What is Coordinate: Definition and 908 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.

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  1. banananaz

    MHB How do I find the Euclidean Coordinate Functions of a parametrized curve?

    I've been given a curve α parametrized by t : α (t) = (cos(t), t^2, 0) How would I go about finding the euclidean coordinate functions for this curve? I know how to find euclidean coord. fns. for a vector field, but I am a bit confused here. (Sorry about the formatting)
  2. V

    I Object in or out of a circular field of view? (celestial coordinate system)

    In celestial coordinate system (right ascension/declination), how to check if an object with position RA and dec is within a given circular field of view of radius R (in arcminutes) and centred at (0,0)? R is small in this case so I assumed that I could compute the distance d of the object from...
  3. Athenian

    Finding the Basis Vectors for a Coordinate System

    To my understanding, to get the basis vectors for a given coordinate system (in this case being the elliptic cylindrical coordinate system), I need to do something like shown below, right? $$\hat{\mu}_x = \hat{\mu} \cdot \hat{x}$$ $$\hat{v}_z = \hat{v} \cdot \hat{z}$$ And do that for...
  4. K

    Conversion between vector components in different coordinate systems

    I am not completely sure what the formulas ##v_j = v^a\frac {\partial x^j} {\partial \chi^a}## and ##v^b = v^a\frac {\partial \chi^b} {\partial x^j}## mean. Is ##v_j## the j:th cartesian component of the vector ##\vec v## or could it hold for other bases as well? What does the second equation...
  5. K

    Curvilinear coordinate system: Determine the standardized base vectors

    How I would have guessed you were supposed to solve it: What you are supposed to do is just take the gradients of all the u:s and divide by the absolute value of the gradient? But what formula is that why is the way I did not the correct way to do it? Thanks in advance!
  6. Markus Hanke

    A Coordinate Infall Time for a Vaidya Black Hole

    Consider an observer starting a purely radial free fall from rest at infinity in outgoing Vaidya spacetime - this being a simple model for a radiating black hole. Does anyone have an explicit expression for the coordinate in-fall time (assuming purely radial motion) from infinity to event...
  7. mcastillo356

    Verifying Coordinate System for Electric and Magnetic Forces

    The attached file is the coordinate system I've used a) $$\vec{E}=\dfrac{\vec{F_e}}{q}=\dfrac{1,10\cdot{10^{-13}}\hat{j}\;N}{1,6\cdot{10^{-19}}\;C}=6,88\cdot{10^5}\hat{j}\;N/C$$ b) $$\sum{\vec{F_{net}}}=\vec{0}=\vec{F_e}+\vec{F_m}$$...
  8. Kaguro

    Flux in a rotated cylindrical coordinate system

    ##\vec F= 2x^2y \hat i - y^2 \hat j + 4xz^2 \hat k ## ## \Rightarrow \vec \nabla \cdot \vec F= 4xy-2y+8xz## Let's shift to a rotated cylindrical system with axis on x axis: ##x \to h, y \to \rho cos \phi, z \to \rho sin \phi ## Then our flux, as given by the Divergence theorem is the volume...
  9. Leo Liu

    Is the length of the arm r in a polar coordinate a function of the angle?

    My textbook says ##\vec r (\theta) = r \hat r (\theta)##, where ##\hat r (\theta)## is the terminal arm (a position vector in some sense). It can be seen that both ##\vec r (\theta)## and ##\hat r (\theta) ## are function of ##\theta##; whereas, the length of the vector ##r## is not. I...
  10. S

    I Tangent space basis vectors under a coordinate change

    I'm studying 'Core Principles of Special and General Relativity' by Luscombe - the chapter on tensors. Quoting: The book goes on to talk about a switch to the spherical coordinate system, in which ##\mathbf{r}## is specified as: $$\mathbf{r}=r\sin\theta\cos\phi\ \mathbf{\hat...
  11. agnimusayoti

    Volume in the first octant bounded by the coordinate planes and x + 2y + z = 4.

