What is Coordinate: Definition and 909 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. zonde

    I Coordinate singularity at Schwarzschild radius

    I would like to ask how rigorous is the statement that Schwarzschild metric has coordinate singularity at Schwarzschild radius. The argument is that singularity at Schwarzschild radius appears because of bad choice of coordinates and can be removed by different choice of coordinates. However...
  2. MichPod

    A Reasonable measurement of both coordinate and momentum?

    Can a reasonable observable operator be defined which measures a two-component observable, first component for the approximate coordinate and the second for the approximate momentum (so that the precision of each measurement do not contradict Heisenberg inequality)? I am actually thinking of...
  3. vibhuav

    I Evaluating metric tensor in a primed coordinate system

    I am trying to learn GR. In two of the books on tensors, there is an example of evaluating the inertia tensor in a primed coordinate system (for example, a rotated one) from that in an unprimed coordinate system using the eqn. ##I’ = R I R^{-1}## where R is the transformation matrix and...
  4. JTC

    I Coordinate systems vs. Euclidean space

    Good Morning I am having some trouble categorizing a few concepts (I made the one that is critical to this post to be BOLD) Remote parallelism: the ability to move coordinate systems and frames around in space. Euclidean Space Coordinate systems: Cartesian vs. cylindrical I am aware that if...
  5. S

    Find Coordinates of Closest Point on y=sqrt(x) to (4,0)

    Homework Statement Find the coordinates of the point ##P(x,y)## on the curve ##y = \sqrt{x}## that is closest to the point ##(4,0)##. Homework EquationsThe Attempt at a Solution The derivative is ##y'(x) = \frac{1}{2\sqrt{x}}##. Do I then find the tangent line to ##y = \sqrt{x}##. A little...
  6. T

    I Evaluating the integral in spherical coordinates - how to do it correctly?

    I should evaluate ##\int d^3 p \ \exp(i \vec{p} \cdot \vec{x}) / \sqrt{|p| + m^2}## over all ##\mathbb{R}^3##. How can I do this in spherical coordinates? Since ##\vec{p}## is a position vector in ##\mathbb{R}^3##, our ##\vec{r}## of the spherical coordinates would be just equal to ##\vec{p}##...
  7. D

    Finding the volume surrounded by a curve using polar coordinate

    Homework Statement I tried to answer the following questions is about the curve surface z= f (x, y) = x^2 + y^2 in the xyz space. And the three questions related to each otherA.) Find the tangent plane equation at the point (a, b, a^2+ b^2) in curved surface z . The equation of the...
  8. E

    Curvilinear coordinate adaptive method in COMSOL

    Hello everyone, In the COMSOL v5.1 doc i haven't found the mathematical description on how does the automatic geometry analysis or the curvilinear coordinates adaptive method (it is mentioned that they are similar) works. It would be convenient to have an idea about how does the Ecoil vector...
  9. S

    I Nearly Lorentz Coordinate Systems: Is h a Tensor?

    Hello! I am reading Schutz A first course in GR and he introduces the Nearly Lorentz coordinate systems as ones having a metric such that ##g_{\alpha\beta} = \eta_{\alpha\beta} + h_{\alpha\beta}##, with h a small deviation from the normal Minkowski metric. Then he introduces the Background...
  10. Frankenstein19

    Calculate the probability that the particle's x coordinate

    Homework Statement For a 1.0 × 10-26 g particle in a box whose ends are at x = 0 and x = 2.000 Å, calculate the probability that the particle's x coordinate is between 1.6000 and 1.6001 Å if n=1 Homework Equations The Attempt at a Solution I know that since the interval between 1.6000 and...
  11. M

    Electrostatics Boundary condition and coordinate choice

    Homework Statement So I have an equation V = Ae(kx)+Be(-kx) And boundary conditons V= V0 when x=0 and V= 0 when x=b 2. Homework Equations I have solved ones where v=0 at x=0 where it nicely simplifies as the exponentials =1 and the Coeffecients A=-B which leads to a sinh function and I...
  12. Q

    I Coordinate transformation - Rotation

    How author derives these old basis unit vectors in terms of new basis vectors ? Please don't explain in two words. \hat{e}_x = cos(\varphi)\hat{e}'_x - sin(\varphi)\hat{e}'_y \hat{e}_y = sin(\varphi)\hat{e}'_x + cos(\varphi)\hat{e}'_y
  13. H

    A Correct coordinate transformation from Poincare-AdS##_3## to global AdS##_3##

    Consider the transformation from Poincare-AdS##_3## geometry to global AdS##_3## geometry: $$ds^{2} = \frac{dr^{2}}{r^{2}} + r^{2}g_{\alpha\beta}dx^{\alpha}dx^{\beta}, \qquad \text{Poincare-AdS$_3$}$$ $$ds^{2} = \frac{dr^{2}}{r^{2}} + r^{2}\left(-dt^{2}+r^{2}d\phi^{2}\right), \qquad...
  14. allanwinters

    Basically solved, Last coordinate does not match?

