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.
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...
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...
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...
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?
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...
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...
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...
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...
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##...
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...
Hi PF!
Can anyone help me define a coordinate system for a circular arc that makes a specified angle ##\alpha## with a 90 degree wedge? See picture titled Geo.
As an example, a circular arc can be parameterized over a straight line by ##s##, making angle ##\alpha##, via $$\vec T = \left\langle...
Let us consider Ashtekar's definition of asymptotic flatness at null infinity:
I want to see how to construct the so-called Bondi coordinates ##(u,r,x^A)## in a neighborhood of ##\mathcal{I}^+## out of this definition.
In fact, a distinct approach to asymptotic flatness already starts with...
I've taken on a new job recently where I'm having to maintain an existing application that generates a points profile to drive a CNC machine and part of it projects points from an axial plane (which represents the machine's working axis) onto another plane which (I think) acts as as a...
Homework Statement :[/B]
The following expression stands for the two angular bisectors for two lines :\frac{a_{1}x+b_{1}y+c_{1}}{\sqrt{a_{1}^{2}+b_{1}^{2}}}=\pm \frac{a_{2}x+b_{2}y+c_{2}}{\sqrt{a_{2}^{2}+b_{2}^{2}}}\qquad
Homework Equations
The equations for the two lines are :
##a_1x + b_1y +...
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...
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...
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...
I'm trying to understand the BMS formalism in General Relativity and I'm in doubt with the so-called Bondi Coordinates.
In the paper Lectures on the Infrared Structure of Gravity and Gauge Theories Andrew Strominger points out in section 5.1 the following:
In the previous sections, flat...
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...
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...
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...
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...
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}##...
Homework Statement
Consider the metric ds2=(u2-v2)(du2 -dv2). I have to find a coordinate system (t,x), such that ds2=dt2-dx2. The same for the metric: ds2=dv2-v2du2.
Homework Equations
General coordinate transformation, ds2=gabdxadxb
The Attempt at a Solution
I started with a general...
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...
Hi everyone!I'm trying to obtain the natural and dual basis of a circular paraboloid parametrized by:
$$x = \sqrt U cos(V)$$
$$y = \sqrt U sen(V)$$
$$z = U$$
with the inverse relationship:
$$V = \arctan \frac{y}{x}$$
$$U = z$$
The natural basis is:
$$e_U = \frac{\partial \overrightarrow{r}}...
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...
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...
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...
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...
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...
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...
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...
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...
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}...
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...
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...
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)...
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...
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...
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...
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...
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...
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...
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...
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?
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...