Coordinate Definition and 910 Threads

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\\...

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...

Can a reasonable observable operator be defined which measures a two-component observable, first component for the approximate coordinate and the second for the approximate momentum (so that the precision of each measurement do not contradict Heisenberg inequality)? I am actually thinking of...

I am trying to learn GR. In two of the books on tensors, there is an example of evaluating the inertia tensor in a primed coordinate system (for example, a rotated one) from that in an unprimed coordinate system using the eqn. ##I’ = R I R^{-1}## where R is the transformation matrix and...

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...

Homework Statement Find the coordinates of the point ##P(x,y)## on the curve ##y = \sqrt{x}## that is closest to the point ##(4,0)##. Homework EquationsThe Attempt at a Solution The derivative is ##y'(x) = \frac{1}{2\sqrt{x}}##. Do I then find the tangent line to ##y = \sqrt{x}##. A little...

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 I tried to answer the following questions is about the curve surface z= f (x, y) = x^2 + y^2 in the xyz space. And the three questions related to each otherA.) Find the tangent plane equation at the point (a, b, a^2+ b^2) in curved surface z . The equation of the...

Hello everyone, In the COMSOL v5.1 doc i haven't found the mathematical description on how does the automatic geometry analysis or the curvilinear coordinates adaptive method (it is mentioned that they are similar) works. It would be convenient to have an idea about how does the Ecoil vector...

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...

Homework Statement For a 1.0 × 10-26 g particle in a box whose ends are at x = 0 and x = 2.000 Å, calculate the probability that the particle's x coordinate is between 1.6000 and 1.6001 Å if n=1 Homework Equations The Attempt at a Solution I know that since the interval between 1.6000 and...

Homework Statement So I have an equation V = Ae(kx)+Be(-kx) And boundary conditons V= V0 when x=0 and V= 0 when x=b 2. Homework Equations I have solved ones where v=0 at x=0 where it nicely simplifies as the exponentials =1 and the Coeffecients A=-B which leads to a sinh function and I...

How author derives these old basis unit vectors in terms of new basis vectors ? Please don't explain in two words. \hat{e}_x = cos(\varphi)\hat{e}'_x - sin(\varphi)\hat{e}'_y \hat{e}_y = sin(\varphi)\hat{e}'_x + cos(\varphi)\hat{e}'_y

Consider the transformation from Poincare-AdS##_3## geometry to global AdS##_3## geometry: $$ds^{2} = \frac{dr^{2}}{r^{2}} + r^{2}g_{\alpha\beta}dx^{\alpha}dx^{\beta}, \qquad \text{Poincare-AdS$_3$}$$ $$ds^{2} = \frac{dr^{2}}{r^{2}} + r^{2}\left(-dt^{2}+r^{2}d\phi^{2}\right), \qquad...

Homework Statement Does the line with equation (x, y, z) = (5, -4, 6) + u(1,4,-1) lie in the plane with equation (x, y, z) = (3, 0, 2) + s(1,1,-1) + t(2, -1, 1)? Justify your answer algebraically. Homework Equations (x,y,z) = (x0,y0,z0) +s(a1.a2,b3) + t(b1,b2,b3) The Attempt at a Solution...

Homework Statement On the diagram, a charged particle of charge 0.000003 C and mass 0.000007 kg moves across the electric field 6760 V/m with initial speed 40 m/s. When its x coordinate is 93.3 cm, its y coordinate is (in cm)? Homework Equations y=(e*Em*x^2)/(2*m*v^2), where Em is electric...

Homework Statement Find the average y coordinate of the points on the semicircle parametrized by C:[0,##\pi##]-->##R^3##, ##\theta##-->(0, a*sin##\theta##, a*cos##\theta##); a>0 Homework EquationsThe Attempt at a Solution I think the answer should be an integral of the circle in the y...

Why should a person prefer irrational coordinate system over rational? My friend stated that its because most lines such as ##y=e## cannot be plotted on a rational grid system. But that cannot be true since ##e## does have a rational number summation ##2+1/10+7/100...## which can be utilised to...

15.3.65 Improper integral arise in polar coordinates $\textsf{Improper integral arise in polar coordinates when the radial coordinate r becomes arbitrarily large.}$ $\textsf{Under certain conditions, these integrals are treated in the usual way shown below.}$ \begin{align*}\displaystyle...

