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. K

    I Coordinate systems on the 2-sphere

    If I define the two dimensional sphere in the usual way, this gives me a metric ##ds^2 = r^2 d\theta^2 + r^2 \sin^2 \theta d\phi^2##. Can I just define a new coordinate system giving a point coordinates ##(\theta', \phi') = (\theta r^2, \phi r^2 \sin^2 \theta)##?. This gives me the metric ##ds^2...
  2. cianfa72

    I About spacetime coordinate systems

    Hi, There is a point that, in my opinion, is not quite emphasized in the context of general relativity. It is the notion of spacetime coordinate systems that from the very foundation of general relativity are assumed to be all on the same footing. Nevertheless I believe each of them has to be...
  3. QuasarBoy543298

    I Understanding 3D Coordinate Rotations with Euler Angles

    I'm trying to wrap my head around the concept. we use 3 rotations to transfer our regular cartesian coordinates (3 x,y,z unit vectors) to other 3 unit vectors. each rotation is associated with an angle. so far I'm good. but now I saw in Landau's and Lifshitz's "mechanics" book this thing...
  4. jk22

    I Coordinate Change & Gravity: A Schwazschild Metric Analysis

    What I understood is that The Schwazschild metric is obtained by setting spherical symmetry in the metric and solves the field equation in vacuum. But is the flat metric a solution too, or does it mean that changing the coordinates induces gravity ?
  5. 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...
  6. 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...
  7. A

    I Tangent vector basis and basis of coordinate chart

    I am learning the basics of differential geometry and I came across tangent vectors. Let's say we have a manifold M and we consider a point p in M. A tangent vector ##X## at p is an element of ##T_pM## and if ##\frac{\partial}{\partial x^ \mu}## is a basis of ##T_pM##, then we can write $$X =...
  8. J

    Torque coordinate transformation

    Hi all, I am currently working on a creating a mathematical model of a longboard and am in need of advice. The pictures describe the sitaution. Side view Top view Back view The pictures depict a simplified longboard - the brown line is the deck and the black lines represent the...
  9. M

    MHB Vectors 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...
  10. A

    A How to find the displacement vector in Spherical coordinate

    Is there a way of subtracting two vectors in spherical coordinate system without first having to convert them to Cartesian or other forms? Since I have already searched and found the difference between Two Vectors in Spherical Coordinates as...
  11. C

    I Force fields in curvilinear coordinate systems

    I am trying to solve problems where I calculate work do to force along paths in cylindrical and spherical coordinates. I can do almost by rote the problems in Cartesian: given a force ##\vec{F} = f(x,y,z)\hat{x} + g(x,y,z)\hat{y}+ h(x,y,z)\hat{z}## I can parametricize my some curve ##\gamma...
  12. R

    I Differential of the coordinate functions

    Hello folks, I'm glad that I discovered this forum. :) You might save me. I'm hearing right now differential geometry and am having some problems with the subject. May you explain me the follwoing. We had the special case of the i-th projection. My lecturer now posited that the differential of...
  13. 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...
  14. sams

    I Explaining Coordinate Rotation in Arfken & Weber Chapter 1

    In Mathematical Methods for Physicists, 6th Edition, by Arfken and Weber, Chapter 1 Vector Analysis, pages 8-9, the authors make the following statement: "If Ax and Ay transform in the same way as x and y, the components of the general two-dimensional coordinate vector r, they are the...
  15. 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...
  16. TheQuestionGuy14

    B Difference Between Proper Time and Coordinate Time Explained

    I was reading up on the nature of time and found this: " In one sense, "time" is the time that is in the equations of physics. That's the t in the equations of the paper, it's the parameter that describes how the states of all systems in the universe change." Does this explain coordinate time...
  17. HCverma

    In coordinate bond, why doesn't the H^+ atom get a negative charge?

