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. T

    I N-spherical coordinate angle intervals

    This is a kind of silly-sounding question I never realized puzzled me until moments ago, when I looked up the algorithm for spherical coordinates in n dimensions. In two dimensions, we have polar coordinates, consisting of r from 0 to ∞, and θ from 0 to 2π. In spherical coordinates, we have a...
  2. K

    Acceleration in a new Coordinate System - General Relativity

    Homework Statement Below: Jac = Jacobian matrix; ξ = d/dφ for some continuous parameter φ which labels different points on the worldline. (I'm sorry for my poor English.) Consider a new coordinate system xµ' which differs from the original Cartesian coordinate system xµ; the Cartesian...
  3. nysnacc

    Double integral coordinate transform

    Homework Statement Homework Equations transformation The Attempt at a Solution u = x-y v = x+y I convert each side in terms of u, v, get: u = 0, u = -2 v = 2, v = 4 Correct?
  4. nysnacc

    What is the average area using polar coordinates?

    Homework Statement Homework Equations Average (area) = 1/Area * integrate of polar The Attempt at a Solution y= r* sin theta x= r* cos theta r^2 = x^2+y^2
  5. Battlemage!

    I Coordinate transformation of a vector of magnitude zero

    Is there some geometry in which a coordinate transformation of a vector of magnitude zero transforms to a vector that does not have a zero magnitude? Since the formula for the magnitude of a vector is √(x12+x22+...xn2), I can see no way for it to have magnitude zero unless every component is...
  6. F

    Trying to understand vector conversion matrices

    Homework Statement This isn't exactly a "problem" per se , but I need to understand it for a course I'm taking. I'm trying to understand the significance and when to use the vector conversion matrices, or just the identities. I'll use an example that I made up, using rectangular to polar...
  7. RJLiberator

    Coordinate representation of vectors?

    Homework Statement Starting from the coordinate representation for the vectors, show the result in Equation 1.16 of Griffith's book. (1.16)A \cdot (B \times C) = \left[ \begin{array}{ccc} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \end{array} \right] Note: Here, I use * to...
  8. Evangeline101

    Applications of Trig: related acute angles, coordinate plane

    Homework Statement Homework Equations none The Attempt at a Solution Two possible locations on the coordinate axis for the terminal arm of angle A: Two possible values for the measure of angle A and the related acute angle: Can someone please tell me if I did this correctly?
  9. B

    A Polar coordinate neighbourhoods in manifolds

    In my introduction to manifolds the following is stated: Polar coordinates (r, phi) cover the coordinate neighborhood (r > 0, 0 < phi < 2pi); one needs at least two such coordinate neighborhoods to cover R2. I do not understand why two are needed. Any point in R2 can be described by polar...
  10. F

    I Vector components, scalars & coordinate independence

    This question really pertains to motivating why vectors have components whereas scalar functions do not, and why the components of a given vector transform under a coordinate transformation/ change of basis, while scalar functions transform trivially (i.e. ##\phi'(x')=\phi(x)##). In my more...
  11. S

    I Coordinate independent version of "gradient"?

    Is the "gradient" vector a concept that that is coordinate independent ? For example, the concept of a vector representing a force is independent of what coordinate system is used to represent the vector. So is a "gradient vector" such a physical vector ? The web page...
  12. F

    A A question about coordinate distance & geometrical distance

    As I understand it, the notion of a distance between points on a manifold ##M## requires that the manifold be endowed with a metric ##g##. In the case of ordinary Euclidean space this is simply the trivial identity matrix, i.e. ##g_{\mu\nu}=\delta_{\mu\nu}##. In Euclidean space we also have that...
  13. D

    I Coordinate System: Understanding Polar Vectors

    Hello! I understand the the polar coordinate system without vectors. But when it is related to vector, it is confusing. Do the unit vectors r and phi keep changing? How do I interpret it as they changes? For example, F = 2 r + 3 phi. Based on the vector addition and scale multiplication, it...
  14. C

    Triple integral in polar coordinate

    Homework Statement why x is p(cosθ)(sinφ) ? and y=p(sinθ)(cosφ)? z=p(cosφ) As we can see, φ is not the angle between p and z ... Homework EquationsThe Attempt at a Solution
  15. F

