Coordinates Definition and 1000 Threads

I am trying to derive the tangential acceleration of a particle. We have tangential velocity, radius and angular velocity. $$v_{tangential}= \omega r$$ then by multiplication rule, $$\dot v_{tangential} = a_{tangential} = \dot \omega r + \omega \dot r$$ and $$a_{tangential} = \ddot \theta r +...

I have a vector in cylindrical Coordinates: $$\vec{V} = \left < 0 ,V_{\theta},0 \right> $$ where ##V_\theta = V(r,t)##. The Del operator in ##\{r,\theta,z\}$ is: $\vec{\nabla} = \left< \frac{\partial}{\partial r}, \frac{1}{r}\frac{\partial}{\partial \theta}, \frac{\partial}{\partial z}...

I need a little help with this problem please

It's a silly example, but hopefully it will help me to understand the maths. Two guys ##A## and ##B## are initially at the same spacetime event ##O##, and then ##B## receives an impulse along the ##x##-direction giving him an initial coordinate velocity ##\dot{x}_B = v_0## as measured by ##A##...

Hello, I hope the equation formatting comes out right but I'll correct it if not. So far, I have attempted to write ##\ddot{a}_k(t) = \sum_{n}(u^{k}_n)^*\ddot{q}_n(t) ##. Then I expand the right hand side with the original equation of motion, and I rewrite each coordinate according to its own...

In physics is usually defined that in cylindrical coordinates ##\varphi \in [0,2 \pi)##. In relation with Deckart coordinates it is usually written that \varphi=\text{arctg}(\frac{y}{x}). Problem is of course because arctg takes values from ##-\frac{\pi}{2}## to ##\frac{\pi}{2}##. What is the...

I'm trying to evaluate the following integral in cylindrical coordinates. $$\int_0^6 \int_0^{\frac{\sqrt{2}}{2}}\int_x^{\sqrt{1-x^2}}e^{-x^2-y^2} \, dy \, dx \, dz$$ After attempting to set the bounds in cylindrical coordinates, I got $$\int_0^6 \int_0^{\frac{\sqrt{2}}{2}}\int_{\rho \cos\varphi...

I just want to visualize the math, any help would be appreciated. TIY

##A^{x'} = T(A^{x})##, where T is a linear transformation, in such way maybe i could express the transformation as a changing of basis from x to x' matrix: ##A^{x} = T_{mn}(A^{x'})##, in such conditions, i could say det ##T_{mn} \neq 0##. But how to deal with, for example, ##(x,y) -> (e^x,e^y)## ?

Doing a review for my SAT Physics test and I'm practicing vectors. However, I am lost on this problem I know I need to use trigonometry to get the lengths then use c^2=a^2+b^2. But I need help going about this.

Hello there.Could coordinates be functions?For example in a n-manifold with (x1,...xn) let be the coordinates could they be functions of a coordinate system not belonging to the n-manifold?Or we could first use a coordinate system then have our results, and then have a second coordinate system...

I'm learing about antennas in a course, and we are using Jin's Electromagnetic text. This isn't a homework problem, I'm just trying to understand what I'm supposed to do in this situation. This part of the text discusses how to evaluate a radiation pattern. One of the steps to evaluate the...

As the subject title states, I am wondering how would one go about transforming Cartesian coordinates in terms of spherical harmonics. To my understanding, cartesian coordinates can be transformed into spherical coordinates as shown below. $$x=\rho \sin \phi \cos \theta$$ $$y= \rho \sin \phi...

Points A ,B and C have coordinates A(1,3) B(5,1) and C(2,-8). point D is such that the vector AD = Vector BC + (2x) vector AB + (3y) vectorAC = vector AB + (2x) vectorAC + (3y) vector BC find coordinates of D

I got a polar function. $$ \psi = P(\theta )R(r) $$ When I calculate the Laplacian: $$ \ \vec \nabla^2 \psi = P(\theta)R^{\prime\prime}(r) + \frac{P(\theta)R^{\prime}(r)}{r} + \frac{R(r)P^{\prime\prime}(\theta)}{r^{2}} $$ Now I need to convert this one into cartesian coordinates and then...

I got the answer for velocity and acceleration. But I don't know how to draw the shape of the particle's motion over time. How to draw it? should we change a,b,c,e into a numbers or not? or we may not to change a,b,c,e? Please help me how to draw the shape of particle's motion over time?