    First, I try to make a sketch and from that I take limit of integration from: 1. ##z_1 = 0## to ##z_2 = 4 - x -2y## 2. ##x_1 = 0## to## x_2 = 4- 2y ## 3. ##y_1 = 0## to ##y_2 = 2## Then, I define infinitesimal volume element in the first octant as ##dV = 1/8 dz dz dy##. Therefore, $$V=1/8...
  12. Z

    A Curvilinear Coordinate System

    Hello, the physical domain in the (y, z) space is mapped to a rectangular computational region in the (ŋ,Ƹ)-space, where (ŋ,Ƹ) are the new coordinates. This technique frees the computational simulation from geometry restriction. after transforming the governing equations ( PDEs) to the...
  13. hilbert2

    A Ground state energy of a particle-in-a-box in coordinate scaling

    The energy spectrum of a particle in 1D box is known to be ##E_n = \frac{h^2 n^2}{8mL^2}##, with ##L## the width of the potential well. In 3D, the ground state energy of both cubic and spherical boxes is also proportional to the reciprocal square of the side length or diameter. Does this...
  14. Arman777

    I The picture of the Comoving coordinate

    I am trying to understand the picture of the metric in terms of the comoving coordinates but it become really confusing for me beacuse every book uses different notation for the same things. So Let's suppose we have a flat 3D Euclidian Space, we can write the metric as, $$dl^2 = dx^2 + dy^2 +...
  15. E

    B Clarification of coordinate fictitious forces

    I was reading through this Wikipedia article and stumbled across a section related to outlining the differences between "state-of-motion" fictitious forces and "coordinate" fictitious forces. I have no idea what the second category is supposed to be, and wondered whether someone could explain...
  16. O

    I Coordinate Systems After Deformation of Axes

    Disclaimer: I am a physics student and I have very little knowledge of topology or differential geometry. I don't necessarily expect a complete answer to this question, but I haven't really found any reference that approaches what I'm trying to ask, so I'd be quite happy to simply be pointed in...
  17. Arman777

    I Understanding the property of the Comoving Coordinate

    In Weinberg's Cosmology, the comoving coordinate described as "A particle at rest in these coordinates will, therefore, stay at rest, so these are co-moving coordinates" Now when we write the proper distance ##s = a(t)\chi## where ##\chi## is the comoving coordinate. Taking the time...
  18. T

    A 11 coordinate system for separation of variables

    Good Morning I have a very vague memory of having read (about 40 years ago) that there are only 11 coordinate systems in which the field equations of physics can be separated. I can no longer be sure if my memory has failed me. But this issue has been in my head for all these years. (Gotta do...
  19. S

    Infinitesimal coordinate transformation of the metric

    I kinda know how to do this problem, it is just that I hit a sign problem. If I take the partial derivative of the coordinate transformation with respect to ##x'^\mu##, I get writing it first in the inverse form, ##x^\alpha = x'^\alpha - \epsilon^\alpha## ##\frac{\partial x^\alpha}{\partial...
  20. T

    I Coordinate dependence of recession velocities

    Superluminal recession velocities of far away galaxies are due to the choice of FRW-coordinates. As @Ibix said here #71 "The key point, in this context, is that you will never see these galaxies overtake a light pulse." But is there any other choice? Riemann normal coordinates don't seem to be...
  21. Ayoub Tamin

    I Lewis H Ryder: Cartesian to Polar Coord Transformations

    The example is about the transformation between the cartesian coordinates and polar coordinates using the definition In lewis Ryder's solution, I got confused in this specific line I really can't see how is that straightforward to find?
  22. S

    Schwarzschild coordinate time integral

    I have tried integration by parts where, ##c dt = -\frac{1}{\sqrt{r*}} \frac{r^{3/2} dr}{r - r*} = \frac{1}{\sqrt{(r*)^3}} \frac{r^{3/2} dr}{1 - \Big(\sqrt{\frac{r}{r*}} \Big)^2}## ##u = r^{3/2} \quad \quad dv = \frac{dr}{1 - \Big(\sqrt{\frac{r}{r*}} \Big)^2}## ##du = \frac{3}{2} r^{1/2} dr...
  23. E

    Weirdness with this relative velocity formula when defining coordinate systems differently

    Wikipedia gives, "The relative velocity ##{\displaystyle {\vec {v}}_{B\mid A}}## is the velocity of an object or observer B in the rest frame of another object or observer A." Suppose the coordinate system being used in the rest frame of ##A## is has its origin slightly displaced from ##A##...
  24. BadgerBadger92

