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.
When I take ##x = 2\cos(t)## and ##y = 2\sin(t)##, the integral becomes ##\int_{t=\frac{\pi}{2}}^0 4(2\cos(t))^2 \cdot 2 dt = -8\pi##. The final answer is ##8\pi##. Why is my method wrong?
I played around with desmos and the parameterisation seems correct...
So i am confused as to what can be parallel transported , can an arbitrary tensor be transported along any curve that we wish , or do we define a curve and then solve the equation of parallel transport (which is a linear first order differential equation ) and then the solutions we get from...
So the solution is obviously given here, I'm just trying to understand it. I thought that integrating f(x) from -5 to 5 would give the area under the curve (including the areas below the "pond" at the edges of the image but above y=0. I don't really understand why we are subtracting the integral...
The Importance of Being Symmetric: Flat Rotation Curves from Exact Axisymmetric Static Vacuum Spacetimes
... Analyzing the low-velocity limitcorresponding to the Newtonian approximation of the Schwarzschild metric, we find an effective logarithmic potential. Thisyields flat rotation curves for...
I had to look this up; will need to read on it.
from my research,
https://byjus.com/question-answer/the-equation-of-straight-line-equally-inclined-to-the-axes-and-equidistant-from-the-points-1-2-and-3-4-is-ax-by-c-0-where/
...
I have noted that at equally inclined; the slope value is ##1##...
Hello everyone,
I am curious, suppose you have a function ##f(x)=x^3## and you to find the area under the curve from 0 to x, the area would be ##\frac {x^4}{4}## but this is units of ##L^4## if x is length, but area is units of ##L^2## so what is going on here?
The reason I'm curious is I...
hi , I want to measure the transmission curve and wavelengths of colored mineral glass , which is the simplest way or which instrument can I use ?
thanks
Degree of freedom along a parabola, or any such tame curve, is one from lagrangian mechanics point of view. It makes sense. However how does degree of freedom accompany a space filling curve. Intuitively degree of freedom is not two, since not all motions are possible along the curve. How would...
Currently when set to 1C, the fridge cools to and maintains 4-5C(As a conversion when set to 34f it cools and maintains approx. 40f. ), all functions like fan/defrost are working correctly. Potentially the main board has an issue but all other zones are working correctly so I would prefer to not...
We recently did an experiment to generate the hysteresis curve of a certain material. The experiment involved switching the current in the wire looped around a ring of the material, and recording the first throw of the ballistic galvanometer. I am not going into the details of calculations...
I'm trying to understand the Maxwell-Boltzmann gas molecule speed distribution. Suppose we have a container of gas such that all the molecules are identical.
At first I was under the mistaken impression that one starts with the premise that the distribution of their translational kinetic...
As you can see in this picture: This explanation "relation between the normal and the slope of a curve" is formulated here:
$$\frac{1}{\rho} \frac{d\rho }{d\psi }=\tan\left(\frac{\theta+\psi}{2}\right)$$
I got confused because I don't have the curve equation(regarding the slope of the curve...
I have a several questions on the following block of codes taken from ganfetex01_aux.in:
solve
save outf="ganfetex01_$'index'.str"
extract init inf="ganfetex01_$'index'.str"
extract name="2DEG" 1e-4 * area from curve (depth, impurity="Electron Conc" material="All" mat.occno=1 x.val=0.5) \...
What should I learn to make astrophysical measurements from open data?
Suppose I want to measure the rotation speed of galaxies to generate galactic rotation curves like these: https://en.wikipedia.org/wiki/Galaxy_rotation_curve
What should I do and what should I learn?
I think I should get...
Find ms solution;
My approach;
##xt=t^2+2## and ##yt=t^2-2##
##xt-2=t^2## and ##yt+2=t^2##
##⇒xt-2=yt+2##
##xt-yt=4##
##t(x-y)=4##
##t=\dfrac{4}{x-y}##
We know that;
##x+y=2t##
##x+y=2⋅\dfrac{4}{x-y}##...
Relatively new area to me; will solve one -at- time as i enjoy the weekend with coffee.
