What is Coordinate: Definition and 907 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.
The problem and solution is,
However, I am confused how they get ##\vec a = (1, 2)## (I convert from column vector to coordinate form of vector). I got ##\vec a = (a_1, a_2) = (a_1, 2a_1) = a_1(1, 2)## however, why did they eliminate the constant ##a_1##?
Thanks for any help!
Hello,
I have watched a really good Youtube video on linear algebra by Dr. Trefor Bazett and it made me think about a question...
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A basis in 2D space is formed by any two independent vectors that are not collinear geometrically. Any vector in the 2D space can then be...
Hi,
I'm puzzled about the definition of smooth coordinate chart for a manifold (e.g. spacetime).
From my point of view there is no invariant way to define a smooth coordinate chart since a coordinate system is smooth only w.r.t. another coordinate system already defined on the given manifold...
Can someone provide me with some free resources (classes, books, notes, sites anything) for co-ordinator geometry?
I want to study it from the basics while understanding the logic of every step and build upto start of collage level.
Note ; non free resources are welcome too, but free resources...
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...
I need to prove that under an infinitesimal coordinate transformation ##x^{'\mu}=x^\mu-\xi^\mu(x)##, the variation of a vector ##U^\mu(x)## is $$\delta U^\mu(x)=U^{'\mu}(x)-U^\mu(x)=\mathcal{L}_\xi U^\mu$$ where ##\mathcal{L}_\xi U^\mu## is the Lie derivative of ##U^\mu## wrt the vector...
I started by expanding ##dx## and ##dt## using chain rule:
$$dt = \frac{dt}{dX}dX+\frac{dt}{dT}dT$$
$$dx = \frac{dx}{dX}dX+\frac{dx}{dT}dT$$
and then expressing ##ds^2## as such:
$$ds^2 =...
I am looking for more books like this one: https://archive.org/details/MethodOfCoordinateslittleMathematicsLibrary
Method of Coordinaes (Little Mathematics Library) by A. S. Smogorzhevsky
I am also interested in papers if you can suggest any. I am interested in texts, that explore the idea of...
Hi everybody, I encountered a problem simulation of permanent magnets (PM) in Ansys Maxwell. There are many PM in my simulation and I need to define for each of them a proper coordinate system (CS). But I could only defined 255 CS. After that I can create new CS but It won't be possible to...
Hi I'm wondering if someone can illustrate with an example what I bracketed in blue? I'm having a hard time visualizing how it is that the accelerations of the components are NOT necessarily equal to the components of the acceleration...Much appreciated!
I have a rocket and it is moving straight from a point P with a velocity ##v##. When I say that ##x'=0## is at the place we sit in the rocket, then when the event happened outside his rocket at the point P, can I say that the coordinate of the event is for him negative, so ##x'=-vt'##, although...
Suppose $$ D=\{ (x,0) \in \mathbb{R}^2 : x \in \mathbb{R}\} \cup \{ (0,y) \in \mathbb{R}^2 : y \in \mathbb{R} \}$$ is a subset of $$\mathbb{R}^2 $$ with subspace topology. Can this be a 1d or 2d manifold?
Thank you!
My Progress:
I tried to perform the coordinate transformation by considering a general function ##f(\mathbf{k},\omega,\mathbf{R},T)## and see how its derivatives with respect all variable ##(\mathbf{k},\omega,\mathbf{R},T)## change:
$$
\frac{\partial}{\partial\omega} f =...
In describing the Galelian or Lorentz transformations, All books I've read keep talking about clocks and meter sticks, but I don't see how an event happening away from the observer could be instantaneously described by a set of coordinates and a point in time; information conveying the event...
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...
Could one derive a set of coordinate transformations that transforms events between different reference frames in the de Sitter metric using the invariant line element, similar to how the Lorentz Transformations leave the line element of the Minkowski metric invariant? Would these coordinate...
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 need to evaluate ##\nabla_{\mu} A^{\mu}## at coordinate basis. Indeed, i should prove that ##\nabla_{\mu} A^{\mu} = \frac{1}{\sqrt(|g|)}\partial_{\mu}(|g|^{1/2} A^{\mu})##.
So, $$\nabla_{\mu} A^{\mu} = \partial_{\mu} A^{\mu} + A^{\beta} \Gamma^{\mu}_{\beta \mu}$$
The first and third terms...
Hello all,
I have some issues understanding the inertial-frame (or global-frame, G-frame) versus the body-frame (B-frame) when it comes to simulating the motion of a rigid body in 2 dimensions (planar body mechanics) in a system of ODEs. I have been self-learning from textbooks on simulating...
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...
When we take the x-axis parallel to incline surface its clear that the horizontal component of weight is causing the block to come down but when we take the standard orientation its not so clear to me. Is horizontal component of ##F_N## causing the block to come down?
<Moderator's note: Use of...
$$L = \frac {mv^2}{2} - mgy$$
It is clear that ##\dot{x}=\dot{\theta}L## and ##y=-Lcos \theta##. After substituting these two equations to Lagrange equation, we will get the answer by simply using this equation: $$\frac {d} {dt} \frac {∂L}{∂\dot{\theta}} - \frac {∂L}{∂\theta }= 0$$
But, What if...
