# What is Coordinate systems: Definition and 117 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. ### A Separation of variables is possible only in 11 coordinate systems?

I vaguely (strong word there because I can no longer remember the source, but the idea sticks in my head for 30 years now) recall reading (somewhere long forgotten) that method of separation of variables is possible in only 11 coordinate systems. I list them below: 1.Cartesian coordinates...
2. ### I Electric Field & Interplay between Coordinate Systems | DJ Griffiths

Hi. I believe I have what may be both a silly and or a weird query. In many Griffiths Problems based on Electric Field I have seen that a coordinate system other than Cartesian is being used; then using Cartesian the symmetry of the problem is worked out to deduce that the field is in (say) z...

4. ### I Exploring Uncommon Coordinate Systems in Physics

I have come across Cartesian, Polar, Cylindrical & Spherical Coordinate Systems so far and was wondering if someone could tell me which are the uncommon systems used in physics which everyone says that they exist but no one explicitly mentions. Is there a "standard reference" or are they just...
5. ### A Kerr Metric: Removing Singularity via Coordinate Transformation

We know that, the singularity of the Schwarzschild metric at ##r = 2M## can be removable via coordinate transformation to Kruskal-Szekers . Can we apply a similar argument to the Kerr metric? If so, what's the name of this coordinate system?
6. ### Frame indifference and stress tensor in Newtonian fluids

During lecture today, we were given the constitutive equation for the Newtonian fluids, i.e. ##T= - \pi I + 2 \mu D## where ##D=\frac{L + L^T}{2}## is the symmetric part of the velocity gradient ##L##. Dimensionally speaking, this makes sense to me: indeed the units are the one of a pressure...
7. ### Free body diagrams, coordinate systems origin/orientation

Hello, When solving statics or dynamics problems, one important step is to draw the free body diagram (FBD) with all the external forces acting ON the system. The "chosen" system may be composed of a single or multiple entities. The external forces have components that must be projects on the...
8. ### 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...
9. ### 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...
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### 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##...
11. ### 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...
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28. ### I Amplitudes of Fourier expansion of a vector as the generalized coordinates

When discussing about generalized coordinates, Goldstein says the following: "All sorts of quantities may be impressed to serve as generalized coordinates. Thus, the amplitudes in a Fourier expansion of vector(rj) may be used as generalized coordinates, or we may find it convenient to employ...
29. ### I Metric Tensor as Simplest Math Object for Describing Space

I've been reading Fleisch's "A Student's Guide to Vectors and Tensors" as a self-study, and watched this helpful video also by Fleisch: Suddenly co-vectors and one-forms make more sense than they did when I tried to learn the from Schutz's GR book many years ago. Especially in the video...
30. ### I Reference Frame Usage in General Relativity

In the book General Relativity for Mathematicians by Sachs and Wu, an observer is defined as a timelike future pointing worldline and a reference frame is defined as a timelike, future pointing vector field Z. In that sense a reference frame is a collection of observers, since its integral lines...
31. ### A How these notions relate to the usual SR approach?

In the context of General Relativity spacetime is a four-dimensional Lorentzian manifold M with metric tensor g, its Levi-Civita connection \nabla and a time orientation vector field T \in \Gamma(TM). In this context I've seem the following three definitions: A coordinate system is a chart...
32. ### I Orientation of the Earth, Sun and Solar System in the Milky Way

I've been tinkering with a few diagrams in an attempt to illustrate the motion of the solar system in its journey around the Milky Way. I also wanted portray how the celestial, ecliptic and galactic coordinate systems are related to each other in a single picture. Note: in the Celestial, or...
33. ### I GPS Data Availability - Positions, Clock Readings etc.

Hi there! Does anyone know where data from the GPS is available? Any data at all - positions. clock readings anything like that. Many thanks!
34. ### 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...
35. ### 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...
36. ### 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...
37. ### 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}...
38. ### 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...
39. ### Non-Euclidean geometry and the equivalence principle

As I understand it, a Cartesian coordinate map (a coordinate map for which the line element takes the simple form ##ds^{2}=(dx^{1})^{2}+ (dx^{2})^{2}+\cdots +(dx^{n})^{2}##, and for which the coordinate basis ##\lbrace\frac{\partial}{\partial x^{\mu}}\rbrace## is orthonormal) can only be...
40. ### Covariant and contravariant basis vectors /Euclidean space

I want ask another basic question related to this paper - http://www.tandfonline.com/doi/pdf/10.1080/16742834.2011.11446922 If I have basis vectors for a curvilinear coordinate system(Euclidean space) that are completely orthogonal to each other(basis vectors will change from point to point)...
41. ### A question concerning Jacobians

Apologies for perhaps a very trivial question, but I'm slightly doubting my understanding of Jacobians after explaining the concept of coordinate transformations to a colleague. Basically, as I understand it, the Jacobian (intuitively) describes how surface (or volume) elements change under a...
42. ### Local parameterizations and coordinate charts

I have recently had a lengthy discussion on this forum about coordinate charts which has started to clear up some issues in my understanding of manifolds. I have since been reading a few sets of notes (in particular referring to John Lee's "Introduction to Smooth Manifolds") and several of them...
43. ### General relativity and curvilinear coordinates

I have just been asked why we use curvilinear coordinate systems in general relativity. I replied that, from a heuristic point of view, space and time are relative, such that the way in which you measure them is dependent on the reference frame that you observe them in. This implies that...
44. ### The speed of a curve in different coordinate systems

Hello, If for a curve in Cartesian coordinates ##||\dot{{\mathbf r}}||=\mbox{const}## (i.e. the curve is constant speed) will the speed of the curve change in cylindrical and spherical coordinates? Could someone experienced share how the transition from flat Euclidian space to curved space...
45. ### Orthogonal coordinate systems - scale factors

Homework Statement Start from the 'relevant equation' below and derive $$(1) \frac{\partial{\bf{\hat{q}}_{i}}}{{\partial{q}}_{j}}={\hat{q}}_{i}\frac{1}{{h}_{i}} \frac{\partial{h}_{i}}{{\partial{q}}_{j}}, {i}\ne{j}$$  (2) \frac{\partial{\bf{\hat{q}}_{i}}}{{\partial{q}}_{i}}= -\sum...
46. ### Event horizon in different coordinate systems

Hi guys, I have a GR question. It is usually said that black holes have event horizons in which time freezes/stops relative to an outside observer. This happens in the Schwarzschild coordinate system. But are there any coordinate systems in which the coordinate time of the black hole and its...
47. ### Converting/Creating Coordinate Systems for Other Planets?

Hi, this may be a very basic concept, but I'm trying to develop coordinate systems for other planets from their right ascension and declination and prime meridians so that, given a location on that planet, you could visualize the sky and its stars.. I've been reading...
48. ### Choice of Origin of Coordinate Systems

I am having a personal discussion with somebody elsewhere (not on Physics Forums) and we are stuck at the moment because of a disagreement that I narrowed down to the question whether, in the context of SR, two observers in different reference frames can choose the origin of their coordinate...
49. ### Why Does General Relativity Use Coordinate Systems?

If you look at Newtonian gravity, there is no major deal with coordinate systems. I am guessing we use coordinate systems because in general relativity we think of coordinate systems as different frames of references and that all frame of references must have the same laws of physics. Is that why?
50. ### Vectors in different coordinate systems

how do i write vectors in polar coordinate? And what will the azimuth coordinate represent? I was trying to figure out the vector connecting a ring to its center using polar coordinates, so that i would perform an integration over d(phi) (finding the electric field due to a semicircle at the...