What is Coordinates: Definition and 1000 Discussions
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.
I think the given solution is wrong.
The lens forms image at ##(+75,0)## which is ##25 cm## from pole of the convex mirror which acts as virtual object for mirror.
It is true that the reflected ray is rotated by ##2\theta## as in case of plane mirror. Rotation of Spherical Mirrors
But that...
I have been using Maple for a decade, but only recently started using the Physics Package.
I unfortunately ran into trouble to contract indices ( Maple calls it SumOverRepeatedIndices ).
Below I give an example that will execute if you paste it into a Maple worksheet.
A metric is defined. A...
Dirac in his "GTR" (Chap 19, page 34-35) finds a coordinate system ##(\tau, \rho)## which has no coordinate singularity at ##r=2m##. Explicitly, the transformation looks like (after some algebra):
$$\tau=t + 4m\sqrt{\frac{r}{2m}} + 2m\log{\frac{\sqrt{r/2m}-1}{\sqrt{r/2m}+1}}$$
$$\rho=t +...
For this problem,
My working for finding the coordinates of the mass is,
##x = x_p + x_m = R\cos(\omega t) + l\sin(\phi)##
##y = -y_p - y_m = -R\sin(\omega t) -l\cos\phi##
However, I am told that correct coordinates of the mass is
##x = x_p + x_m = R\cos(\omega t) + l\sin(\phi)##
##y = y_p -...
For this problem,
The correct coordinates are,
However, I am confused how they got them.
So here is my initial diagram. I assume that the point on the vertical circle is rotating counterclockwise, that is, it is rotating from the x-axis to the y-axis.
Thus ## \omega t > 0## for the point...
For this problem,
I correctly got the same coordinates for the pendulum mass using another coordinate system. The coordinate system I used was the other coordinate system rotated counterclockwise by 90 degrees. Why is the pendulum mass coordinates invariant in my cartesian coordinate system...
I think I have completed the exercise but since I have seldom used polar coordinates I would be grateful if someone would check out my work and tell me if I have done everything correctly. Thanks.
My solution follows.
Since ##\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2=1## it follows...
A question that might occur to anyone reading about the Bell Spaceship Paradox is, can we construct a coordinate chart in which all of the Bell observers (i.e., observers following worldlines like those of the spaceships in the "paradox" scenario) are "at rest"? I put "at rest" in scare-quotes...
Learning Galilean transformation and just want to see if I understand the concept well.
both frames are moving relative to some other frame(me standing all the time, not moving). frame A moving 5m/s, frame B moving 7m/s, which in turn means frame B moving 2m/s relative to frame A. Galilleo...
Studying and tinkering with some solutions, I've come to some realizations and questions regarding the regularization of coordinate singularities, so I'd like to see if my conclusions are good, and I guess I have some questions as well. There are two questions/conclusions, but since they...
Has it ever happened that your professor has used left handed co-ords instead of right handed co-ords, which we generally use? Has this ever caused you any confusion.
I found this on 'David J. Griffiths - Introduction to Electrodynamics-Pearson (2013)' and I was wondering if this was something I...
We were taught that in cylindrical coodrinates, the position vector can be expressed as
And then we can write the line element by differentiating to get
.
We can then use this to do a line integral with a vector field along any path. And this seems to be what is done on all questions I've...
Problem:
Solution:
When I looked at an example problem, they started writing the potential in terms of the Legendre polynomials.
The example problem:
This is what I did:
$$V_0 \alpha P_2 (\cos(\theta)) \Rightarrow \frac{\alpha 3 \cos ^2 (\theta)}{2} - \frac{\alpha}{2} \Rightarrow \frac{\alpha...
I calculated the speed using the information provided through the above equation and finding V' = 1.2 m/s.
However, the first solution must be -1,2 m/s. I don't know how to reach it, any suggestion?
Hi. I hope everyone is well. I'm just an old person struggling to make sense of something I've read and I would be very grateful for some assistance. This is one of my first posts and I'm not sure all the LaTeX encoding is working, sorry. Your help pages suggested I add as much detail as...
In physics there is a notation ##\nabla_i U## to refer to the gradient of the scalar function ##U## with respect to the coordinates of the ##i##-th particle, or whatever the case may be.
A question asks me to prove that
$$\nabla_1U(\mathbf{r}_1- \mathbf{r}_2 )=-\nabla_2U(\mathbf{r}_1-...
Would method of separation of variables lead to a solution to the following PDE?
$$ \frac{1}{r} \frac{ \partial}{\partial r} \left( kr \frac{ \partial T}{ \partial r}\right) = \rho c_p \frac{\partial T }{ \partial t }$$
This would be for the transient conduction of a hollow cylinder, of wall...
So given the Helmholtz equation $$\nabla^2 u(x,y,z) + k^2u(x,y,z)=0$$ we do the separation of variables $$u=u_x(x)u_y(y)u_z(z)= u_xu_yu_z$$ and ##k^2 = k_x^2 + k_y^2 +k_z^2## giving three separate equations; $$\nabla^2_x u_x+ k_x^2 u_x=0$$ $$\nabla^2_y u_y+ k_y^2 u_y=0$$ $$\nabla^2_z u_z+ k_z^2...
My issue is in deriving the coordinates of a point on a wheel that rotates without slipping. In Morin's solution he says that:
My attempt at rederiving his equation:
I do not understand how the triangle on the bottom with sides indicated in green is the same as the triangle on top that is...
so I was wondering. there is this normal force on the can from the path. And there's this formula to find the angle between the radial line and the tangent or also between the normal force and either the radial or theta axis. the formula is ##\psi = r/dr/d\theta##. The thing is that here they...
