Parametrization Definition and 90 Threads

Let the origin be where the pendulum string is attached to the ceiling. $$\sin{\theta(t)}=\frac{x(t)}{L}\tag{1}$$ $$\cos{\theta(t)}=\frac{y(t)}{L}\tag{2}$$ $$\theta(t)=\sin^{-1}{\frac{x(t)}{L}}\tag{3}$$ $$\dot{\theta}(t)=\frac{\dot{x}(t)}{\sqrt{L^2-x^2(t)}}\tag{4}$$...

The example goes like this: The group SO(2) is specified by angles ##\theta##. Let's parametrize a path by ##0 \leq t \lt 1## and consider the path ##\theta (t) = 2 \pi t##. Then it says, "There is no smooth function ##\theta (t,u)## for ##0 \leq u \leq 1##, such that ##\theta (t,0) = \theta...

I have studied up to now about forecasts to constrain cosmological parameters in the context of CPL( Chevallier-Polarski-Linder ) parametrization with w_0, ,w_a parameters in equation of state for cosmic fluid. For this, I have used Matter power spectra ("fake data") generated by CAMB and CLASS...

How can i move from this expression: $$\frac{4}{\pi^{4}} \int dk \frac{1}{k^2} \frac{1}{(1+i(k-k_{f}))^3} \frac{1}{(1+i(k-k_{i}))^3}$$ to this one: $$\frac{4}{\pi^{4}} \int dk \frac{1}{k^2} \frac{1}{(1+|k-k_{i}|^2)^2} \frac{1}{(1+|k-k_{f}|^2)^2}$$ using Feynman parametrization (Integration by...

I had been doing some calculations involving propagators with both a quadratic and a linear power of loop momentum in the denominator. In the context of HQET and QCD with strategy of regions. The texts which I am following sometimes tend to straightaway use Schwinger and I am just wondering if...

Let ##x=(x^1,\ldots,x^m)## be local coordinates in a manifold ##M##; and let ##\{\Gamma^i_{jk}(x)\}## be a connection. Assume that we have a curve ##x=x(t),\quad \dot x\ne 0##. Is this curve geodesic or not? My guess is that the answer is "yes" iff for all ##k,n## the function ##x(t)##...

I tried to looking at lower-dimensional cases: For ##n=2## we have $$(x(t),y(t))=(cos(t),sin(t))$$ For ##n=3## we define two orthogonal unit vectors ##\vec{a}## and ##\vec{b}## that are orthogonal to ##\vec{u}##, leading to $$(x(t),y(t),z(t))=(cos(t)(a_1,a_2,a_3)+sin(t)(b_1,b_2,b_3))$$ It seems...

Let us consider a sphere of a unit radius . Therefore, by choosing the canonical spherical coordinates ##\theta## and ##\phi## we have, for the differential length element: $$dl = \sqrt{\dot{\theta}^2+sin^2(\theta)\dot{\phi}^2} $$ In order to find the geodesic we need to extremize the...

Hello, I need the natural parametrization or a geodesic curve contained in the surface z=x^2+y^2, that goes through the origin, with x(s=0)=0, y(s=0)=0, dx/ds (s=0)=cos(a) and dy/ds(s=0)=sin(a), with "a" constant, expressed as a function of the arc length, i.e., I need r(s)=r(x(s),y(s)). Thank...

i) I approximate the solenoid as a cylinder with height L and radius R. I am not sure how I am supposed to place the solenoid in the coordinate system but I think it must be like this, right? The surface occupied by the cylinder can be described by all vectors ##\vec x =(x,y,z)## so that...

To describe the equation of a line, in 2 dimensions, we need a (point on the line + slope to measure slantiness) or two points. Another way: The trajectory of a moving point along the line. Suppose that the moving point initially is at a point of know coordinates r0=(x(t=0), y(t=0), z(t=0)) and...