Hi, I was trying to gain an understanding of a proof of the divergence theorem in curvilinear coordinates. I have found these online notes here and am looking at the proof on pages 4-5. The method intuitively makes sense to me as opposed to other proofs which fiddle around with vector...

there is a problem in a book that asks to find the orthogonal trajectories to the curves described by the equation : $$r^{2} = a^{2}\cos(\theta)$$ the attempt of a solution is as following : 1- i defferntiate with respect to ##\theta## : $$2r \frac{dr}{d\theta} = -a^{2}\;\sin(\theta)$$ 2- i...

Why the velocity in spherical coordinates equal to ## v^2 = v \dot{} v = \dot{r}^2 + \dot{r}^2\dot{\theta}^2## maybe ## v^2 = v \dot{} v = (\hat{ \theta } \dot{ \theta } r +\hat{r} \dot{r} + \hat{ \phi } \dot{\phi } r \sin{ \theta}) \dot{} (\hat{ \theta } \dot{ \theta } r +\hat{r} \dot{r} +...

Hi, I was just working on a homework problem where the first part is about proving some formula related to Stokes' Theorem. If we have a vector \vec a = U \vec b , where \vec b is a constant vector, then we can get from Stokes' theorem to the following: \iint_S U \vec{dS} = \iiint_V \nabla...

FBD Block 1 FBD Block 2 FBD Pulley B I'm mainly concerned with the coordinate system direction in this problem, but just to show my attempt, here are the equations I got from the system. ##-T_A + m_1g = m_1a_1## ##T_B - m_2g = m_2a_2## ##T_A - 2T_B = 0## Using the fact that the lengths...

I encountered a question which asked me to describe the rose petal sketched below in polar coordinates. The complete answer is R = {(r, θ): 0 ≤ r ≤ 6 cos(3θ), 0 ≤ θ ≤ π}. That makes sense to me for the right petal. What about the other two on the left?

Recently, I was tasked to find the surface area of the Schwarzschild Black Hole. I have managed to do so using spherical and prolate spheroidal coordinates. However, my lecturer insists on only using Weyl canonical coordinates to directly calculate the surface area. The apparent problem arises...

The equations of motion are: \ddot{r}-r{\dot{\theta}} ^{2} = -\frac{1}{r^{2}} for the radial acceleration and r\ddot{\theta} + 2\dot{r}\dot{\theta}= 0 for the transverse acceleration When I integrate these equations I get only circles. The energy of the system is constant and the angular...

If I use cartesian co-ordinates, I get: ##\bar{x}=\frac{1}{A}\iint x\, dA=\frac{1}{A} \iint r^2\cos\theta\, dr\, d\theta= \frac{2a\sin\theta}{3\theta}## ##\bar{y}=\frac{1}{A}\iint y\, dA=\frac{1}{A}\iint r^2\sin\theta\, dr\, d\theta= \frac{2a(1-\cos\theta)}{3\theta}## But if I use polar...

In the example above, the authors claim that when ##r=r_0e^{\beta t}##, the radial acceleration of the particle is 0. I don't quite understand it because they did not assume ##\beta=\pm \omega##. Can anyone please explain it to me? Many thanks.

I am learning to use polar coordinates to describe the motions of particles. Now I know how to use polar coordinates to solve problems and the derivations of many equations. However, the big picture of polar coordinates remains unclear to me. Would you mind sharing your insight with me so that I...

Hey there, I've been recently been going back over the basics of GR, differential geometry in particular. I was watching one of Susskind's lectures and did not understand the argument made here (26:33 - 35:40). In short, the argument goes as follows (I think): we have some generic metric ##{ g...

[Ref. 'Core Concepts in Special and General Relativity' by Luscombe] Let ##M,M'## be manifolds and ##\psi:M\to M'## a diffeomorphism. Even if ##\psi## weren't a diffeomorphism, and instead just a smooth map, the coordinates of the pushback of ##\mathbf{t}\in T_p(M)##, would be related to the...

If a particle is in a magnetic field ##\vec{B} = B\hat{z}## with velocity ##\vec{v} = v_x \hat{x} + v_y \hat{y} + v_z \hat{z}##, then in Cartesian coordinates we can obtain the pair of differential equations $$\ddot{x} = \frac{qB}{m}\dot{y}$$$$\ddot{y} = -\frac{qB}{m}\dot{x}$$which give the...

Hi Everyone, I am wondering how to prove an ideal of a ring $R$ which is defined as a coordinates. Let $R$ be the ring of $\mathbb{Z} \times \mathbb{Z}$. Let $I={(a,a)| a\in \mathbb{Z}}$. I determine that the $I$ is a subring of $R$. Next step is to show the multiplication between the elements...