1. Unit tangent
##r=xi+yj+zk##
##r=(t-\dfrac{t^3}{3})i+t^2j+(t+\dfrac{t^3}{3})k##
##T=\dfrac{dr}{dt} ⋅\dfrac{dt}{ds}##
##\dfrac{dr}{dt}=(1-t^2)i+2tj+(1+t^2)k##...
I was interviewing an AI chatbot and was wondering how good it is at physics… can anyone confirm whether the answer it gave is true or false? This was through a chatbot called ChatGPT.
My prompt:
What’s the formula for the Brachistochrone curve?
The AI answer:
The Brachistochrone curve is a...
Fix points ##p,q\in \mathbb{R}^n##, and let ##\gamma : [a,b] \to \mathbb{R}^n## be a continuously differentiable curve from ##p## to ##q## whose arclength equals the Euclidean distance between the points, ##|q - p|##. Prove that ##\gamma## lies on the straight line passing through ##p## and ##q##.
ooops...this was a bit tricky but anyway my approach;
...
##\dfrac{dy}{dx}=-2x##
therefore;
##\dfrac{y-7}{x+1}=-2x##
and given that, ##y=4-x^2## then;
##4-x^2-7=-2x^2-2x##
##x^2+2x-3=0##
it follows that, ##(x_1,y_1)=(-3,-5)## and ##(x_2,y_2)=(1,3)##.
There may be another approach...
The implicit curve in question is ##y=\operatorname{arccoth}\left(\sec\left(x\right)+xy\right)##; a portion of the equations graph can be seen below:
In particular, I'm interested in the area bound by the curve, the ##x##-axis and the ##y##-axis. As such, we can restrict the domain to ##[0...
Hi...
i want to draw Equilibrium curve for SO2 oxidation to SO3, i found following relations but don't know to use them... kindly tell me how to draw Equilibrium curve using these equations or any source on web from where i can get directly SO2 oxidation data Vs Temp?
Equations are following...
Hi, good evening/morning/night!
I have a problem with this 2 homework problems about curved beams.
In both I must calculate the stress at points A and B.
My procedure is correct, but for some reason in:
First problem - the effort in A is very close to that of the teacher, but the effort in B...
This question is from a Further Maths paper;
Part (a) is pretty straight forward...No issue here...one has to use chain rule...
Let ##U=\dfrac{e^x+1}{e^x-1}## to realize ##\dfrac{du}{dx}=\dfrac{-2e^x}{(e^x-1)^2}##
and let
##y=\ln u## on taking derivatives, we shall have...
IMPORTANT: NO CALCULATORS
I assumed two points, (a, f(a)) and (b, f(b)) where b is greater than a. Since the tangent line is shared, I did
f'(a) = f'(b):
1) 4a^3 - 4a - 1 = 4b^3 - 4b - 1
2) 4a^3 - 4a = 4b^3 - 4b
3) 4(a^3 - a) = 4(b^3 - b)
4) a^3 - a = b^3 - b
5) a^3 - b^3 = a - b
6) (a...
This is the question...hmmmm it stressed me a little bit.:cool:
Find the textbook solution here; no. 6
Now my approach to this was as follows;
On integration,
##y=\dfrac{(kx-1)^6}{3k} +c##
on using the point ##(0,1)## and ##(1,8)##, we end up with
##1=\dfrac{1}{3k} +c##...
How do we define tangent line to curve accurately ?
I cannot say it is a straight line who intersect the curve in one point because if we draw y = x^2 & make any vertical line, it will intersect the curve and still not the tangent we know. Moreover, tangent line may intersect the curve at other...
I have two 1D matrices X(1,j) and Y(1,j) of equal length. To fit Y to a model asin(bx) I tried:
fit = fittype(@(a,b,X), a*sin(b*X));
[fitted, gof] = fit(X, Y, fit)
coefficients = coeffvalues(fitted)
this gives the message: Error using fittype>iDeduceCoefficients
The independent variable x does...
Hello :
Have a question regarding the mathematical model of reflective curve where could i find information on it ? (pdf , webpages , ebooks ,...etc )
Other than Wikipedia
Best Regards
HB
So I get the exercise and all and have just solved it. But .. I kind of very very intuitively determined ##\theta## to also be the angle for the circular sector.
