Coordinates Definition and 1000 Threads

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.

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  1. F

    A Aberration with explicit dependance on object coordinates

    Hello, In order to get the coefficients of the aberration expansion with no explicit dependance on object coordinates I fit the optical path difference with the Zernike basis and convert with the paper of Robert K. Tyson "Conversion of Zernike aberration coefficients to Seidel and higher-order...
  2. M

    I Origin of coordinates in multipole expansion

    I am reading Griffiths chapter 3.4.3 on origin of coordinates in multipole expansion (can be found online here https://peppyhare.github.io/r/notes/griffiths/ch3-4/) And I got stuck at this: For the figure 3.22: the dipole moment $p = qd\hat{y}$ and has a corresponding dipole term in the...
  3. kirito

    I Appropriate coordinates for a given electric field

    this is the field I was provided and this is the charge density that I have reached I tried to use this yet the output was different I also used Cartesian it gave me the same output as the spherical ones
  4. Rick16

    I How to find inverse coordinates?

    Is there a systematic way to do it? In particular, I have the coordinates ##x=au \sin v \cos w##, ##y=bu\sin v\sin w##, ##z=cu\cos v##, where a, b, c are constants, and I want to find ##u(x,y,z)##, ##v(x,y,z)##, ##w(x,y,z)##. I could solve the three equations for u, v, and w and then try to...
  5. C

    I Line element in Kerr coordinates at singularity

    From paper 'A brief introduction to Kerr spacetime' ( https://arxiv.org/pdf/0706.0622 ) setting m->0 in the line element in Kerr coordinates gives, equation 7 : $$ \text{d}s_0^{2} = -\left( \text{d}u + a\sin^{2}\theta \text{d}\phi \right)^{2}+2\left( \text{d}u + a\sin^{2}\theta \text{d}\phi...
  6. kal8578

    I Change of coordinates in a potential energy field

    Hello, I am having some confusions in what should be basic pointwise Newtonian mechanics, and would like to get some help with that. It is all about changing coordinates in potential energies. Let us start by considering a point particule in a 2d world with an axis x (left-right) and an axis z...
  7. K

    I A question about Curved Spaces: Gauss and Riemann (Einstein Gravity in a Nutshell by Zee)

    In p. 84, Zee says “In the new coordinates, M is replaced by M’ = R[-1]MR.” However, I figure out M is replaced by M’ = RMR[-1]. Why is M replaced by M’ = R[-1]MR?
  8. cianfa72

    I Frobenius theorem applied to frame fields

    Frobenius's theorem gives necessary and sufficient conditions for smooth distributions ##\mathcal D## defined on a ##n##-dimensional smooth manifold to be completely integrable. Now consider a smooth frame field given by ##n## linearly independent smooth vector fields. I suppose Frobenius's...
  9. P

    I Noethers theorem, transformations of the Lagrange density

    I'm so confused here. If we make the transformation of the coordinates x -> x', are we not suppose to consider the transformation of the coordinates only $$ \phi(x) \rightarrow \phi(x') $$ ? Then why are they writing $$ \phi(x) \rightarrow \phi'(x') $$ ? If $$ \phi(x) $$ is a scalar function...
  10. A

    Is the Solution for Rotation of Spherical Mirrors Incorrect?

    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...
  11. R

    Maple Physics Package Question: Trouble contracting indices

    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...
  12. Kostik

    A Dirac's coordinates ##(\tau, \rho)## for the Schwarzschild metric with ##r \le 2m##

    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 +...
  13. M

    Planar pendulum with rotating pivot

    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 -...
  14. M

    Simple pendulum with moving support

    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...
  15. M

    Pendulum attached to a rotating vertical disk

    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...
  16. L

    Elliptical motion in polar coordinates

    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...
  17. PeterDonis

    A Can We Construct a Coordinate Chart Where Bell Observers Are At Rest?

    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...
  18. H

    I Interpretation of (X,T) coordinates in Kruskal diagram

    These are the points in the book: What is "naturally used"? Does it hold only as the observer crosses the event horizon? How can they "use" them?
  19. G

    I Galilean relativity for 2 frames

    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...
  20. Antarres

    A Continuation of Coordinates Across Black Hole Horizons

    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...
  21. Slimy0233

    Right handed co-ordinates and Left handed co-ordinates

    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...
  22. CaliforniaRoll88

    Find the coordinates of a point in 3-space

    ##\hat a_B=\frac 2 3\hat a_x-\frac 2 3\hat a_y+\frac 1 3\hat a_z## ##\left| \vec r\right |=\sqrt {(x_B-6)^2+(y_B+2)^2+(z_B+4)^2}=10## ##\left |\vec A\right |=\sqrt {6^2+2^2+4^2}=\sqrt {5}{6}## Not sure where to go from here. Please help! Source: Problem 1.3; Engineering Electromagnetics, 8th...
  23. chiyu

    I Vector calculus: line element dr in cylindrical coordinates

    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...
  24. milkism

    Separation of variables in spherical coordinates (electrostatics)

    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...
  25. Atabold

    Einstein relativity between 2 coordinates systems

    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?
  26. DaraRychenkova

    Optimizing Polar Axis for Dipole in Polar Coordinates

    I don't know how to get the result referring to the previous task. Is my decision correct?
  27. MathMan2022

    Find the coordinates of D which lies on the vector BC

    Not sure on howto proceed here?
  28. O

    I Dot product of two vector operators in unusual coordinates

    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...
  29. cwill53

    I Gradient With Respect to a Set of Coordinates

    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-...
  30. E

    I PDE - Heat Equation - Cylindrical Coordinates.

    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...
  31. bob012345

    I Helmholtz Equation in Cartesian Coordinates

    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...
  32. R

    Coordinates of a point on a rotating wheel

    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...
  33. A

    A Stretching Coordinates System/Reductive perturbation theory

  34. S

    Angle between normal force and radial line for cylindrical coordinates

    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...
  35. PeterDonis

    A Two Questions about Novikov Coordinates

    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...
  36. phos19

    Solving Curl A in Spherical Coordinates: Tips & Hints

    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...
  37. Addez123

    How to calculate a sink using spherical coordinates

    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...
  38. M

    Using Right-Handed Coordinates: Exploring Solutions

    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!
  39. Onyx

    B Calc. Christoffel Symbols of Hiscock Coordinates

    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...
  40. nav888

    Can the positive direction be different for each particle in a system?

    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...
  41. chwala

    Find the coordinates of intersection between tangents and given curve

    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...
  42. D

    I Equation of motion: choice of generalized coordinates

    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...
  43. homeworkhelpls

    Question about vector coordinates

    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?
  44. josephsanders

    B Method of images and spherical coordinates

    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...
  45. Ahmed1029

    I Instantaneous coordinates of an event in space (special relativity)

    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...
  46. T

    Integration of acceleration in polar coordinates

    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...
  47. K

    I Understand 4-Vectors & Spacetime: Hartle Gravity Chapter 5

    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...
  48. R

    Troubleshooting Coordinates System from Cone

    I tried using coordinates system from cone, but not got what actually want to get. Any idea from you will greatly appreciated. Thanks
  49. VVS2000

    A Independence of generalized coordinates and generalized velocities

    How can I make sense of this and further how to think of this in the context of phase space diagrams?
  50. A

    I Formula for integration of natural coordinates over an element

    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...
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