What is Coordinate system: Definition and 200 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. MasterOgon

    I Navier-Stokes equation in a triangular coordinate system

    The Navier-Stokes equation is solved in a vector grid in a Cartesian coordinate system. That is, rectangular. But does a rectangular mesh relate to what happens in a gas or liquid, and is it better to use a triangular mesh? Undoubtedly, it is incredibly difficult to take into account all the...
  2. cianfa72

    I General Relativity and the precession of the perihelion of Mercury

    Hi, as test of GR I'm aware of there is the "anomalous" precession of the perihelion of Mercury. My question is: in which coordinate system are the previsions of GR verified concerning the above ? Thanks.
  3. N

    I Changes under a rotation around the z-axis by an angle α

    Hi, I'm trying to solve the problem from here: https://www.physics.uoguelph.ca/chapter-1-newtonian-mechanics Exercise 1.1: Determine how the coordinates $$x$$ and $$y$$, as well as the basis vectors $$\hat{x}$$ and $$\hat{y}$$, change under a rotation around the $$z$$ axis by an angle $$α$$...
  4. Vanilla Gorilla

    B Solving for the Nth divergence in any coordinate system

    Preface We know that, in Cartesian Coordinates, $$\nabla f= \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} + \frac{\partial f}{\partial z}$$ and $$\nabla^2 f= \frac{\partial^2 f}{\partial^2 x} + \frac{\partial^2 f}{\partial^2 y} + \frac{\partial^2 f}{\partial^2 z}$$ Generalizing...
  5. Trysse

    Geometry Looking for books (or papers) on the Cartesian coordinate system

    I am looking for more books like this one: https://archive.org/details/MethodOfCoordinateslittleMathematicsLibrary Method of Coordinaes (Little Mathematics Library) by A. S. Smogorzhevsky I am also interested in papers if you can suggest any. I am interested in texts, that explore the idea of...
  6. Y_G

    Ansys Maxwell - Coordinate System Limitations

    Hi everybody, I encountered a problem simulation of permanent magnets (PM) in Ansys Maxwell. There are many PM in my simulation and I need to define for each of them a proper coordinate system (CS). But I could only defined 255 CS. After that I can create new CS but It won't be possible to...
  7. N

    I Triple equation for integral on a graph

    Hi, so I'm trying to find the volume of a shape using integral, I found the equation of one plane in 3D space but the second one is something like that, which I cannot write in integral as a function: ##\frac{2(2x-a)}{a}=-\frac{2(6y-a\sqrt3)}{a\sqrt3}=\frac{2z-a\sqrt3}{a\sqrt3}## In the 3D...
  8. B

    A Visualizing Arbitrary Coordinate System - Example Needed

    Hi I'm wondering if someone can illustrate with an example what I bracketed in blue? I'm having a hard time visualizing how it is that the accelerations of the components are NOT necessarily equal to the components of the acceleration...Much appreciated!
  9. G

    I Coordinate System Transformation: Lowering/Raising Indices Explained

    In《Introducing Einstein's Relativity Ed 2》on page 106"lowering the first index with the metric,then it is easy to establish,for example by using geodesic coordinates..." In 《A First Course in General Relativity - 2nd Edition》on page 159 "If we lower the index a,we get(in the locally flat...
  10. Trysse

    B: Calculate the distance between two points without using a coordinate system

    Dear all, the following problem is not a home-work problem. I have come up with this question for myself. Nevertheless, I am stuck and need your help. The question is: Can I calculate the distance between points A and B from this information? And if yes, how? I think it should be possible...
  11. K

    I Rate of change of ##L## in a rotating coordinate system

    * We've a vector ##\mathbf{A}## lying in space, changing according to some rule. * We introduce an inertial frame and find ##\left(\frac{d}{d t} \mathbf{A} \right)_{i n}## in it. * We also introduce a co located frame rotating with ##\mathbf{\omega}##. In this rotating frame I find...
  12. K

    A Change of a vector in a rotating coordinate system

    Goldstein 3 ed, pg 171, under" rate of change of a vector " : The author derives the relationship between the change of a vector in a stationary and rotating coordinate system. In the process he uses this assumption :>It is no loss of generality to take the space and body axes as...
  13. K