    Homework Statement Does the line with equation (x, y, z) = (5, -4, 6) + u(1,4,-1) lie in the plane with equation (x, y, z) = (3, 0, 2) + s(1,1,-1) + t(2, -1, 1)? Justify your answer algebraically. Homework Equations (x,y,z) = (x0,y0,z0) +s(a1.a2,b3) + t(b1,b2,b3) The Attempt at a Solution...
  15. M

    Y coordinate of a charged particle

    Homework Statement On the diagram, a charged particle of charge 0.000003 C and mass 0.000007 kg moves across the electric field 6760 V/m with initial speed 40 m/s. When its x coordinate is 93.3 cm, its y coordinate is (in cm)? Homework Equations y=(e*Em*x^2)/(2*m*v^2), where Em is electric...
  16. F

    Average y coordinate of points on parametrized semicircle

    Homework Statement Find the average y coordinate of the points on the semicircle parametrized by C:[0,##\pi##]-->##R^3##, ##\theta##-->(0, a*sin##\theta##, a*cos##\theta##); a>0 Homework EquationsThe Attempt at a Solution I think the answer should be an integral of the circle in the y...
  17. F

    I Use of irrational numbers for coordinate system

    Why should a person prefer irrational coordinate system over rational? My friend stated that its because most lines such as ##y=e## cannot be plotted on a rational grid system. But that cannot be true since ##e## does have a rational number summation ##2+1/10+7/100...## which can be utilised to...
  18. karush

    MHB Evaluating Improper Integrals in Polar Coordinates

    15.3.65 Improper integral arise in polar coordinates $\textsf{Improper integral arise in polar coordinates when the radial coordinate r becomes arbitrarily large.}$ $\textsf{Under certain conditions, these integrals are treated in the usual way shown below.}$ \begin{align*}\displaystyle...
  19. S

    Cartesian to curvilinear coordinate transformations

    Homework Statement Is there a more intuitive way of thinking or calculating the transformation between coordinates of a field or any given vector? The E&M book I'm using right now likes to use the vector field ## \vec F\ = \frac {\vec x} {r^3} ## where r is the magnitude of ## \vec x...
  20. J

    Change in a vector upon rotation of the coordinate frame

    Homework Statement Hi everyone. We were discussing conservation of angular momentum as a consequence of rotational invariance in class. There was one point where we needed to compute the change in a vector A when the coordinate frame is rotated by angle Δ(Φ). Homework Equations The teacher...
  21. OcaliptusP

    Particle's Motion in XY Coordinate System

    Homework Statement Coordinates of a particle which moves on a xy coordinate system given with: x=-(5m)sinωt y=(4m)-(5m)cosωt In these equlations t's unit given as second, and ω's unit second^-1. A-) Found velocity and acceleration components when t=0 B-) Write equlations for position and...
  22. U

    Rectangular to Spherical Coordinate conversion....

    Homework Statement Convert from rectangular to spherical coordinates. (-(sqrt3)/2 , 3/2 , 1) Homework Equations We know the given equations are ρ = sqrt(x^2 + y^2 + z^2) tan theta = y/x cos φ = z / ρ The Attempt at a Solution My answer was (2, -pi/3, pi/3) It should be a simple plug and go...
  23. Tursinbay

    I Metric transformation under coordinate transformation

    In the second volume, Field Theory, of popular series of Theoretical Physics by Landau-Lifschitz are given following equations as in attached file from the book. Here is considered metric change under coordinate transformation. How is the new, prime metric expressed in original coordinates is...
  24. donaldparida

    Find the angle between this vector and the coordinate axes

    Homework Statement : [/B] r vector = 3t i + (4t-5t2)j. Find the angle made by the vector with respect to the x-axis and the y-axis. 2. Homework Equations : A.B=AxBx+AyBy 3. The Attempt at a Solution : I tried to take the dot product of the unit vector along x-axis and r. I did the same...
  25. M