Hello, In 3D, the trajectory, which is a curve, represents all the points that an object occupies during its motion. Given a certain basis (Cartesian, cylindrical, spherical, etc.), the instantaneous position of the moving object, relative to the origin, along its trajectory can always be...

I'm reading a book on Lie groups and one of the first examples is on SL(2,R). It says that every element of it can be written as the product of a symmetric matrix and a rotation matrix, which I can see, but It also makes the assertion that the symmetric matrix can be parameterized by a...

The equation is $$\|\left(\begin{array} &a\\b\\c\\d\end{array}\right)\|^2=1$$ I was wondering if the number of parameters is 6 and not 3, since we can consider rotations in the differents planes : we choose 2 directions among 4 hence $$C^4_2=6$$ possibilities ?

Hey! :o I want to show that $\iint_{\Sigma}(\nabla\times f)\cdot d\Sigma=0$ for the function $f(x,y,z)=(1,1,1)\times g(x,y,z)$ when $\Sigma$ is the surfcae that is defined by the relations $x^2+y^2+z^2=1$ and $x+y+z\geq 1$. I have done the following: Let $g(x,y,z)=(g_1, g_2, g_3)$. Then...

Hi everyone! I'm a student of electrical engineering. At my math class, we were given a problem to solve at home. Now, from what I've managed to gather, this is a trick question, but I would like to get someone else's opinion on the task. It's also worth mentioning that parametrization is a...

I am reading "Complex Analysis for Mathematics and Engineering" by John H. Mathews and Russel W. Howell (M&H) [Fifth Edition] ... ... I am focused on Section 1.6 The Topology of Complex Numbers ... I need help in fully understanding a remark by M&H ... made just after Example 1.22 ... Example...

I will use an example: -The surface is given by the intersection of the plane: y+z=2 -And the infinite cilinder: x2+y2<=1 We want to parametrize this surface, it could be done easily with: x=r cosθ y=r sin θ z=2 - r cos θ Then this surface could be written using vector notation: S= r...

I have been reading through "Complex Analysis for Mathematics & Engineering" by J. Matthews and R.Howell, and I'm a bit confused about the way in which they have parametrised the opposite orientation of a contour ##\mathcal{C}##. Using their notation, consider a contour ##\mathcal{C}## with...

Hello Forum, In kinematics, the important variables are the velocity v, the acceleration a, and the object's position x. These variables are usually presented as functions of time: x(t), v(t) and a(t). The acceleration can either be constant, or vary with time, i.e. a(t), or vary with position...

An area A in the (x,y) plane is limited by the y-axis and a parabola with the equation x=6-y^2. Further, is a surface F given by the part of the graph for the function h(x,y)=6-x-y^2 which satisfies the conditions x>=0 and z>=0. Determine a parametrization for A and for F. So far I've got the...

I'm given that: S is the surface z =√(x² + y²) and (x − 2)² + 4y² ≤ 1 I tried parametrizing it using polar coordinates setting x = 2 + rcos(θ) y = 2rsin(θ) 0≤θ≤2π, 0≤r≤1 But I'm not getting the ellipse that the original equation for the domain describes So far I've tried dividing everything...

Hello! (Wave) Is $r(t)=(t^2,t^4)$ a parametrization of the parabola $y=x^2$? I have written the following: For $x(t)=t^2$ and $y(t)=t^4$ we have that $y(t)=t^4=(t^2)^2=x^2(t)$, so $r(t)=(t^2,t^4)$ a parametrization of the parabola $y=x^2$. Is it right? How could we say it more formally...

Homework Statement y^2 + 3x - x^3 = C, C\in\mathbb{R}\setminus\{0\} Homework EquationsThe Attempt at a Solution Keeping in mind that ##\cos ^2\alpha + \sin ^2\alpha = 1## I would go about it \left (\frac{y}{\sqrt{C}}\right )^2 + \left (\frac{\sqrt{3x-x^3}}{\sqrt{C}}\right )^2 = 1 would then...