    A Rotation matrix and rotation of coordinate system

    If we change the orientation of a coordinate system as shown above, (the standard eluer angles , ##x_1y_1z_1## the initial configuration and ##x_by _b z_b## the final one), then the formula for the coordinates of a vector in the new system is given by ##x'=Ax## where...
  14. L

    I Reference frame vs coordinate system

    Just want to clarify some concepts. There seems to be difference between reference frame and coordinate system. See https://en.wikipedia.org/wiki/Frame_of_reference#Definition . A reference frame is something has physical meaning and is related to physical laws, whereas coordinate system...
  15. cianfa72

    I Clock synchronization for ring-riding observers on rotating disk

    Hello, reading the wiki entry for Langevin observers on rotating disk - Born_coordinates I'm struggling with the following quoted sentence: But as we see from Fig. 1, ideal clocks carried by these ring-riding observers cannot be synchronized. I do not grasp why, starting from the figure...
  16. Halc

    I Coordinate System for Minkowskian Spacetime Relative to Event

    I have been using a coordinate system that is anchored on an event (rather than a speed reference) in Minkowskian spacetime. This makes it sort of a special case (no gravity or dark energy, just like special relativity) of the cosmological (or CMB-isotropic) coordinate system used to foliate the...
  17. A

    Divergence in Spherical Coordinate System by Metric Tensor

    The result equation doesn't fit with the familiar divergence form that are usually used in electrodynamics. I want to know the reason why I was wrong. My professor says about transformation of components. But I cannot close to answer by using this hint, because I don't have any idea about "x"...
  18. K

    I Axes of the 2-d coordinate system used in vector resolution

    Hello, This question is with regards to the discussion around page 56 (1971 Edition) in Anthony French's Newtonian Mechanics. He is discussing the choice of a coordinate system where the axes are not necessarily perpendicular to each other. Here is the summary of what I read (as applied to...
  19. Martian2020

    B Entanglement and action at a distance -- What coordinate system?

    Maybe my question is naive and due to my not deep enough knowledge of particle physics. I imagine we entangle two particles on Earth and then send one on spaceship going from Earth - two coordinate frames moving in relation to one another. Moments of simultaneity are different for them. When...
  20. 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...
  21. 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...
  22. 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!
  23. B

    Choosing proper coordinates in a complex 2 pulley system

    FBD Block 1 FBD Block 2 FBD Pulley B I'm mainly concerned with the coordinate system direction in this problem, but just to show my attempt, here are the equations I got from the system. ##-T_A + m_1g = m_1a_1## ##T_B - m_2g = m_2a_2## ##T_A - 2T_B = 0## Using the fact that the lengths...
  24. 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) $$\sum{\vec{F_{net}}}=\vec{0}=\vec{F_e}+\vec{F_m}$$...
  25. 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...
  26. Z

    A How Do You Determine the Mapping Functions in 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...
  27. 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...
  28. Adesh

    Why does the current have no ##\phi## component in a toroidal coil?

    These are images from the book Introduction to Electrodynamics by David J. Griffiths . . . My problem is that I'm unable to understand how the current has zero ##\phi## component (I have underlined it in the first image)? I do understand cylindrical coordinates, I know...
  29. 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.]
  30. 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...
  31. 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...
  32. L

    I Understanding the definition of derivative

    As far as I understand, when we want to differentiate a vector field along the direction of another vector field, we need to define either further structure affine connection, or Lie derivative through flow. However, I don't understand why they are needed. If we want to differentiate ##Y## in...
  33. 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...
  34. K

    B Are Polar Coordinates on ##\mathbb R^2## a 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...
  35. K

    I Definition of Cartesian Coordinate System

    I was asking myself what is the definition of a Cartesian Coordinate System. Can we say that it's a coordinate system such that - the basis vectors are the same ##\forall x \in R^n## - the basis vectors are orthonormal at each ##x \in R^n## So for instance, normalized polar coordinates do not...
  36. D

    I The Supergalactic plane and a coordinate system

    There is remarkably little information on the internet including Wikipedia on this topic. Can someone point me in the right direction as I want to build a visualization software that illustrates the supergalactic plane and the coordinate system with any kind of celestial sphere involved. All I...
  37. M

    MHB How to Visualize Vectors and Subspaces in a Coordinate System?