    B Time in Cartesian Coordinate Systems: Math Q&A

    I am teaching myself math and have a question about cartesian coordinate systems. How is time illustrated in such a graph? [Moderator's note: Moved from a math forum after post #13.]
  25. P

    Show that the metric tensor is independent of coordinate choice

    I need to use some property of the relalation between the coordinate systems to prove that g_{hk} is independent of the choice of the underlying rectangular coordinate system. I will try to borrow an idea from basic linear algebra. I expect any transformation between the rectangular systems to...
  26. L

    Graphing θ=π/4 on a Polar Coordinate System

    When you graph something like ##θ=\frac{π}{4}## on a Polar Coordinate System: Why does the line go into the opposite quadrant as well? I can intuitively understand why it is in the first quadrant: ##θ = 45°## there and so all possible values of ##r## would apply there, giving you a straight line...
  27. K

    I Questions on Galactic Coordinate Systems

    Please refer to article in Wikipedia https://en.wikipedia.org/wiki/Galactic_coordinate_system The following questions are related to the galactic coordinate system: Is the galactic center located on the galactic plane? Since our Sun is above the center of the galactic disk, is the galactic...
  28. C

    I Two Dimensional Coordinate Plane with Distance as Third Dimension

    Imagine we draw a two dimensional finite plane with coordinate axes; for simplicity, let's make it a square. Now, suppose we add a third dimension that represents the possible distances between any two points on the square. Now we have a three dimensional space. What shape will that space have...
  29. A

    New coordinate system and point location in new system

    hi all, firstly i need to calculate point location in new coordinate system. ı have 2 line segments and a point(x,y,z) in word cartesian coordinates system. For example, my first line segment is (0,0,5) , (50,0,3) locations and second line segment is (0,6,3),(0,-6,7) locations in cartesian...
  30. M

    I Coordinate systems parameterized by pseudo arc-length

    Hi PF! Can anyone help me define a coordinate system for a circular arc that makes a specified angle ##\alpha## with a 90 degree wedge? See picture titled Geo. As an example, a circular arc can be parameterized over a straight line by ##s##, making angle ##\alpha##, via $$\vec T = \left\langle...
  31. A

    Coordinate transformations on the Minkowski metric

    The line element given corresponds to the metric: $$g = \begin{bmatrix}a^2t^2-c^2 & at & 0 & 0\\at & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{bmatrix}$$ Using the adjugate method: ##g^{-1}=\frac{1}{|g|}\tilde{g}## where ##\tilde{g}## is the adjugate of ##g##. This gives me...
  32. D

    I What is the ambiguity surrounding coordinate differentials in Relativity?

    There is an ambiguity in certain texts that I want to clarify, atleast it seems ambiguous to me. When describing the differential line element in Relativity by the differential dx's, are they to be infinitesimal vectors or just infinitesimal increments. They are labeled coordinate differentials...
  33. L

    Differentiating with coordinate transformations

    T = (x+\frac{1}{\alpha}) sinh(\alpha t) X = (x+\frac{1}{\alpha}) cosh(\alpha t) - \frac{1}{\alpha} Objective is to show that ds^2 = -(1 +\alpha x)^2 dt^2 + dx^2 via finding dT and dX and inserting them into ds^2 = -dT^2 + dX^2 Incorrect attempt #1: dT= (dx+\frac{1}{\alpha})...
  34. glmhd

    I How to make something independent of the coordinate frame?

    In page 49, chap 8 of the book "classical mechanics point particles and relativity" of Greiner, there is the following sentence: "In order to become independent of the coordinate frame, a set of orthogonal unit vectors is put at the point of the trajectory of the mass point given by ##s##."...
  35. P

    I Coordinate time between spatially separated events in Schwarzschild spacetime

    Edit: I'm leaving the original post as is, but after discussion I'm not confused over coordinate time having a physical meaning. I was confused over a particular use of a coordinate time difference to solve a problem, in which a particular coordinate time interval for a particular choice of...
  36. torito_verdejo

    Advantages of Polar Coordinate System & Rotating Unit Vectors

    What is the advantage of using a polar coordinate system with rotating unit vectors? Kleppner's and Kolenkow's An Introduction to Mechanics states that base vectors ##\mathbf{ \hat{r}}## and ##\mathbf{\hat{\theta}}## have a variable direction, such that for a Cartesian coordinates system's base...
  37. S

    MHB Partially unpicking coordinate rotation

    I measured a vector many times, and then processed the data using a computer program. The program did a great many useful things, including rotate the coordinate system about all three axes. I have measured values for x, y, and z along the original axes. The program helpfully gave me the values...
  38. bob012345

    I Proper vs. coordinate acceleration

    [Moderator's note: Spun off from previous thread due to topic change.] Can you show me a situation where Newton says things accelerate and Einstein says they don't? Being in free fall doesn't mean things don't accelerate. I drop a ball towards the Earth and it not only accelerates but it has a...
  39. B

    Why Does d/dx Not Equal d/dx' Even When x Equals x'?

