Coordinate Definition and 868 Threads

Hi everyone, sorry for the basic question. But I was just wondering how one does the explicit coordinate change from dxdy to dr in the polar-coordinates method for solving the gaussian. I can appreciate that using the polar element and integrating from 0 to inf covers the same area, but how do...

Homework Statement A small bead of mass m slides on a frictionless cylinder of radius R which lies with its cylindrical axis horizontal. At t = 0 , when the bead is at (R,0), vz = 0 and the bead has an initial angular momentum Lo < mR sqrt(Rg) about the axis of the cylinder where g is the...

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'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...

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 ?

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...

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...

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 =...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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 $$ x = r\cos(\theta - \omega t)\\ y = r\sin(\theta - \omega t) $$...

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...

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...

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...

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 $(u,v,w,p)$...

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...

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...

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...

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...

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...

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...

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...

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...

If I move a coordinate system by an angle theta, why does the vector still have the same direction, but the components are different?

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...

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...

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...

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...

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...

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...

$\textsf{Evaluate the spherical coordinate integrals}$ \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 ...

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...

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)

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} } \\...

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...

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...

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...

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...

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

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\\...