    A Brain Overload: Comparing Proper Time, Ephemeris Time & Coordinate Time

    Currently reading the following document which is a bit of a brain overload at the minute! Im considering Equation (4.61). It is the general relativistic correction due to the Schwarzschild field for a near Earth satellite when the parameters \beta, \;\gamma \equiv 1. However, as you will...
  26. C

    I Coordinate Charts: Clarifying Misunderstanding

    I feel embarrassed to ask this, but I may have a misunderstanding in my understanding of some basics. I was told that ##\psi: U \rightarrow \psi(U)##, where ##U = (0, \infty) \times (0, \pi) \times (-\pi, \pi)## and ##\psi(\rho, \varphi, \theta) = (\rho\cos\theta\sin\varphi...
  27. Bunny-chan

    Potential gravitational energy coordinate axis

    Homework Statement I know that potential gravitational energy is relative to the reference point that I decide to choose (like in the picture below). But then if, for instance, I set my reference point in the ceiling and my vertically down y-axis to be positive. What would the potential...
  28. chinmay

    Handling Rotational Degrees of Freedom in Coordinate Transformations

    I am trying to analyse response of a dynamic system. The source disturbance is about x,y,theta (rotation about x ) & Phi of one coordinate system (red coloured coordinate system in the attached figure). I need to get the response in another coordinate system ( green coloured coordinate system...
  29. C

    I Coordinate system vs ordered basis

    I have an issue with the definition of coordinate system in differential geometry vs the definition of coordinate system in linear algebra. The post is a bit long, but it's necessary so that I get my point across. Let ##V## be an ##n##-dimensional normed space over the reals and equip ##V##...
  30. J

    A Hyperbolic Coordinate Transformation in n-Sphere

    ##x= r Cosh\theta## ##y= r Sinh\theta## In 2D, the radius of hyperbolic circle is given by: ##\sqrt{x^2-y^2}##, which is r. What about in 3D, 4D and higher dimensions. In 3D, is the radius ##\sqrt{x^2-y^2-z^2}##? Does one call them hyperbolic n-Sphere? How is the radius defined in these...
  31. Grimble

    I Proper (and coordinate) times re the Twin paradox

    This was straying from the point in the original thread, but I thought it made a point... The stay-at -home twin is at rest in her frame and her clock must therefore measure proper time. The traveling twin, carries his clock with him; it is therefore at rest in his frame and must also measure...
  32. vanhees71

    I Tensor Invariance and Coordinate Variance

    <This thread is a spin-off from another discussion. Cp. https://www.physicsforums.com/threads/wedge-product.914621/#post-5762138> Also again, be warned about this sloppy notation of indizes. You should put the prime on the symbol (or in addition to the symbol). Otherwise the equations don't...
  33. M

    Framework Problem: Constructing Engine Mount for Airplane

    Hi! I'm currently doing a project where I'm constructing a framework for an engine mount connecting an airplane with an engine. The project involves both calculations by hand and with CAD(Creo), and i have no problem with the CAD part as i have done the simulations. The part with doing...
  34. S

    A Is Non-Zero Metric Determinant Enough for a Global Coordinate System?

    Is there a universal criteria to determine if a coordinate system is global? I think that it is sufficient for the determinant of the metric to be non-zero in order for a coordinate system to be global. Is this so? For example, take the metric ##ds^{2} = \ell^{2}(-\cosh^{2}\rho\ dt^{2} +...
  35. K

    I Linear transformation of a given coordinate

    I have a question about weights of a basis set with respect to the other basis set of one specific vector space. It seems the weights do not covert linearly when basis sets convert linearly. I've got this question from the video on youtube "linear transformation" Let's consider a vector space...
  36. 2

    Switching coordinate system of a field

    Homework Statement Say I have some sort of a vector field in the cylindrical coordinate system \vec{F}(r, \Theta, z) = f(\vec{A}(r,\Theta,z),\vec{B}(r,\Theta,z)) How do I switch to the Cartesian coordinates? More precisely, how do I transform A_r = g(A_x,A_y,A_z), A_\Theta = h(A_x,A_y,A_z)...
  37. parshyaa

    Find the third coordinate of a vertex of an equilateral triangle

    Homework Statement Q. Prove that If (x1,y1) and (x2,y2) are the coordinates of the two vertices of an Equilateral Triangle then the coordinates of the 3rd vertex (X,Y) are $$X=\frac{x1+x2\pm\ √3(y1-y2)}{2},$$ $$Y=\frac{y1+y2\pm\ √3(x1-x2)}{2},$$ The Attempt at a Solution I used distance...
  38. ytht100

