does anyone know how I can do this? I would like a basic plot of stopping power (MeV/cm) vs path length (cm).
Thank you
Hi all
I was wondering if someone could help explain what the area under a load vs deflection curve tells you.
I have a concrete sample which I loaded until it failed.
I plotted the load (kN) and deflection (mm) as shown below.
My question is; if the curve in red can be represented as a...
Homework Statement
Let ##U## be a plane given by ##\frac{x^2}{2}-z=0##
Find the curve with the shortest path on ##U## between the points ##A(-1,0,\frac{1}{2})## and ##B(1,1,\frac{1}{2})##
I have a question regarding the answer we got in class.
Homework Equations
Euler-Lagrange
##L(y)=\int...
is it possible to draw a random curve on a piece of graph paper and find the function that defines that curve? Assuming smooth curves.
And if so,is it possible to do so with complex curves?
tex
I drew a diagram in order to help figure out something for a tabletop game I'm putting together.
My question is about the physics of materials, and is not directly about the fictitious psychic/magic abilities in my game world.
I drew a diagram consisting of dots representing particles and...
Hi people,
I am looking for a database where I can get raw data to plot the Bragg's curve for alpha particles energy loss vs range in Silicon, Germanium and CZT.
Your help will be much appreciated.
Cheers,
Shri
Homework Statement
Parameterize the part of the curve which allows an equilateral triangle, with the height 3R, to roll from one vertex to the next one, while its center travels at a constant height.
Homework Equations
I will include some pictures to show what I'm doing
The Attempt at a...
Please see attached image.
When we want to find the area under a curve, we can use the formula
##A = \int_a^b\left|g(x)-f(x)\right|dx## where g(x) is greater than f(x) and both are continuous over the closed interval ##[a,b]##
My text, as seen in the picture, described the area under the curve...
I know that scanning tunneling microscopy (STM) I-V curve is exponential for a conducting sample.
How would it look like if I had a semiconductor and a non-conductor sample?
May I know how to solve the equation as below:
(1) y2 = x3 + x + 1 mod 17
Finding Inverses
Finding Points on the Curve
(2) y2 = x3 + 3x + 1 mod 13
Finding Inverses
Finding Points on the Curve
Please help in the explanation of BH curve (See attached file).
At the starting point, more current is required to bring the material to its magnetic saturation point( in fig form origin to pt.a shown by purple line ) as compared to when the material is fully demagnetized and then brought to...
Homework Statement
Write the equations of tangent lines to the curve of the implicit function x2+2x+2y2-4y=5
that are normal to the line y=x+122. The attempt at a solution
I know that the slope has to be m=-1
I found the derivative using implicit differentiation:
dy/dx=(-2x-2)/(4y-4)
Now I am...
Homework Statement
If I have the two curves
##\phi (t) = ( \cos t , \sin t ) ## with ## t \in [0, 2\pi]##
##\psi(s) = ( \sin 2s , \cos 2s ) ## with ## s \in [\frac{\pi}{4} , \frac{5 \pi}{4} ] ##
My textbook says that they are equivalent because ##\psi(s) = \phi \circ g^{-1}(s) ## where ##...
Homework Statement
I need to obtain the equation of 2 paraboloids separated by a distance L.
Homework Equations
I think that the equations should be:
z_1=x^2+y^2
z_2=x^2+y^2-L
The Attempt at a Solution
The problem is that when I plot the region between two inequations,
x^2+y^2>=z and...
I am very new too Matlab and how it all works but I am having trouble understanding at what axis the numerical integration is occurring from on the graph that I plotted.
So I am currently doing an experiment in gamma ray spectroscopy and due to issue with the software we found it hard to...
Hi all
I was hoping someone could shed some light on the following:-
I am trying to understand what Yield strength is and understand the exact limit of where elastic and plastic deformation occurs on a stress strain curve.
Correct me if I am wrong but I define:-
Yield strength as the amount...
Hi, I'm stuck on a homework problem in my Calculus III class.
I solved 3a really easily, but 3b is giving me a lot of trouble. I know that to find the tangent line, I first have to find the slope, which is represented by the vector:
<3cos^2(t)(-sin(t)), 3sin^2(t)(cos(t))>.
I know the formula...
Homework Statement
A car rounds a level curve of radius 100 meters at a constant speed of 30 m/s.
What is the magnitude of the acceleration of the car?
Homework Equations
C = 2piRThe Attempt at a Solution
I calculated the circumference as 628m. Besides that, I'm ignorant of the procedure to...
Homework Statement
Variables are all given as arbitrary constants
Homework Equations
W=∫F⋅dl
y=½bx^2
μ=cx^2
The Attempt at a Solution
The obvious force action on the object is the downward gravity force. I don't see how to calculate the force that 'moves' the object down the curve or the...
When two magnets already sticked to each other being pulled apart, energy is applied, so I assume that the energy is stored like some sort of potential energy which will be turned back into kinetic energy when they accelerate towards each other to stick back together. So the magnet itself...
Homework Statement
The graph of ##y^2 + 2xy + 40 \mid x \mid =400## divides the plane into regions. The area of the bounded region is?