The problem here is that my geometry bag is very weak, I didn't have any geometry in HS, will fix that sooner or later but anyway...
If C is the simple closed curve in the xy plane not enclosing the origin, how to prove that $\displaystyle\int_C \vec{F}\cdot d\vec{r} =0 $ where $$ F= \frac{yi +xj}{x^2+y^2}$$
How to answer this question? Any math help will be accepted. I am working on this question. If any member of Math...
For part (a);
$$\int e^{3y} \,dy=\int 3x^2\ln x \,dx$$$$\frac{e^{3y}}{3}=x^3\ln x-\frac{x^3}{3}+k$$$$\frac{e^{3}}{3}=e^3-\frac{e^3}{3}+k$$$$\frac{e^{3y}}{3}=x^3\ln x-\frac{x^3}{3}-\frac{e^3}{3}$$$$e^{3y}=3x^3 \ln x-x^3-e^3$$
You may check my working...i do not have the solution.
Find question here;
Find solution here;
I used the same approach as ms- The key points to me were;
* making use of change of variables...
$$A_{1}=\int_0^\frac{π}{4} {\frac {4\cos 2x}{3-\sin 2x}} dx=-2\int_3^{-2} {\frac {du}{u}}= 2\int_2^3 {\frac {du}{u}}=2\ln 3-2\ln2=\ln 9 - \ln 4=\ln...
I have a chemistry project and my research question roughly translates to:
What is the effect of the different metals electrolytic nature on iron corrosion?
(In french: Quel est l’effet de la nature électrolytique de différents métaux (cuivre, aluminium, magnésium) sur la corrosion humide du...
Hello forum,
I am reading this article on quantum machine learning. At one point in the article (page 7) they plot the ROC curve as background rejection vs. signal efficiency. Researching these concepts (since I did not understand them fully), I read that ROC curves should be plotted as TPR...
If there is a car resting on a banked curve with angle theta, velocity v = 0, but N (normal)*sin(theta) > 0. So N*sin(theta) =/= (m*v^2)/r with v = 0. But my physics textbook just defined N*sin(theta) = (m*v^2)/r in banked curve. What is going on here?
I am trying to solve the problem below. I have previously calculated from 0 to 4 seconds how far the rocket will travel in each second. I am stuck now as to how to start this problem. I have searched but unable to find the answer. Do i need to rearrange this? A is currently 14 which does not get...
Given the parameterization of an inverted cycloid:
$$x(t)=r(t-\sin t)$$
$$y(t)=r(1+\cos t)$$
where $$t \in [0, 2\pi]$$.
I am asked to parameterize the curve in its natural parameter. To do it:
$$s=\int_{t_0}^{t} ||\vec{x}'(t*)||dt*$$
The modulus of the squared velocity is...
So it's basically a half circle with radius a.
y = asin(t)
$$\int_0^{\pi} asin(t) dt = -acos(t) |_0^{\pi} = 2a$$
The book says the answer is ##2a^2##, but maybe that's wrong?
What I do is set the two equations equal to one another and solve for z.
This gives:
$$z = \sqrt{x^2+2y^2-4x}$$
which is a surface and not a curve.
What am I doing wrong?
This is just to recall a nice fact:
Any two points ##A,B\in\mathbb{R}^n\backslash\mathbb{Q}^n,\quad n>1## can be connected with a ##C^\infty##-smooth curve that does not intersect ##\mathbb{Q}^n##.
The proof is surprisingly simple: see the attachment
I'm making a program that generates lines in 3D space. One feature that I need is to have an incrementally increasing angle on a line (a bending line / curve).
The problem is simple if the line exists in the xy-plane, then it would be a case of stepping say 1m, increase the azimuthal angle φ...
I have a formula y=log(x)/log(0.9) which has this graph:
I want to find the intersection of this curve and a tangent line illustrated in this rough approximation:
The axes have very different scales, so the line isn't actually a slope of -1, it's just looks that way.
How can I figure out:
1)...