Coordinates Definition and 1000 Threads

Homework Statement If a staight ε: y=(-λ+μ)x +2λ -μ , (where μ and λ are real numbers) passes through point A(0,1) and is parallel to an other straight lin. ζ: y= -2x + 2008 find λ and μ Homework EquationsThe Attempt at a Solution It is clear that when x=0 we know that 2λ-μ=1 which is one of...

Homework Statement In structural dynamics of multiple degrees of freedom structures, the solution of the following PDE varies with the respect of the applied load, however in numerous literature I have read, the solution is a combination of modal coordinates and modal shapes: $$m \ddot v + c...

Hello, I get that both polar unit vectors, ##\hat{r}## and ##\hat{\theta}##, are unit vectors whose directions varies from point to point in the plane. In polar coordinates, the location of an arbitrary point ##P## on the plane is solely given in terms of one of the unit vector, the vector...

Homework Statement A small bead of mass m slides on a frictionless cylinder of radius R which lies with its cylindrical axis horizontal. At t = 0 , when the bead is at (R,0), vz = 0 and the bead has an initial angular momentum Lo < mR sqrt(Rg) about the axis of the cylinder where g is the...

Hello, For 2D motion, I understand that velocity, position and acceleration of a point object can be described using the fixed basis vector ##\hat {i}## and ##\hat {j}## and the rectangular coordinates ##x(t)## and ##y(t)## which which are functions of time ##t##. Another option is to use...

Homework Statement Homework EquationsThe Attempt at a Solution I have found expressions for the unit vectors for cylindrical coordinates in terms of unit vectors in rectangular coordinates. I have also found the time derivatives of the unit vectors in cylindrical coordinates. However, I am...

The formula for the angle required for you to launch a projectile with a given velocity, gravity, distance and height difference is, taking g as gravity, v as total velocity, x as total distance on the horizontal plane and y as how high the target is above you (Negative value means the target is...

Homework Statement I have phase space coordinates (x0,y0,z0,vx,vy,vz)=(1,0,0,0,1,0). I need to analytically show that these phase space coordinates correspond to a circular orbit. Homework Equations r=sqrt(x^2+y^2+z^2) maybe? The Attempt at a Solution My core problem here is maybe that I...

Homework Statement An electrostatic field ## \mathbf{E}## in a particular region is expressed in cylindrical coordinates ## ( r, \theta, z)## as $$ \mathbf{E} = \frac{\sin{\theta}}{r^{2}} \mathbf{e}_{r} - \frac{\cos{\theta}}{r^{2}} \mathbf{e}_{\theta} $$ Where ##\mathbf{e}_{r}##...

From the FLRW metric Proper distance can be derived like this, $$ds^2=-c^2dt^2+a^2(t)[dr^2+S_k(r)^2d\Omega^2]$$ Let us fixed the time at ##t=t_0## for the measurement and assume that the object has only radial component, then the metric equation turns out to be, $$ds^2=a^2(t_0)dr^2$$...

Homework Statement find the surface area of a sphere shifted R in the z direction using spherical coordinate system. Homework Equations $$S= \int\int \rho^2 sin(\theta) d\theta d\phi$$ $$x^2+y^2+(z-R)^2=R^2$$ The Attempt at a Solution I tried to use the sphere equation mentioned above and...

I have the coordinates of a hurricane at a particular point defined on the surface of a sphere i.e. longitude and latitude. Now I want to transform these coordinates into a axisymmetric representation cylindrical coordinate i.e. radial and azimuth angle. Is there a way to do the mathematical...

This question concerns inertial frames. I am aware that an inertial frame is one that is not accelerating. I am aware of an alternative definition: it is one on which no forces are applied. (Yes, they are the same thing.) I am also aware of the d'Alembert "forces" that appear when a frame is...

I am modeling some dynamical system and I came across integral that I don't know how to solve. I need to integrate vector function f=-xj+yi (i and j are unit vectors of Cartesian coordinate system). I need to integrate this function over a part of spherical shell of radius R. This part is...

Dear all, I am implementing a fluid analysis on cerebral aneurysms and it is obligatory to apply the origin of my coordinate system at the inlet of the aneurysm.I use Ansys Mechanical for the meshing of my geometry which comes with a pre-applied Global coordinate system. I am able to create a...

Dear all, given a dataset of (x,y) coordinates, how can someone determine the equation of a plot created with Origin or Excel that passes through all these points? Depending on the dataset, it is safe to use a trendline ( i.e logarithmic, polynomial) in Excel. The problem is that the graph...