    A Manifolds: local & global coordinate charts

    I'm fairly new to differential geometry (learning with a view to understanding general relativity at a deeper level) and hoping I can clear up some questions I have about coordinate charts on manifolds. Is the reason why one can't construct global coordinate charts on manifolds in general...
  16. Y

    Spherical Harmonics Change of Coordinate System

    Homework Statement Let $$\vec H = ih_4^{(1)}(kr)\vec X_{40}(\theta,\phi)\cos(\omega t)$$ where ##h## is Hankel function of the first kind and ##\vec X## the vector spherical harmonic. a) Find the electric field in the area without charges; b) Find both fields in a spherical coordinate system...
  17. mertcan

    I Jacobian matrix generalization in coordinate transformation

    hi, I always see that jacobian matrix is derived for just 2 dimension ( ıt means 2x2 jacobian matrix) in books while ensuring the coordinate transformation. After that kind of derivation, books say that you can use same principle for higher dimensions. But, I really wonder if there is a proof...
  18. baby_1

    Fourier Series in cylindrical coordinate

    Homework Statement Here is my question Homework Equations I don't know with what formula does the book find Fourier series? The Attempt at a Solution
  19. ShayanJ

    Derivation of rotation formula in a general coordinate system

    Homework Statement [/B] In a set of axes where the z axis is the axis of rotation of a finite rotation, the rotation matrix is given by ## \left[ \begin{array}{lcr} &\cos\phi \ \ \ &\sin\phi \ \ \ &0 \\ & -\sin\phi \ \ \ &\cos\phi \ \ \ &0 \\ &0 \ \ \ &0 \ \ \ &1 \end{array} \right]##...
  20. B

    A What is the coordinate free stress-energy-momentum tensor

    Without regard to a coordinate system (I only wish to consider special relativity) the stress-energy-momentum tensor defines a linear transformation from a 4-vector to a 4-vector. Let T be the linear transformation then b = T(a), a and b are 4-vectors. What is the physical meaning of a and b...
  21. D

    I Defining a generalized coordinate system

    (Note that the title of this thread might be incorrect - I'm just drawing on the vocabulary people use when discussing Lagrangian Mechanics...) Hi, I'm trying to set up a coordinate system to represent points in space where one of the coordinates is the distance along a parametric curve, one is...
  22. W

    A Normal velocity to the surface in Spherical Coordinate System

    Let's say we have r=R( theta, phi, t) on the surface of the particle and need to find the normal vector in Spherical Coordinate system. We know that, the unit vector =grad(r-R( theta, phi, t)) / |grad((r-R( theta, phi, t))| where grad is Spherical gradient operator in term of e_r, e_\theta...
  23. powerof

    I Curl from requiring invariance under inertial coordinate changes

    While investigating about the curl I have found this interesting perspective: http://mathoverflow.net/a/21908/69479 I lack the knowledge to do the derivation on my own so I would like to ask for your help. I am an undergraduate. I do not understand what a "first order differential operator"...
  24. Philosophaie

    Find the Y-Axis in a Coordinate System with Given X and Z-Axis Values

    MENTOR note: moved from General Math hence no template What would be the Y-Axis if: X-Axis: theta=266.4 phi=-28.94 Z-Axis: theta=192.85 phi=27.13 where: theta=atan(Y/X) phi=asin(Z/R) My thinking, theta is +90 from X-Axis and phi is -90 from the Z-Axis. Is the Y-Axis theta=356.4 phi=-62.87?
  25. arpon

    I Infinitesimal area element in polar coordinate

    We know, that the infinitesimal area element in Cartesian coordinate system is ##dy~dx## and in Polar coordinate system, it is ##r~dr~d\theta##. This inifinitesimal area element is calculated by measuring the area of the region bounded by the lines ##x,~x+dx, ~y,~y+dy## (for polar coordinate...
  26. S

    Tangent vectors in the coordinate basis

    Homework Statement In Euclidean three-space, let ##p## be the point with coordinates ##(x,y,z)=(1,0,-1)##. Consider the following curves that pass through ##p##: ##x^{i}(\lambda)=(\lambda , (\lambda -1)^{2}, -\lambda)## ##x^{i}(\mu)=(\text{cos}\ \mu , \text{sin}\ \mu , \mu - 1)##...
  27. binbagsss