Hi,
I was reading this insight schwarzschild-geometry-part-1 about the transformation employed to rescale the Schwarzschild coordinate time ##t## to reflect the proper time ##T## of radially infalling objects (Gullstrand-Painleve coordinate time ##T##).
As far as I understand it, the vector...
* 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...
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...
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...
I'd like to get some help on checking my understanding of special relativity, specifically I'm trying to clarify the idea of coordinates. Any comment is really appreciated!
The spacetime is an affine space ##M^4##, which is associated with a 4 dimensional real vector space ##\mathbb{R}^4##...
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...
If a system is represented by a set of generalized coordinates ##q_i## in which one coordinate say ##\theta## is such that ##d \theta## represents a rotation of the system about a fixed axis( an axis whose orientation remains fixed in space) then the kinetic energy ##T## shouldn't depend on it...
Hi,
I know there is actually no way to set up a global coordinate chart on a 2-sphere (i.e. we cannot find a family of 2-parameter curves on a 2-sphere such that two nearby points on it have nearby coordinate values on ##\mathbb R^2## and the mapping is one-to-one).
So, from a formal...
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...
The answer in the textbook are options A, C and D.
I understand why it is option A, because it is a scalar, I also get that option D is correct because the magnitude of a vector doesn't depend on the coordinate axes. I don't get how option C could be correct. If option C is correct why not D as...
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...
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?
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"...
I posted a thread yesterday and I think that I did not formulated it properly.
So I have a metric ##{ds}^{2}=-{dt}^{2}+{dx}^{2}+2{a}^2(t)dxdy+{dz}^{2}##
I was asked to find the the coordinate transformation so that I can get a diagonalized metric.
so what I've done is I assumed a coordinate...
hey there :)
So I had a homework, and I was asked to diagonalize the metric ##{ds}^2=-{dt}^2+{dx}^2+2a^2(t)dxdy+{dz}^2## and to find the coordinate transformation for the coordinates of the new metric.
so I found the coordinate transformation but the lecturer said that what I found is a...
Let us suppose we have a covariant derivative of a contravariant vector such as
$$\nabla_{\mu}V^{\nu}=\partial_{\mu}V^{\nu} + \Gamma^{\nu}_{\mu \lambda}V^{\lambda}$$
If ##\Delta_{\mu}V^{\nu}## is a (1,1) Tensor, it must be transformed as
$$\nabla_{\bar{\mu}}V^{\bar{\nu}} = \frac{ \partial...
Hello,
This question is with regards to the discussion around page 56 (1971 Edition) in Anthony French's Newtonian Mechanics. He is discussing the choice of a coordinate system where the axes are not necessarily perpendicular to each other. Here is the summary of what I read (as applied to...
Hello,
here on PF I've seen many threads about the concepts of 'reference frame' and 'coordinate system'.
In the context of SR my 'envision' about the concept of 'frame of reference' is basically the 'rods & clocks latticework' as introduced in the book Spacetime physics (Taylor, Wheeler)...
The Earth is moving with respect to the CBR at a speed of 390 kilometers per second, I read in the article https://www.scientificamerican.com/article/how-fast-is-the-earth-mov/. Does FLWR metric coordinate space coincides with integrated local FRs where CBR is homogeneous, and the Earth is...
I am studying @Orodruin's Insight "Explore Coordinate Dependent Statements in an Expanding Universe". It looks pretty interesting. About three pages in it reads "expanding ##x^a## to second order in ##\xi^\mu## generally leads to$$
x^a=e_\mu^a\xi^\mu+c_{\mu\nu}^a\xi^\mu\xi^\nu+\mathcal{O}_3...
So while reading T. Frankel's "The Geometry of Physics", I was going through the part on cotangent bundles which ended with the definition of Poincare 1-form. The author argued that cotangent bundles are better suited than tangent bundles for some problems in physics and that there is no natural...
As in Bernard Schutz's A first course in general relativity, page 220, we suppose a gravitational wave travels in the z-direction with pure "+" polarization, so that the metric in the TT coordinate system is given by$$ds^2=-dt^2+[1+h_{+}(z-t)]dx^2+[1-h_{+}(t-z)]dy^2+dz^2 .$$ Suppose that two...
Summary:: x
Question:
Book's Answer:
My attempt:
The coordinate vectors of the matrices w.r.t to the standard basis of ## M_2(\mathbb{R}) ## are:
##
\lbrack A \rbrack = \begin{bmatrix}1\\2\\-3\\4\\0\\1 \end{bmatrix} , \lbrack B \rbrack = \begin{bmatrix}1\\3\\-4\\6\\5\\4 \end{bmatrix}...
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...
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've been given a curve α parametrized by t :
α (t) = (cos(t), t^2, 0)
How would I go about finding the euclidean coordinate functions for this curve? I know how to find euclidean coord. fns. for a vector field, but I am a bit confused here.
(Sorry about the formatting)