Homework Statement A rocket is to rendezvous with a satellite and needs to make a course adjustment. the rocket has a velocity = (10 + 0 + 0) ms−1 relative to the satellite and mission control has sent a command to the rocket side thruster to exert a thrust = (0 − 100 + 0) N for 100 seconds...

Homework Statement [/B] A particle is moving along a curve described by ##p(t) = Re^{\omega t}## and ##\varphi (t) = \omega t##. What is the particles transverse acceleration? Homework Equations [/B] None The Attempt at a Solution [/B] The position vector is ##Re^{\omega t} \vec{e_p}##...

Homework Statement I am studying co- and contra- variant vectors and I found the video at youtube.com/watch?v=8vBfTyBPu-4 very useful. It discusses the slanted coordinate system above where the X, Y axes are at an angle of α. One can get the components of v either by dropping perpendiculars...

Hi All. I'm new here so I hope you will bear with me and please tell me if this is out of context or in the wrong part of the forum. I am trying to programm a falling pipe system with corners in ArchiCAD (GDL). I have been programming in different languages for a while now, but new to 3D vector...

Homework Statement Find the eigenfunctions and eigenvalues of the isotropic bidimensional harmonic oscillator in polar coordinates. Homework Equations $$H=-\frac{\hbar}{2m}(\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial...

I've started on "Noether's Theorem" by Neuenschwander. This is page 35 of the 2011 edition. We have the Lagrangian for a central force: ##L = \frac12 m(\dot{r}^2 + r^2 \dot{\theta}^2 + r \dot{\phi}^2 \sin^2 \theta) - U(r)## Which gives the canonical momenta: ##p_{\theta} = mr^2...

Hi, I have a general question. How do I show that an operator expressed in spherical coordinates is self adjoint ? e.g. suppose i have the operator i ∂/∂ϕ. If the operator was a function of x I know exactly what to do, just check <ψ|Qψ>=<Qψ|ψ> But what about dr, dphi and d theta

Hello, I am in need of some clarification on tangential velocity in polar coordinates. As far as I know, the tangential velocity vector is ##\vec{v} = v\vec{e_t}##, where ##\vec{e_t} = \frac{\vec{v}}{v}##. This gives us the ##\vec{e_r}## and ##\vec{e_\varphi}## coordinates of the tangential...

Homework Statement I want to convert R = xi + yj + zk into cylindrical coordinates and get the acceleration in cylindrical coordinates. Homework Equations z The Attempt at a Solution I input the equations listed into R giving me: R = i + j + z k Apply chain rule twice: The...

So, let me derive the curl in the cylindrical coordinate system so I can showcase what I get. Let ##x=p\cos\phi##, ##y=p\sin\phi## and ##z=z##. This gives us a line element of ##ds^2 = {dp}^2+p^2{d\phi}^2+{dz}^2## Given that this is an orthogonal coordinate system, our gradient is then ##\nabla...

If the radius unit vector is giving us some direction in spherical coordinates, why do we need the angle vectors or vice versa?

Homework Statement Convert the vector given in spherical coordinates to cylindrical coordinates: \vec{F}(r,\theta,\varphi) = \frac{F_{0}}{arsin\theta}\bigg{[}(a^2 + arsin\theta cos\varphi)(sin\theta \hat{r} + cos\theta \hat{\theta}) - (a^2 + arsin\theta sin\varphi - r^2 sin^2\theta)...

Homework Statement Graph the set of points whose polar coordinates satisfy the given equation or inequality. 0 ≤ θ ≤ , 0 ≤ r ≤ 4 Homework Equations - The Attempt at a Solution Is it correct ?

In the recent thread about the gravitational field of an infinite flat wall PeterDonis posted (indirectly) a link to a mathpages analysis of the scenario. That page (http://www.mathpages.com/home/kmath530/kmath530.htm) produces an ansatz for the metric as follows (I had to re-type the LaTeX -...

Consider an "unidimensional elevator" of size L accelerating w.r.t. a given inertial reference frame. Suppose each elevator's point accelerates with a constant proper acceleration ##g## according Rindler acceleration profile. In the given inertial frame with coordinates ##(x,t)## the elevator...

My question is why isn't the radial component e→r of acceleration in cylindrical coords simply r'' ? If r'' is the rate at which the rate of change of position is changing in the radial direction, wouldn't that make it the radial acceleration? I.e, the acceleration of the radius is the...