The general intention of Novikov coordinates on Schwarzschild spacetime is to construct a "comoving" coordinate chart for purely radial timelike geodesics, i.e., every such geodesic should have a constant radial coordinate, and the time coordinate should be the same as proper time for observers...
I've tried writing the curl A (in spherical coord.) and equating the components, but I end up with something that is beyond me:
\begin{equation}
{\displaystyle {\begin{aligned}{B_r = \dfrac{1}{4 \pi} \dfrac{-3}{r^4} ( 3\cos^2{\theta} - 1) =\frac {1}{r\sin \theta }}\left({\frac {\partial...
The issue is that the singularity is not in the center of the sphere.
So how would I calculate it?
I have a few questions:
1. Can I calculate the terms separately like so:
$$A = grad(a+b) = grad(a) + grad(b)$$
2. If I use a spherical coordinate system with the center being at the singularity I...
Hi!
For this problem,
Why did the solutions choose to use a different coordinate system? I choose to use the right-handed coordinate system.
Many thanks!
The Hiscock coordinates read:
$$d\tau=(1+\frac{v^2(1-f)}{1-v^2(1-f)^2})dt-\frac{v(1-f)}{1-v^2(1-f)^2}dx$$
##dr=dx-vdt##
Where ##f## is a function of ##r##. Now, in terms of calculating the christoffel symbol ##\Gamma^\tau_{\tau\tau}## of the new metric, where ##g_{\tau\tau}=v^2(1-f)^2-1## and...
I'm not really struggling with the question but the coordinate systems involved more so. So due to the modelling assumptions we know that the tension will be equal throughout the rope so we can use f = ma on each particle respectively and solve the resulting equation (as acceleration will be...
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...
I am looking at a textbook solution to the following problem of finding the equation of motion of a half disk. In the solution, the author considers the half disk has a COM at the black dot, and to find the instantaneous translational velocity of the center of mass (he considers rotational...
here i found AB to be (-3, 2) and then i thought to do 2/5 multiplied by AB to find AC, however this is incorrect and instead i would have to involve the origin. Why and how can i involve the origin?
I am finding the potential everywhere in space due to a point charge a distance 'a' on the z-axis above an infinite xy-plane held at zero potential. This problem is fairly straight forward; place an image charge q' = -q at position -a on the z-axis. I have the solution in cartesian coordinates...
In relativity, no signal travels faster than light, and hence if something happened away from me, I will only know about it after some time. This means I cannot measure instantly the position and time of something as it happens; this would violate special relativity. I however imagine that I...
I made this exercise up to acquire more skill with polar coordinates. The idea is you're given the acceleration vector and have to find the position vector corresponding to it, working in reverse of the image.
My attempts are the following, I proceed using 3 "independent" methods just as you...
Hartle, gravity. Chapter 5
"A four-vector is defined as a directed line segment in four-dimensional flat spacetime in the same way as a three-dimensional vector (to be called a three-vector in this chapter) can be defined as a direcied line segment in three-dimensional Euclidean Space"For...
In a textbook I own a formula is given for the integration of natural coordinates over an element. In this case it is a 1 dimensional element (i.e. a line segment) with coordinates ##x_i## and ##x_j##. The coordinate ##x## over the element is written as:
$$
x = L_1(x) x_i + L_2(x) x_j
$$
with...
It's a topic that's been giving be a headache for some time. I'm not sure if/why/whether I can always consider velocities and (independent) coordinates to be independent, whether in case of cartesian coordinates and velocities or generalized coordinates and velocities.
In the ADM decomposition, like in the construction of the FRW metric, the coordinates are defined to be co-moving, so we know $$d\tau = dt$$ (i.e. the lapse function is normalized away)
Starting from a five-dimensional embedded hyperboloid (as in carroll pg. 324) ## -u^2 + x^2 + y^2 + z^2 + w^2...
Hi PF!
I have a function ##f(s,\theta) = r(s,\theta)\hat r + t(s,\theta)\hat \theta + z(s,\theta)\hat z##. How can I plot such a thing in Mathematica? Surely there's an easier way than decomposing ##\hat r, \hat \theta## into their ##\hat x,\hat y## components and then using ParametricPlot3D?
Hello, I am trying to solve the following problem:
If ##z=f(x,y)##, where ##x=rcos\theta## and ##y=rsin\theta##, find ##\frac {\partial z} {\partial r}## and ##\frac {\partial z} {\partial \theta}## and show that ##\left( \frac {\partial z} {\partial x}\right){^2}+\left( \frac {\partial z}...
I calculate that \mbox{curl}(\vec{e}_{\varphi})=\frac{1}{\rho}\vec{e}_z, where ##\vec{e}_{\rho}##, ##\vec{e}_{\varphi}##, ##\vec{e}_z## are unit vectors of cylindrical coordinate system. Is there any method to spot immediately that ##\mbox{curl}(\vec{e}_{\varphi}) \neq 0 ## without employing...
I’m wondering if there is a formula for calculating the coordinate points of a polygon given the following
- Center point is known
- area is known
- Point A is known
- Points B, C, and D are UNKNOWN
I am NOT a math pro - this is for a puzzle I’m trying to solve and I can’t remember if this...
The wavefunction of ##|\psi\rangle## is given by the bra ket
##\psi (x,y,z)=
\langle r| \psi\rangle##
I can convert the wavefunction from Cartesian to polar and have the wavefunction as ## \psi (r,\theta,\phi)##
What bra should act on the ket ##|\psi\rangle## to give me the wavefunction as ##...