Homework Statement Is there a more intuitive way of thinking or calculating the transformation between coordinates of a field or any given vector? The E&M book I'm using right now likes to use the vector field ## \vec F\ = \frac {\vec x} {r^3} ## where r is the magnitude of ## \vec x...

Homework Statement Hi everyone. We were discussing conservation of angular momentum as a consequence of rotational invariance in class. There was one point where we needed to compute the change in a vector A when the coordinate frame is rotated by angle Δ(Φ). Homework Equations The teacher...

Homework Statement Coordinates of a particle which moves on a xy coordinate system given with: x=-(5m)sinωt y=(4m)-(5m)cosωt In these equlations t's unit given as second, and ω's unit second^-1. A-) Found velocity and acceleration components when t=0 B-) Write equlations for position and...

Homework Statement Convert from rectangular to spherical coordinates. (-(sqrt3)/2 , 3/2 , 1) Homework Equations We know the given equations are ρ = sqrt(x^2 + y^2 + z^2) tan theta = y/x cos φ = z / ρ The Attempt at a Solution My answer was (2, -pi/3, pi/3) It should be a simple plug and go...

In the second volume, Field Theory, of popular series of Theoretical Physics by Landau-Lifschitz are given following equations as in attached file from the book. Here is considered metric change under coordinate transformation. How is the new, prime metric expressed in original coordinates is...

Homework Statement : [/B] r vector = 3t i + (4t-5t2)j. Find the angle made by the vector with respect to the x-axis and the y-axis. 2. Homework Equations : A.B=AxBx+AyBy 3. The Attempt at a Solution : I tried to take the dot product of the unit vector along x-axis and r. I did the same...

Currently reading the following document which is a bit of a brain overload at the minute! Im considering Equation (4.61). It is the general relativistic correction due to the Schwarzschild field for a near Earth satellite when the parameters \beta, \;\gamma \equiv 1. However, as you will...

I feel embarrassed to ask this, but I may have a misunderstanding in my understanding of some basics. I was told that ##\psi: U \rightarrow \psi(U)##, where ##U = (0, \infty) \times (0, \pi) \times (-\pi, \pi)## and ##\psi(\rho, \varphi, \theta) = (\rho\cos\theta\sin\varphi...

Homework Statement I know that potential gravitational energy is relative to the reference point that I decide to choose (like in the picture below). But then if, for instance, I set my reference point in the ceiling and my vertically down y-axis to be positive. What would the potential...

I am trying to analyse response of a dynamic system. The source disturbance is about x,y,theta (rotation about x ) & Phi of one coordinate system (red coloured coordinate system in the attached figure). I need to get the response in another coordinate system ( green coloured coordinate system...

I have an issue with the definition of coordinate system in differential geometry vs the definition of coordinate system in linear algebra. The post is a bit long, but it's necessary so that I get my point across. Let ##V## be an ##n##-dimensional normed space over the reals and equip ##V##...

##x= r Cosh\theta## ##y= r Sinh\theta## In 2D, the radius of hyperbolic circle is given by: ##\sqrt{x^2-y^2}##, which is r. What about in 3D, 4D and higher dimensions. In 3D, is the radius ##\sqrt{x^2-y^2-z^2}##? Does one call them hyperbolic n-Sphere? How is the radius defined in these...

This was straying from the point in the original thread, but I thought it made a point... The stay-at -home twin is at rest in her frame and her clock must therefore measure proper time. The traveling twin, carries his clock with him; it is therefore at rest in his frame and must also measure...

<This thread is a spin-off from another discussion. Cp. https://www.physicsforums.com/threads/wedge-product.914621/#post-5762138> Also again, be warned about this sloppy notation of indizes. You should put the prime on the symbol (or in addition to the symbol). Otherwise the equations don't...

Hi! I'm currently doing a project where I'm constructing a framework for an engine mount connecting an airplane with an engine. The project involves both calculations by hand and with CAD(Creo), and i have no problem with the CAD part as i have done the simulations. The part with doing...

Is there a universal criteria to determine if a coordinate system is global? I think that it is sufficient for the determinant of the metric to be non-zero in order for a coordinate system to be global. Is this so? For example, take the metric ##ds^{2} = \ell^{2}(-\cosh^{2}\rho\ dt^{2} +...