    Coordinate transformation: derivative of spherical coordinate with cartesian coordinate

    I have the following equations: \left\{ \begin{array}{l} x = \sin \theta \cos \varphi \\ y = \sin \theta \cos \varphi \\ z = \cos \theta \end{array} \right. Assume \vec r = (x,y,z), which is a 1*3 vector. Obviously, x, y, and z are related to each other. Now I want to calculate \frac{{\partial...
  39. I

    Geometry Alternative to SL Loney's Coordinate Geometry

    Hello In India, SL Loney's Elements of Coordinate Geometry is very popular for entrance examinations. I wanted to refresh my coordinate geometry, so tried reading through the book. But I found that the language used is old. I found myself referring to current material on the topic to properly...
  40. yecko

    Tangent slope with polar coordinate

    Homework Statement http://i.imgur.com/4FPnTNS.jpg Homework Equations (Written in above photo) The Attempt at a Solution (Written in above photo) I have tried hard in figuring out what's wong I have done done, but what I finally got is still option d instead of the model answer e. Are there...
  41. N

    B Coordinate of side of a irregular polgyon

    Hey all, I have a list of line lengths and angles, but only the angles between line n and n-1, can't find a single expression to get the coordinate that works for all cases, i tried \sum{\sqrt{\frac{L^2 -c^2}{tan(\sum{\theta})^2+1}}} and similar expressions but they all assume triangles can be...
  42. M

    Coordinate geometry - centroid (SL LONEY exercise problem)

    Homework Statement If G be the centroid of ΔABC and O be any other point, prove that , ## 3(GA^2 + GB^2 + GC^2)=BC^2+CA^2+AB^2## ##and,## ##OA^2 + OB^2 + OC^2 = GA^2.GB^2+GC^2+3GO^2## Homework Equations i m practising from S L LONEY coordinate geometry first chapter ... only the equation...
  43. J

    Changing the coordinate system of the hands of clock

    I want to understand what changing coordinate system means for hands of clock. Let's say the clock only has hour and minute hand. It can move let's say just in the upper 180 deg. of the clock (as shown in the figure). The area between the two hands is V1, and the rest is V2. Depending on the...
  44. M

    I Chart coordinate maps of topological manifolds

    Hello every one . first of all consider the 2-dim. topological manifold case My Question : is there any difference between $$f \times g : R \times R \to R \times R$$ $$(x,y) \to (f(x),g(y))$$ and $$F : R^2 \to R^2$$ $$(x,y) \to (f(x,y),g(x,y))$$ Consider two topological...
  45. L

    I Determining Coordinate Transformations for Cloaking Device

    I was reading this article: https://arxiv.org/ftp/arxiv/papers/1005/1005.5206.pdf , regarding the mathematical description of a diamond-shaped cloaking device, and am struggling to understand how the authors found the coordinate transformations in equations (1) and (5). What is the process for...
  46. binbagsss

    I Solve Null Geodesic: Affine Parameter & Coordinate Time - Q1,Q2,Q3

    I am asked a question about how far a light ray travels, the question is to be solved by solving for the null goedesic. I am given the initial data: the light ray is fired in the ##x## direction at ##t=0##. The relvant coordinates in the question are ##t,x,y,z##, let ##s## be the affine...
  47. CheeseSandwich

    I Conceptual Question About Polar Coordinate System

    I am learning about the polar coordinate system, and I have a few conceptual questions. I understand that in Cartesian coordinates there is exactly one set of coordinates for any given point. However, in polar coordinates there is an infinite number of coordinates for a given point. I see how...
  48. Jonathan Scott

    A Reference for coordinate view of equations of motion

    Some time in the 1980s when I first started studying relativistic gravity, for ease of comparison with Newtonian and Special Relativity gravity I worked through pages of geodesic equations for a general isotropic coordinate system with spherical symmetry, converting everything to terms relating...
  49. K

    I Coordinate vs. Non-Coordinate Basis in General Relativity

    Is it correct, at least in the context of general relativity, to say that in a coordinate basis, the inner product between space-like basis vectors will be 1, and in a non-coordinate basis the inner product will be defined by the corresponding component of the metric? Can I take this conditions...
  50. doktorwho

    Solving for the trajectory in the polar coordinate system

    Homework Statement On the surface of a river at ##t=0## there is a boat 1 (point ##F_0##) at a distance ##r_0## from the point ##O## (the coordinate beginning) which is on the right side of the coast (picture uploaded below). A line ##OF_0## makes an angle ##θ_0=10°## with the ##x-axis## whose...
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