Well, it's physics friday! (carpe diem etc, what else) :) 1. Homework Statement I present to you this (not so) pleasant expression that seemingly appeared on a page out of nowhere. \vec{F}(r, \theta, \varphi) = \frac{F_0}{ar \sin\theta}[(a^2 + ar \sin\theta \cos\varphi)(\sin\theta \hat{r} +...

I've been looking at the torus parametrization \begin{equation} \phi(u,v)=((r\cos u+a)\cos v, (r\cos u +a)\sin v, r\sin u) \end{equation} with \begin{equation}a>0, r\in(0,a)\end{equation}. I want to invert this map to get a chart map for the torus. Can anyone give me a hand with this? Thanks!

Homework Statement I am looking to find the parametrization of the curve found by the intersection of two surfaces. The surfaces are defined by the following equations: z=x^2-y^2 and z=x^2+xy-1 Homework EquationsThe Attempt at a Solution I can't seem to separate the variables well...

A while ago, I read a proof in a book on GR that when using the proper-time parametrization, the two conditions ## \delta \int_{\lambda_1}^{\lambda_2} \sqrt{-g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu} d\lambda=0## and ## \delta \int_{\lambda_1}^{\lambda_2} (-g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu)...

I am reading "Differential Geometry of Curves and Surfaces" by Manfredo do Carmo and on page 156 he gives the following parameterization of the torus x(u,v) = ((a + r cos u )cos v, (a + r cos u)sin v, r sin u) 0 < u < 2*pi, 0 < v < 2*pi Doesnt this leave out some of the torus,? I know that...

Homework Statement The question is completely described in the photo. Homework Equations Trigonometric translation properties The Attempt at a Solution The problem is in two dimensions, so I'm ignoring the z coordinates. For a circle centered at (0,a), the position vector of P is ##(a##...

Hello! There is a parametric way of defining a spiral curve: z = a*t; x = r1*cos(w*t) y = r2*sin(w*t). Is there a way to define the thickness of spiral?

Well if I have a worldline given by x^{\mu}(\tau) And I want to make a Poincare transformation: x^{\mu} (\tau) \rightarrow \Lambda^{\mu}_{\nu} x^{\nu}(\tau) + a^{\mu}. I have one question,why can't a, \Lambda explicitly depend on \tau? that is to have: x^{\mu}(\tau) \rightarrow...

Homework Statement Find the arc length parametrization of the curve r = (3t cost, 3tsint, 2sqrt(2)t^(3/2) ) . Homework Equations s(t)=integral of |r'(t)| dt The Attempt at a Solution I was able to get the integral of the magnitude of the velocity vector to simplify to: s(t) = integral of...

A particle moves along the curve 9x^2 + 16y^2 = 144 a)Find a parametrization of the curve which corresponds to the particle making one trip around the curve in a clockwise direction starting at (4,0) so I know that cos^2t + sin^2t = 1 which is a circle. I also know that x^2 + y^2 = 1 is a...

I have been stuck on the following double integral for some time: ∫(0 to inf) dα1 ∫(0 to inf) dα2 a1^(n1) * exp(-i (α1+α2) m^2) * (α1+α2)^(n2) which arose after using alpha paremetrization on a Feynman integral. I was advised by my supervisor to use the substitution α1 = 1/2 (t+u) and α2 = 1/2...

I'm facing some doubts regarding the parametrization of a given matrix. Let's say, the following matrix is reduced. From: $\begin{bmatrix}0 & 2 & -8\\0 & 2 & 0\\0 & 0 & 2\end{bmatrix}$ To: $\begin{bmatrix}0 & 1 & 0\\0 & 0 & 1\\0 & 0 & 0\end{bmatrix}$ To Parametrize that I would do the...

The geodesic equation for a path X^\mu(s) is: \frac{d}{d s} U^\mu + \Gamma^\mu_{\nu \tau} U^\nu U^\tau = 0 where U^\mu = \frac{d}{ds} X^\mu But this equation is only valid for affine parametrizations of the path. For a timelike path, being affine means that the parameter s must be...