    In a coordinate bond, why H^+ atom don't get the negative charge? as an example [NH4]^+ If we split [NH4]^+, we get NH3 + H^+. In NH3, N and 3H atoms have completed their octet and H^+ accepts the lone pair of electrons from the N, As we know H^+ has no any electrons but a proton. If it receives...
  18. Gene Naden

    I Connection forms and dual 1-forms for cylindrical coordinate

    I ran across exercise 2.8.4 in Oneill's Elementary Differential Geometry. It says "Given a frame field ##E_1## and ##E_2## on ##R^2## there is an angle function ##\psi## such that ##E_1=\cos(\psi)U_1+\sin(\psi)U_2##, ##E_2=-\sin(\psi)U_1+\cos(\psi)U2## (where ##U_1##, ##U_2##, ##U_3## are the...
  19. E

    I Coordinate transformations in GR

    Hi there I'm studying GR and I am confused about coordinate transformations. In my understanding, if I want to study a rotating reference system this is what I do. In my inertial system the object trajectory is described by $$ x = r\cos(\theta - \omega t)\\ y = r\sin(\theta - \omega t) $$...
  20. C

    Coordinate singularity in Schwarzschild solution

    Hi! I have the following problem I don't really know how to approach. Could someone give me a hand? The line element of a black hole is given by: ds^2=\Bigg(1-\frac{2m}{r}\Bigg)d\tau ^2+\Bigg(1-\frac{2m}{r}\Bigg)^{-1} dr^2+r^2\Big(d\theta ^2+\sin^2(\theta)d\phi ^2\Big) It has an apparent...
  21. DarkStar42

    A Do any coordinate systems include self magnification?

    Take a neutron star, its surface will be gravitationally self magnified so that it looks bigger to the distant observer, than it 'really' is, plus you can see some of the rear facing surface. If you take the centre of the neutron star, then this process must go on there also, although unseen...
  22. 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...
  23. shahbaznihal

    B Proving Linearity in Coordinate Transformation: A Mathematical Approach

    This is intuitively very simple problem but I am unable to complete it with Mathematical rigor. Here is the deal: A coordinate system $(u,v,w,p)$ in which the metric tensor has the following non-zero components, $g_{uv}= g_{ww}=g_{pp}=1$. Find the coordinate transformation between $(u,v,w,p)$...
  24. S

    I Coordinate functions of a many-to-1 function

    How many coordinate functions of a many-to-1 function must also be many-to-1 ? Let ##F## be a function from ##\mathbb{R}_n## into ##\mathbb{R}_n##. Represented as an ##n##-tuple in a particular (not necessarily Cartesian) coordinate system ##h##, ##F## is given by ##n## coordinate functions...
  25. 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...
  26. 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...
  27. Abhishek11235

    A Finding the unit Normal to a surface using the metric tensor.

    Let $$\phi(x^1,x^2...,x^n) =c$$ be a surface. What is unit Normal to the surface? I know how to find equation of normal to a surface. It is given by: $$\hat{e_{n}}=\frac{\nabla\phi}{|\nabla\phi|}$$However the answer is given using metric tensor which I am not able to derive. Here is the answer...
  28. 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...
  29. 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...
  30. binbagsss

    I Coordinate and proper time, null geodesic

    I have a question which asks show that a null geodesic to get to r> R , r some constant, given the space time metric etc, takes infinite coordinate time but finite proper time. ( It may be vice versa ). I just want to confirm that, ofc there is no affine parameter for a null geodesic and so you...
  31. C

    I Nonlinear relation between coordinate time and proper time

    For Schwarzschild geomery $$ds^2=-(1-\frac{2GM}{r})dt^2+(1-\frac{2GM}{r})^{-1}dr^2+r^2d\Omega^2$$ For a Schwarzschild observer , the proper time and coordinate time are related by $$d\tau=(1-\frac{2GM}{r})^{1/2}dt$$ There is a often used relation between proper time and coordinate time $$d\tau...
  32. 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?
  33. stevendaryl

    A Coordinate representation of a diffeomorphism

    I'm trying to understand diffeomorphisms, and I thought I basically understood them, but when I tried to work out a problem I created for myself, I realized I didn't know how to answer it. So let's consider a diffeomorphism generated by a vector field ##V##. If ##X## is a point on our manifold...
  34. Aleoa

    Mapping beetween affine coordinate functions

    Homework Statement Homework Equations As the book says , an affine function of a line is A\rightarrow \mathbb{R} and represent the real number that, multiplied for a basis and starting from an origin of the line gives a certain point of the line, so a origin of the line and a basis is...
  35. Antarres