    Hey! :o We have the basis $B=\left \{\begin{pmatrix}1 \\ 1 \\ 1\end{pmatrix},\begin{pmatrix}2 \\ 1 \\ 0\end{pmatrix}, \begin{pmatrix}1 \\ 2 \\ 1\end{pmatrix} \right \}$ of $\mathbb{R}^3$ and the vector $v$ can we written as a linear combination of the elements of the basis as follows...
  38. R

    A How do I pass a magnetic field to any coordinate system?

    Hello, For example, an electric field vector, such as the gradient of the potential, passes in the following way to any coordinate system:$$E = -\triangledown{}V = - \frac{{\partial V}}{{\partial x^i}}g^{ij}e_j$$ But what about a vector of a magnetic field? How would it be expressed in any...
  39. sams

    I Do we consider a point in a coordinate system to be a scalar?

    Knowing that a scalar quantity doesn't change under rotation of a coordinate system. Do we consider a point in a Cartesian coordinate system (i.e. A (4,5)) a scalar quantity? If yes, why do the components of point A change under rotation of the coordinate system? According to my understanding...
  40. S

    I Nearly Lorentz Coordinate System Explained

    Hello! I am reading "A First Course in General Relativity" by Schutz and in chapter 8 (second edition) he introduces Nearly Lorentz coordinate system. He says that we can always find some coordinates such that the metric is: $$g_{\alpha\beta}=\eta_{\alpha\beta}+h_{\alpha\beta}$$ with...
  41. pobro44

    Converting Coordinate Systems: Exploring the Force on a Semicircular Conductor

    1. The problem statement, all variables and given/known dana I was revisiting University physics textbook and came across this problem. We learned new coordinate systems in classical mechanics classes so I wanted to see if I can apply this to the problem of force on semicircular part of the...
  42. Dreezy

    I Help creating a coordinate system for a robotic arm

    Hey guys, I would appreciate some help with the math behind creating a working coordinate system for a robotic arm. I am currently trying to determine what servo angles are necessary to align a robotic arm's claw to the given coordinates. Geometrically simplified, the robotic arm is a...
  43. S

    Coordinate Systems and Components of a Vector

    Homework Statement Two points in a plane have polar coordinates P1(2.500m, pie/6) and P2(3.800m, 2pie/3) . Determine their Cartesian coordinates and the distance between them in the Cartesian coordinate system. Round the distance to a nearest centimeter.Homework Equations Ax=Acosθ Ay=Asinθ...
  44. M

    Maxwell equation are derived in which coordinate system

    Ignoring special relativity theory,maxwell equation are deduced in which coordinate system?In most electrodynamics textbook,maxwell equation are deduced without specifying which coordinate we are using.For example,when we are solving poisson equation in static case,it seems we can freely choose...
  45. M

    How does rotating a coordinate system affect vector direction and components?

    If I move a coordinate system by an angle theta, why does the vector still have the same direction, but the components are different?
  46. Antarres

    Is Every 2D Riemannian Manifold with Signature (0) Conformally Flat?

    So, I've been studying some tensor calculus for general theory of relativity, and I was reading d'Inverno's book, so out of all exercises in this area(which I all solved), this 6.30. exercise is causing quite some problems, so far. Moreover, I couldn't find anything relevant on the internet that...
  47. Robin04

    Divergence of a vector field in a spherical polar coordinate system

    Homework Statement I have to calculate the partial derivative of an arctan function. I have started to calculate it but I wonder if there is any simpler form, because if the simplest solution is this complex then it would make my further calculation pretty painful... Homework Equations $$\beta...
  48. Pushoam

    Jacobian of a coordinate system wrt another system

    Homework Statement Homework EquationsThe Attempt at a Solution Jacobian of the coordinate- system (## u_1, u_2##) with respect to another coordinate- system (x,y ) is given by J = ## \begin{vmatrix} \frac { \partial {u_1 } } {\partial {x } } & \frac { \partial {u_1 } } {\partial {y} } \\...
  49. D

    Transformation of Vectors in a Rotated Coordinate System

    Homework Statement With respect to a given Cartesian coordinate system S , a vector A has components Ax= 5 , Ay= −3 , Az = 0 . Consider a second coordinate system S′ such that the (x′, y′) x y z coordinate axes in S′ are rotated by an angle θ = 60 degrees with respect to the (x, y) coordinate...
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