Homework Equations
All relevant to calculus
The Attempt at a Solution
My question is how to predict the graph without using a grapher here? I put x and y=0 to...
Homework Statement
f
f
Homework Equations
##a = {v^2}{r}##
##d = vt##
The Attempt at a Solution
The way I do it is break it down into the curved and straight segment.
The bicycle is always traveling at a constant speed.
Curved portion:
distance ##\frac{\pi R^2}{4}##
acceleration ##=...
Hi guys.
Anyone knows a article showing the method of extrapolation curve of the phase transition's temperature by the inverse of lattice size, applied at low-dimensional lattices, like nanotube and nanowire, for example?
Thanks a lot!
Hello.
The curve y = x2 is a parabola that looks like this:
I have a shape Square that looks like this:
What I am noticing is that if I consider the equation y = x2 and also the shape Square, I find that there is no connection between them but the equation y = x2 is pronounced as x-square...
For my understanding there should be no big difference in i-v curve between regular p-n junction and p-i-n junction, the only difference i can think about is do to the bigger resistance of the intrinsic layer. The curve should look the same only the current will rise little bit slower, because...
Hi,
The main question revolves around the Rhodonea curve AKA rose curve. The polar equation given for the curve is r=cos(k). The parametric equation is = cos(k(theta)) cos (theta), = cos(k(theta)) sin(theta) . Can anyone show me the conversion from the general parametric form to the general...
Can anyone recommend papers that directly curve-fit redshift as a function of luminosity distance for type Ia supernova and gamma ray bursts? I am looking for papers that do not curve-fit the data via an assumed model, even one as simple as Friedmann–Lemaître–Robertson–Walker (FLRW) metric. I...
Homework Statement
Find the length of the curve:
##x=\frac{t}{1+t}##
##y=\ln(1+t)##
where ##0 \leq t \leq 2##
Length of curve integral ##\int^a_b \sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2} dt##
Homework Equations
##(\frac{dx}{dt})^2 = \frac{1}{(1+t)^4}## ##(\frac{dy}{dt})^2 =...
Let $$\phi(x^1,x^2...,x^n) =c$$ be a surface. What is unit Normal to the surface?
I know how to find equation of normal to a surface. It is given by:
$$\hat{e_{n}}=\frac{\nabla\phi}{|\nabla\phi|}$$However the answer is given using metric tensor which I am not able to derive. Here is the answer...
I have been teaching undergrad students informally, and one of the math problems that I have always enjoyed introducing them to is how to compute the area under a gaussian curve, or to keep it simple, the area under the curve ##z=e^{-x^2}##
One of my students asked me a question that has...
I am working on a synthesizer project and have reached a point that I am stumped on.
I am in this part trying to work from a basic curve of y=1/x^c (where x≥1):
As I understand, area under the curve between x=1 and x=100,000,000 (ie. more than I need for a rough approximation) would be...
Homework Statement
Question attached in attachments
Homework Equations
Area enclosed by polar graph is ∫0.5r^2
where r is the radius as a function of angle theta
The Attempt at a Solution
I attempted to use the formula above and I subtracted the area of the inside from the outside but it...
I was hoping this c++ file could be adapted to use as data compression or encryption
#include <iostream>
#include <cmath>
using namespace std;
double asil(double t)
{
cout<<"asil\n";
cout<< cos(t) << endl;
double two = 2.0;
cout<< sin(sqrt(two)*t) << endl; return sin(sqrt(two)*t);
}
int...
Hello! (Wave)
Using Green's theorem, I want to compute the integral
$$\oint_C ydx+xdy$$
where $C$ has the parametric representation $r(t)=2 \cos^3 t i+ 2 \sin^3 t j, (0 \leq t \leq 2 \pi)$.
Using Green's theorem, we get that $\oint_C ydx+xdy=\iint_U (1-1)dxdy=0$.
I am wondering if we could...
Hi all,
I performed a resonance experiment over the past two weeks, in which I collected the intensity of a Fabry-Perot cavity whilst adjusting the mirror distance with a piezo-element (the specific setup of the experiment is fairly detached from the question I will ask). My raw data is...
Hey guys, I've got this problem I can't seem to get past. I need to find the tangent line to a parametric curve at t=\frac{\pi}{4}
I thought I solved the equation, but my answer doesn't seem to be registered as correct. I'm guessing that means I stuffed up the equation, but I can't see where...
Hello all,
I am trying to build a simulation to understand what power is required by an electric car in order to accelerate from 0-50kmh. As per my understanding the total power required by engine is calculated using Total Power required at given velocity formula as stated below. I am...
I've been in discussions on another board with a physicist and he contends that a pilot in a plane will need to put a 1 degree downward direction input to the controls to account for the 1 degree of curvature rate at 500mph.
as small as those control inputs are required, he says that the...
Homework Statement
Find a piecewise smooth parametric curve to the astroid. The astroid, given by $\phi(\theta) = (cos^3(\theta),sin^3(\theta))$, is not smooth, as we see singular points at 0, pi/2, 3pi/2, and 2pi. However is there a piecewise smooth curve?
Homework Equations
$\phi(\theta) =...