Homework Statement Find the arc length parameterization of r(t) = <(e^t)sin(t),(e^t)cos(t),10e^t>The Attempt at a Solution so I guess i'll start by taking the derivative of r(t)... r'(t) = <e^t*cos(t) + e^t*sin(t), -e^t*sin(t) + e^t*cos(t), 10e^t> ehh... now do I do ds = |r'(t)|dt and...

Problem: Use an appropraite parametrization x=f(r,\theta), y=g(r,\theta) and the corresponding Jacobian such that dx \ dy \ =|J| dr \ d\theta to find the area bounded by the curve x^{2/5}+y^{2/5}=a^{2/5} Attempt at a Solution: I'm not really sure how to find the parametrization. Once I...

Any two dimensional state can be written as: |\phi\rangle=\cos\frac{\theta}{2}|0\rangle+e^{i\phi}\sin\frac{\theta}{2}|1\rangle where 0\leq\theta\leq\pi and 0\leq\phi\leq 2\pi, and 0\leq\theta\leq\pi. To pick one such state uniformly at random it suffices to draw \phi at random from its...

An area A in the xy-plane is defined by the y-axis and by the parabola with the equation x=6-y^2. Furthermore a surface S is given by that part of the graph for the function h(x,y)=6-x-y^2 that satisfies x>=0 and z>=0. I have to parametrisize A and S. Could this be a...

Homework Statement Let ##S## be the portion of the sphere ##x^2 + y^2 + z^2 = 9##, where ##1 \le x^2 + y^2 \le 4## and ##z \ge 0##. Give a parametrization of ##S## using polar coordinates. Homework Equations The Attempt at a Solution ##1 \le r \le 2 \\ 0 \le \theta \le 2\pi \\ 0...

1. The function f(x,y) = x + y 2. The area A is formed by the lines : x = 0 and x = pi/4 and by the graphs : x + cos(x) and x + sin(x) 3. I have to parametrize A 4. 'Formula' : r(u,v) = (u,v*f(u)+(1-v)*g(u)) Could this be a parameterization of A : assuming f(u) = u+ cos(u)...

I have a trivial mathematical problem with SU(2) parametrization. In www.mat.univie.ac.at/~westra/so3su2.pdf , section 3, there is a sentence starting with "We first assume b = 0 and find then(...)". My question is: doesn't assuming that b = 0 reduce generality of our parametrization? If not, why?

Hello. I just wonder if anybody know if there are any rules, when to use parametrization to greens theorem in a vector line integral over a plane. Becouse, it seems sometimes, you have to parametrizice, and other places you dont. I get confused.

I was wondering if anyone knew of a common technique for parametrizing a regular polygon with an arbitrary number of sides. I figured such a problem would be easy or at least be well documented online, but that doesn't seem to be the case. I started by assuming that the polygon was centered...

Homework Statement Find a parametrization of the vertical line passing through the point (7,-4,2) and use z=t as a parameter. Homework Equations r(t) = (a,b,c) + t<x,y,z> The Attempt at a Solution I used (7,-4,2) as (a,b,c) (the point) and used <0,0,1> for the vector since it had...

Hey guys, how come when you have a parametric equation with two free variables it creates a surface, but when you have a parametric equation with one free variable it draws out a line? I sort of get it intuitively, one dimension is just a line, two dimensions is a plane, so I guess this sort of...

Homework Statement Find a parametrization of the equation of the line formed by the points A, B, and P. A(2,-1,3) B(4,3,1) P(3,1,2)Homework Equations x=x_0+v_1*t y=y_0+v_2*t z=z_0+v_3*tThe Attempt at a Solution Alright, so, I've already determined that P is equidistant from the points A and...

Two objects A and B are traveling in opposite direction on a straight line. At t=0 A and B are at positions P(A)=(-40, -20) and P(B)=(190, 980), respectively. If additionally, their paths are parameterized by directions V(A)=(3,5) and V(B)=(-24, -40), respectively. Then, a) find the point...