I have a question about weights of a basis set with respect to the other basis set of one specific vector space. It seems the weights do not covert linearly when basis sets convert linearly. I've got this question from the video on youtube "linear transformation" Let's consider a vector space...

Homework Statement Say I have some sort of a vector field in the cylindrical coordinate system \vec{F}(r, \Theta, z) = f(\vec{A}(r,\Theta,z),\vec{B}(r,\Theta,z)) How do I switch to the Cartesian coordinates? More precisely, how do I transform A_r = g(A_x,A_y,A_z), A_\Theta = h(A_x,A_y,A_z)...

Homework Statement Q. Prove that If (x1,y1) and (x2,y2) are the coordinates of the two vertices of an Equilateral Triangle then the coordinates of the 3rd vertex (X,Y) are $$X=\frac{x1+x2\pm\ √3(y1-y2)}{2},$$ $$Y=\frac{y1+y2\pm\ √3(x1-x2)}{2},$$ The Attempt at a Solution I used distance...

I have the following equations: \left\{ \begin{array}{l} x = \sin \theta \cos \varphi \\ y = \sin \theta \cos \varphi \\ z = \cos \theta \end{array} \right. Assume \vec r = (x,y,z), which is a 1*3 vector. Obviously, x, y, and z are related to each other. Now I want to calculate \frac{{\partial...

Hello In India, SL Loney's Elements of Coordinate Geometry is very popular for entrance examinations. I wanted to refresh my coordinate geometry, so tried reading through the book. But I found that the language used is old. I found myself referring to current material on the topic to properly...

Homework Statement http://i.imgur.com/4FPnTNS.jpg Homework Equations (Written in above photo) The Attempt at a Solution (Written in above photo) I have tried hard in figuring out what's wong I have done done, but what I finally got is still option d instead of the model answer e. Are there...

Hey all, I have a list of line lengths and angles, but only the angles between line n and n-1, can't find a single expression to get the coordinate that works for all cases, i tried \sum{\sqrt{\frac{L^2 -c^2}{tan(\sum{\theta})^2+1}}} and similar expressions but they all assume triangles can be...

Homework Statement If G be the centroid of ΔABC and O be any other point, prove that , ## 3(GA^2 + GB^2 + GC^2)=BC^2+CA^2+AB^2## ##and,## ##OA^2 + OB^2 + OC^2 = GA^2.GB^2+GC^2+3GO^2## Homework Equations i m practising from S L LONEY coordinate geometry first chapter ... only the equation...

I want to understand what changing coordinate system means for hands of clock. Let's say the clock only has hour and minute hand. It can move let's say just in the upper 180 deg. of the clock (as shown in the figure). The area between the two hands is V1, and the rest is V2. Depending on the...

Hello every one . first of all consider the 2-dim. topological manifold case My Question : is there any difference between $$f \times g : R \times R \to R \times R$$ $$(x,y) \to (f(x),g(y))$$ and $$F : R^2 \to R^2$$ $$(x,y) \to (f(x,y),g(x,y))$$ Consider two topological...

I was reading this article: https://arxiv.org/ftp/arxiv/papers/1005/1005.5206.pdf , regarding the mathematical description of a diamond-shaped cloaking device, and am struggling to understand how the authors found the coordinate transformations in equations (1) and (5). What is the process for...

I am asked a question about how far a light ray travels, the question is to be solved by solving for the null goedesic. I am given the initial data: the light ray is fired in the ##x## direction at ##t=0##. The relvant coordinates in the question are ##t,x,y,z##, let ##s## be the affine...

I am learning about the polar coordinate system, and I have a few conceptual questions. I understand that in Cartesian coordinates there is exactly one set of coordinates for any given point. However, in polar coordinates there is an infinite number of coordinates for a given point. I see how...

Some time in the 1980s when I first started studying relativistic gravity, for ease of comparison with Newtonian and Special Relativity gravity I worked through pages of geodesic equations for a general isotropic coordinate system with spherical symmetry, converting everything to terms relating...

Is it correct, at least in the context of general relativity, to say that in a coordinate basis, the inner product between space-like basis vectors will be 1, and in a non-coordinate basis the inner product will be defined by the corresponding component of the metric? Can I take this conditions...