    Summary: I'm stuck on this simple excersize, to show that in this coord transform, despite x = x', d/dx != d/dx' From "Intro to Smooth Manifolds" (this is a calculus excersize), The Problem I have is with showing d/dx != d/dx' When I write out the Jacobian matrix, I get exactly d/dx = d/dx'...
  40. RichardWattUK

    I "Map" point between coordinate systems

    I've taken on a new job recently where I'm having to maintain an existing application that generates a points profile to drive a CNC machine and part of it projects points from an axial plane (which represents the machine's working axis) onto another plane which (I think) acts as as a...
  41. M

    Convert cylindrical coordinate displacement to Cartesian

    Summary: I can't figure out how the solver carries out the conversions from cartesian to cylindrical coordinates and vice-versa. I have a set of points of a finite element mesh which when inputted into a solver (ansys) gives the displacement of each node. I can get the displacement values of...
  42. colemc20

    Hollow Sphere Inertia in Cartesian Coordinates

    Problem Statement: How do you calculate the rotational inertia of a hollow sphere in cartesian (x,y) coordinates? Relevant Equations: I=Mr^2 My physics teacher said its his goal to figure this out before he dies. He has personally solved all objects inertias in cartesian coordinates but can't...
  43. olgerm

    The derivative of velocity with respect to a coordinate

    Why ##\frac{\partial (0.5*m*\dot{x}^2-m*g*x)}{\partial x}=-mg##? why ##\frac{\partial \dot{x}}{\partial x}=0##? Why ##\frac{\partial (0.5*m*\dot{x}^2-m*g*x)}{\partial \dot{x}}=m*\dot{x}## ? why ##\frac{\partial x}{\partial \dot{x}}=0##? Does it assume that speed is same at every location? I...
  44. V

    I Geodesics under coordinate transformation

    I will start with an example. Consider components of metric tensor g' in a coordinate system $$ g'= \begin{pmatrix} xy & 1 \\ 1 & xy \\ \end{pmatrix} $$ We can find a transformation rule which brings g' to euclidean metric g=\begin{pmatrix} 1 & 0 \\ 0 & 1\\ \end{pmatrix}, namely...
  45. W

    A Coord Transf. in Linearized GR: Understanding Metric Transformation

    I was studying linearized GR where we make the following coordinate transformation ## \tilde{x}^{a} = x^{a} + \epsilon y^{a}(x) ## This coordinate transformation is then meant to imply ## g_{ab}(x) = \tilde{g}_{ab}(x) + \epsilon \mathcal{L}_{Y} g_{ab} ## Would anyone be kind enough to explain...
  46. F

    MHB Find area of a triangle in coordinate plane

  47. K

    B Definition of coordinate system

    In light of the modern definition of what is a coordinate system, namely it's a pair (U, f) with U a region of a m-dimensional manifold, and f a bijection from U to ##\mathbb R^m##, can we say that the polar coordinates on ##\mathbb R^2## are a coordinate system? I was thinking about this and...
  48. looseleaf

    I Gaussian Integral Coordinate Change

    Hi everyone, sorry for the basic question. But I was just wondering how one does the explicit coordinate change from dxdy to dr in the polar-coordinates method for solving the gaussian. I can appreciate that using the polar element and integrating from 0 to inf covers the same area, but how do...
  49. C

    Denavit–Hartenberg Coordinate System/Parameters

    Homework Statement Identify the links/joints, coordinate system, and DH parameters for the robot shown in the picture. *See attached figure Homework Equations Basic knowledge of DH conventions. The Attempt at a Solution *See attached attempts In terms of identifying the joints and links I'm...
  50. Z

    Kinematics in Cylindrical Coordinates

    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...
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