I come across the adjective 'comoving' quite often. I understand comoving coordinates for the Universe. They are coordinates which expand with the expansion of the Universe(?) but I'm confused about what it means in essence. Here are some examples: In Sean Carrol's book there is a question...

In order to compute de lagrangian in spherical coordinates, one usually writes the following expression for the kinetic energy: $$T = \dfrac{1}{2} m ( \dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \sin^2 \theta \dot{\phi}^2 )\ ,$$ where ##\theta## is the colatitud or polar angle and ##\phi## is the...

I am trying to solve the following problem from my textbook: Formulate the vector field $$ \mathbf{\overrightarrow{a}} = x_{3}\mathbf{\hat{e_{1}}} + 2x_{1}\mathbf{\hat{e_{2}}} + x_{2}\mathbf{\hat{e_{3}}} $$ in spherical coordinates.My solution is the following: For the unit vectors I use the...

I know the divergence of any position vectors in spherical coordinates is just simply 3, which represents their dimension. But there's a little thing that confuses me. The vector field of A is written as follows, , and the divergence of a vector field A in spherical coordinates are written as...

Hi all, I can't find a single thing online that translates a cartesian velocity vector directly to spherical vector coordinate system. If I am given a cartesian point in space with a cartesian vector velocity and I want to convert it straight to spherical coordinates without the extra steps of...

$$-2\sin\theta=1\Leftrightarrow\theta=-\frac{\pi}{6},\,-\frac{5\pi}{6}\\ \begin{align*} \int_{-\frac{\pi}{6}}^{-\frac{5\pi}{6}}\frac 12\left(4\sin^2\theta-1\right)d\theta &=\int_{-\frac{\pi}{6}}^{-\frac{5\pi}{6}}\frac 12\left(1-2\cos2\theta\right)d\theta\\...

What I've done so far: From the problem we know that the curve c is a half-circle with radius 1 with its center at (x,y) = (0, 1). We can rewrite x = r cos t and y = 1 + r sin t, where r = 1 and 0<t<pi. z stays the same, so z=z. We can then write l(t) = [x(t), y(t), z ] and solve for dl/dt...

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Hi folks, See below for a solved question finding the down slope distance of an arrow. How easy would it be to solve this question by making the x-axis the slope direction?

The velocity of a particle below is expressed in polar coordinates, with bases e r and e theta. I know that the length of a vector expressed in i,j,k is the square of its components. But here er and e theta are not i,j,k. Plus they are changing as well. Can someone help convince me that the...

For me is not to easy to understand volume element ##dV## in different coordinates. In Deckart coordinates ##dV=dxdydz##. In spherical coordinates it is ##dV=r^2drd\theta d\varphi##. If we have sphere ##V=\frac{4}{3}r^3 \pi## why then dV=4\pi r^2dr always?

If I have a physical problem, say, a particle which is constrained to move in the ##y## direction, which means that its ##x## coordinate remains fixed, does it make sense to write ##y## in terms of polar coordinates? That is, ##y = r \sin\theta##. Since now I have two parameters ##r,\theta##...

[Moderator's note: Thread spun off from previous discussion due to topic change.] Does the observed quadrapole moment change over time when considering a relatively moving object, for certain choices of observer coordinates? My suspicion is that it does (Terrell-Penrose rotation implies...

I was solving a problem and got stuck in two aspects: 1) Geometric issue. Alright, I understand that the coordinates of the lower cylinder are $$( -R \theta_1, R)$$ The coordinates of the upper cylinder are: $$( x_1 + 2R \sin \theta, 3R - 2(R-\cos \theta))$$ I get that the ##x##...

I'm looking for material about the following approach : If one suppose a function over complex numbers ##f(x+iy)## then ##\frac{df}{dz}=\frac{\partial f}{\partial x}\frac{1}{\frac{\partial z}{\partial x}}+\frac{\partial f}{\partial y}\frac{1}{\frac{\partial z}{\partial y}}=\frac{\partial...

This is what I have so far, please need urgent help. I don't understand and know what to do. For the first part, I got a really long answer, for the second part I am trying in terms of mv^2/r = mg, or mg = m*(answer to first), but I am getting nowhere. PLease help

Suppose that I have the coordinates of x and y on a plane. I am writing a piece of software where the user can select a polygon of 3, 4, 5, 6 or 8 sides. All of the polygon points are equidistant from the x, y point. In other words, if you drew a circle where the center was the x, y point, all...