    I Basic question, harmonic coordinate condition algebra

    where ##□=\nabla^{\mu}\nabla_{\mu}## is the covariant D'Alembertian. ##□x^{\mu}=0## ##g^{\rho\sigma}\partial_{\rho}\partial_{\sigma}x^{\mu}-g^{\rho\sigma}T^{\lambda}_{\rho\sigma}\partial_{\lambda}x^{\mu}=0## So this line is fine by subbing in the covariant derivative definition and lowering...
  28. K

    B How to calculate the distance between a Star and Zenith

    I was reading through some questions online and one asked the reader to calculate the distance between a star and Zenith given Sidereal Time 17hrs, RA*=16hr30mins, DEC*=50degrees. Could someone explain to me how you would do this please? There were no examples and so far I haven't managed to...
  29. S

    Is polar coordinate system non inertial?

    Studying the acceleration expressed in polar coordinates I came up with this doubt: is this frame to be considered inertial or non inertial? (\ddot r - r\dot{\varphi}^2)\hat{\mathbf r} + (2\dot r \dot\varphi+r\ddot{\varphi}) \hat{\boldsymbol{\varphi}} (1) I do not understand what is the...
  30. Logic Cloud

    I General notion of coordinate change

    I'm trying to understand the nature of coordinate transformations in physics. In classical mechanics, we can transform to a different coordinate frame by means of a Galilean transformation. In special relativity, this is replaced by a Lorentz transformation. I am now wondering whether there...
  31. AllenFaust

    Finding perpendicular vector in a skewed coordinate system

    Homework Statement I have an a-b coordinate system which is skewed with an angle = 60 deg. I also have a particle position defined by vector V1 (a1, b1, 0) which follows the coordinate system. The problem I have is that I need to get V2 (a2, b1, 0) which is perpendicular to V1. Homework...
  32. AndresPB

    Coordinate Rotation: Find Transformation Matrix for 120° Rotation

    Homework Statement [/B]Hello, I am seeking help solving the following problem: find the transformation matrix that rotates a rectangular coordinate system through an angle of 120° about an axis making equal angles with the original three coordinate axes.Homework Equations none, we need to find...
  33. Jezza

    Div and curl in other coordinate systems

    My question is mostly about notation. I know the general definitions for divergence and curl, which can be derived from the divergence and Stokes' theorems respectively, are: \mathrm{div } \vec{E} \bigg| _P = \lim_{\Delta V \to 0} \frac{1}{\Delta V} \iint_{S} \vec{E} \cdot \mathrm{d} \vec{S}...
  34. frostfat

    Finding a New GPS Coordinate Between Two Lines on a Sphere

    Hi, I have an interesting problem. I have three GPS coordinates, creating two lines across the surface of a sphere (assuming the Earth is spherical). I want to be able to create a new line (across the surface of a sphere) with a gradient that is in between the gradient of the two existing...
  35. S

    Understand Contravariant Transformations b/w Coordinate Systems

    I am trying to make sure that I have a proper understanding of contravariant transformations between coordinate systems. The contravariant transformation formula is: Vj = (∂yj/∂xi) * Vi where Vj is in the y- frame of reference and Vi is in the x-frame of reference. Einstein summation...
  36. W

    Find the coordinate transformation given the metric

    Homework Statement Given the line element ##ds^2## in some space, find the transformation relating the coordinates ##x,y ## and ##\bar x, \bar y##. Homework Equations ##ds^2 = (1 - \frac{y^2}{3}) dx^2 + (1 - \frac{x^2}{3}) dy^2 + \frac{2}{3}xy dxdy## ##ds^2 = (1 + (a\bar x + c\bar y)^2) d\bar...
  37. J

    How to Transform Angular Velocity Vector of a Satellite from ECI to LVLH System?