If one rotates a tangent plane on a curved surface, this point can be expressed as follows: x' = x cos(theta) - y sin(theta) y' = x sin(theta) + y cos(theta) One solves for x and y and computes based on the deviation of the deviation. My question is: would the answer differ if you choose a...

I wrote the equations of the Nabla, the divergence, the curl, and the Laplacian operators in cylindrical coordinates ##(ρ,φ,z)##. I was wondering how to define the direction of the unit vector ##\hat{φ}##. Can we obtain ##\hat{φ}## by evaluating the cross-product of ##\hat{ρ}## and ##\hat{z}##...

I have 2 points expressed in (latitude,longitude) and I want to calculate the angle with respect to the north pole. Since the two points are very near (like hundred of meters), is it possible to consider the two points in the carthesian system simply as: x=longitude y=latitude Then...

I'm trying to understand the BMS formalism in General Relativity and I'm in doubt with the so-called Bondi Coordinates. In the paper Lectures on the Infrared Structure of Gravity and Gauge Theories Andrew Strominger points out in section 5.1 the following: In the previous sections, flat...

Hello everyone. This was originally a homework problem but I realized my misunderstanding stems from the explanation given before the problem so here we are. The thread deals with these 3 pages from Spivak's Calculus: https://ibb.co/kAKyVU https://ibb.co/jXVSPp https://ibb.co/kwRdVU I'm pretty...

Homework Statement Is ##F=(F_r, F_\theta, F_\varphi)## a conservative force? ##F_r=ar\sin\theta\sin\varphi## ##F_\theta=ar\cos\theta\sin\varphi## ##F_\varphi=ar\cos\varphi## Homework Equations ##\nabla\times F=0## The Attempt at a Solution In this case we have to use the curl for spherical...

I am really confused about coordinate transformations right now, specifically, from cartesian to polar coordinates. A vector in cartesian coordinates is given by ##x=x^i \partial_i## with ##\partial_x, \partial_y \in T_p \mathcal{M}## of some manifold ##\mathcal{M}## and and ##x^i## being some...

How do I calculate the resultant of three component vectors set mutually at 60 degrees to each other (not in the same plane)?

Someone told me that I don't need the whole mechanics of GR to be able to calculate the proper time in an accelerated frame of reference. I can just use SR but with curved coordinates and then integrate for time. But he didn't give me a reference where I could find the formula to do this. How do...

Hi. I was wondering what is the relationship between Gaussian normal coordinates and Riemann normal coordinates. Thanks.

While deriving the volume of sphere formula, I noticed that almost everyone substitute the limits 0 to 360 for the angle (theta) i.e the angle between the positive x-axis and the projection of the radius on the xy plane.Why not 0to 360 for the angle fi (angle between the positive z axis and...

What are the coordinates for the sun? ?

Hello, I come across a problem in programming and I do not find a lot of help on the internet, so I hope I can find an answer here. I have a 3D array representing a function, say f(i,j,k) and a basis function u(i,j,k). I would like to perform a general rotation of the basis function u so that I...

Let's take two orthogonal curves in polar coordinates of the form ##\langle r,\theta \rangle##, say ##\langle r,0\rangle## and ##\langle r,\pi/2\rangle##. Cleary both lines are orthogonal, but the dot product is not zero. This must be since I do not have these vectors in the form ##\langle...

1. The problem statement, all variables and given/known dana I was revisiting University physics textbook and came across this problem. We learned new coordinate systems in classical mechanics classes so I wanted to see if I can apply this to the problem of force on semicircular part of the...

I am beginning to study the mathematics of curvilinear coordinates and all textbooks and web sites do not have realistic examples of non-othogonal systems. What are some examples of non-orthoganal curvilinear coordinates so that I can practice on actual systems rather than generalized examples...

I am having trouble understanding the Kerr metric. One of the things which helped me understand the Schwarzschild metric is the Kruskal–Szekeres coordinates. In particular, the fact that light cones were still at 45 degrees was very helpful, and it was helpful to see that the singularity was a...

Working through an online course "Introduction to General Relativity." They give the metric for, Kottler-Moller coordinates, i.e. $$ds^2=(1+ah)^2d\tau^2-dh^2-dy^2$$ and say that it "covers" the Rindler wedge in flat space time, which is defined by $$0<x<\infty,-x<t<x$$ I am having difficulty...