Curve Definition and 1000 Threads

##\mathbb{Q}(\zeta_{p^\infty})##, also written as ##\mathbb{Q}(\mu_{p^\infty})## or ##\mathbb{Q}(p^\infty)##, denotes ##\mathbb{Q}## adjoined with the ##p^{n}##th roots of unity for all ##n##. It's the union of a cylotomic tower, and it's studied in subjects like Iwosawa theory and class field...

I created a problem for myself, which I have documented in LaTeX. I hope it's acceptable that I included snapshots of my pages. I would like to calculate the Work [FONT=courier new]W along curve [FONT=courier new]C1. I have solved the problem as shown below. Is my solution correct (see...

For example, let's say the quadratic curve ax^2 + bx + c intersects the x-axis at x=-5, x = 3 Why is it we can say the equation of the curve is then (a)(x+5)(x-3) ? how does this manipulation come about, in particular, the (a) coefficient. Why can we just slot (a) into the factors (x+5)(x-3) ...

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...

The concentration is the Y axis, and the values are the x axis. What is the equation for finding a target value on a curve between set of points?

this is what i created so far, but not quite sure how to get the resonance curve, i tried to enter it in settings and do an AC sweep

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...

Is a worldline a curve or a trajectory? Why?

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) \...

data I'm trying to fit

i integrated y to get (1/3x^3 + 2x) with upper limit 5 / lower limit 2 but got 45 not 175 / 3

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...

Do all galaxies with dark matter halo have flat rotation curve?

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...

What do we know of a curve(/what can it look like) in R^n if we know that κ1,κ2,...,κn-1 is constant?

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...