    Hi.. I am new to this forum and not sure whether this is the right place to ask for a help. I have to transform angular velocity vector of a satellite from Earth Centered Inertial (ECI) coordinate system to Local Vertical Local Horizontal(LVLH) system. How can I do that..? Any help appreciated..
  38. T

    Coordinate transformation from spherical to rectangular

    Iam having trouble understanding how one arrives at the transformation matrix for spherical to rectangular coordinates. I understand till getting the (x,y,z) from (r,th ie., z = rcos@ y = rsin@sin# x = rsin@cos# Note: @ - theta (vertical angle) # - phi (horizontal angle) Please show me how...
  39. S

    Nonlinear coordinate transformation

    Homework Statement Evening all, I'm trying to solve the 2-D diffusion equation in a region bounded by y = m x + b, and y = -m x -b. The boundary condition makes it complicated to work with numerically, and I recall a trick that involves a coordinate transformation so that y = m x + b, and y =...
  40. S

    Nonlinear coordinate transformation

    Evening all, I'm trying to solve the 2-D diffusion equation in a region bounded by y = m x + b, and y = -m x -b. The boundary condition makes it complicated to work with numerically, and I recall a trick that involves a coordinate transformation so that y = m x + b, and y = -m x -b are mapped...
  41. S

    Maximum Positive Coordinate Reached by a Particle

    The problem gives you a function describing the position of a particle moving along an x axis: x(t) = 12t2 - 2t3 With this function, one must determine the maximum positive coordinate reached by a particle and the maximum positive velocity. The first step to the problem is to take the...
  42. grav-universe

    Producing a metric with only a single coordinate function

    Given the metric c^2 d\tau^2 = c^2 B(r) dt^2 - A(r) dr^2 - C(r) r^2 d\phi^2 and solving only for a static, spherically symmetric vacuum spacetime, I want to reduce the number of coordinate functions A, B, and C from three to only one using the EFE's. We can then make a coordinate choice for...
  43. N

    Self organising map classifying everything to one coordinate

    Hey all,just coded this SOM based on the explination on the ai-junkie website, it seems to place all the input vectors in one single common coordinate, what am I doing wrong? I suspect its the training loop as that was the one part that the site was unclear about, Any help appreciated. static...
  44. W

    Coordinate Transformation for Projectile Motion Calculations

    Homework Statement I'm trying to find the direction and magnitude of Earth's gravity on some projectile. The question states that I can ignore z, and that the origins of the x and y axes should be on the surface of the planet. I should then use Newton's law of Gravity to find the direction and...
  45. DiracPool

    Metric in polar coordinate derivation

    At time 1:11:20, Lenny introduces the metric for ordinary flat space in the hyperbolic version of polar coordinates? Is that what he is doing here? d(tau)^2 = ρ^2 dω^2 - dρ^2. He then goes on to say that this metric is the hyperbolic version of the same formula for Cartesian space, i. e...
  46. Msilva

    Find infinitesimal displacement in any coordinate system

    I am wondering how can I find the infinitesimal displacement in any coordinate system. For example, in spherical coordinates we have the folow relations: x = \, \rho sin\theta cos\phi y = \, \rho sin\theta sin\phi z = \, \rho cos\theta And we have that d\vec l = dr\hat r +rd\theta\hat \theta...
  47. H

    Online polar coordinate plotter

    Tried to post this in the resources forum but could not start a new thread. I can not find graphing software to plot data in polar coordinates. Anyone got links??
  48. O

    How to draw a coordinate system

    Homework Statement If i want to show which direction is "positiv" I can do like this right? (Or is it wrong) 2. But if the figure would look like this, could i draw a coordinate system rather? Is this way to show which way i say as positive? or should i rather draw like this? Or Is...
  49. K

    Curvilinear coordinate, derivative

    https://www.particleincell.com/2012/curvilinear-coordinates/ http://www.jfoadi.me.uk/documents/lecture_mathphys2_05.pdf Hi, I have a question about the curvilineare coordinate system. I wonder why is normal to the isosurfaces?isnt ei a tangent vector to the surface ui since "With these...
  50. A

    MHB Solving for U & P in a Coordinate Change

    Assume that you are given a coordinate change on a line which changes the coordinate x to a new coordinate z given by the formula z=U⋅x+P where U,P are real numbers with U non zero. If the new coordinate of the point -13 is 12 and the new coordinate of the point -7 is 6 then we must have U= ...
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