    Null curve coordinate system

    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...
  36. S

    I Difference between proper time and coordinate time

    Hello! I found this questions in several places, but no answer made me fully understand it, so I decided to give it one more try here. I am not sure I understand the difference between them in GR. I have a feeling of the proper time as the time measured by the clock of someone moving with a...
  37. K

    I Coordinate systems in ##\mathbb{R}^2##

    I want to show some of my current understanding/findings involving vector spaces. The reason is two fold: to ask whether my current understanding is ok and to give context for a specific question in the end. The set ##\{(x,0), (0,y) \}##, with ##x,y \in \mathbb{R}##, spans ##\mathbb{R}^2##. For...
  38. J

    Split Kinetic and Potential Term in Action in Independt. Var

    Homework Statement [/B] I have the following expression: $$S=T+V$$ $$T=\frac{m}{\tau_0+it}((x_1-x_0)^2+(x_2-x_1)^2)+\frac{m}{2(\tau_1-it)}(x_2-x_0)^2$$ $$V= \frac{(\tau_0+it)}{2}(\frac{k_0 x_0}{2}+\frac{k_0 x_2}{2}+k_0 x_1)+(\tau_1-it)(\frac{k_1 x_0}{2}+\frac{k_1 x_2}{2})$$ The main goal...
  39. 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...
  40. karush

    MHB Evaluate the spherical coordinate integrals

    $\textsf{Evaluate the spherical coordinate integrals}$ \begin{align*}\displaystyle DV_{22}&=\int_{0}^{2\pi}\int_{0}^{\pi/4}\int_{0}^{2} \, (\rho \cos \phi) \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta \\ %&=\color{red}{abc} \end{align*} so then next ...
  41. L

    MATLAB Plotting Coordinate Transformations in Matlab

    I was reading this article (https://www.nature.com/articles/srep40083) about designing a panoramic lens with transformation optics, and wanted to try to play around with modifying the coordinate transformations. I contacted one of the authors of the article, and she mentioned that she plotted...
  42. A

    MHB How to find volume of cone+hemisphere on the cone using spherical coordinate

    i have a problem about find volume of hemisphere I do not know the true extent of radius r (0 to ?) i think... cone ( 0 < r < R cosec(\theta) ) hemisphere (0 < r < R)
  43. 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} } \\...
  44. C

    No prefix: Understanding the Total Force on a Mass Attached by Two Springs

    1. Homework Statement The following problem is an example from the book ' Berkely - Waves by Frank S. Crawford Jr '. Mass 'M' slides on a frictionless surface. It is connected to rigid walls by means of two identical springs, each of which has zero mass, spring constant 'K' and relaxed length...
  45. Okinawa Rikenata

    How to get x-y coord. from x-z coord. and y-z coord.?

    I have data from tiltmeter. It has 3 components (time, x, and y) and stations. I use Matlab for calculating the data (.csv) until i get some plots. I have two plots, time-x and time-y. Each plot has a trendline. So that means i have two trendlines in two plots. Assume that time is z and x is...
  46. C

    Constraint of Two blocks on an inclined plane

    Hello, I have an issue regarding a constraint related to an angle: Suppose I have masses 'A' and 'B' on an inclined plane ( of mass 'C') attached by a pulley. I place my origin as shown and I want to find a constraint relating angle β. so, I saw my classmate writing as follows to find...
  47. 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...
  48. Orodruin

    Insights Coordinate Dependent Statements in an Expanding Universe - Comments

    Greg Bernhardt submitted a new PF Insights post Coordinate Dependent Statements in an Expanding Universe Continue reading the Original PF Insights Post.
  49. M

    MHB How to Compute Coordinate Column Vectors in Different Bases?

    Hey! :o We have the matrices $E_{k\ell}\in \mathbb{R}^{2\times 2}$ with $1$ iin the position $(k,\ell)$ and $0$ in the other positions and \begin{equation*}\sigma_0=\begin{pmatrix}1&0\\ 0&1\end{pmatrix}, \ \sigma_1=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}, \ \sigma_2=\begin{pmatrix}0&-i\\...
  50. 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...
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