Coordinates Definition and 1000 Threads

Homework Statement Transform given integral in Cartesian coordinates to one in polar coordinates and evaluate polar integral. ##\int_{0}^3 \int_{0}^x \frac {dydx}{\sqrt(x^2+y^2)}## Homework EquationsThe Attempt at a Solution I drew out the region in the ##xy## plane and I know that ##0 \leq...

Homework Statement $$ U_{tt}=\alpha^2\bigtriangledown^2U$$ in polar coordinates if solution depends only on R, t. Homework EquationsThe Attempt at a Solution So, the books solution is $$U_{tt}=\alpha^2[U_{rr}+\frac{1}{r}U_r]$$. I am getting stuck along the way can't figure out this last step I...

Hello, I have a question about polar coordinates. It is \vec r = \begin{pmatrix}r cos\phi \\ rsin\phi \\ z\end{pmatrix}=r\cdot \vec e_r + z\cdot \vec e_z and than is \ddot{\vec r} = (\ddot{r}-r\dot{\phi}^2)\vec e_r + (r\ddot{\phi} +2\dot{r}\dot{\phi})\vec e_{\phi} + \ddot{z}\vec e_z The...

3. A(a,b), B(-a,-b), and C is plane XOY. P moves along with curve C. If the multiplication product of PA's and PB's gradients are always k, C is a circle only if k = ...? 4. The radius of a circle which meets X-axis at (6,0) and meets the curve $$y=\sqrt{3x}$$ at one point is ... 5. A circle...

Homework Statement [/B] (a) Verify explicitly the invariance of the volume element ##d\omega## of the phase space of a single particle under transformation from the Cartesian coordinates ##(x, y, z, p_x , p_y , p_z)## to the spherical polar coordinates ##(r, θ, φ, p_r , p_θ , p_φ )##. (b) The...

Homework Statement Consider the 'ice cream cone' bounded by z = 14 − x2 − y2 and z = x2 + y2 .(a) Find the equation of the intersection of the two surfaces in terms of x and y. (b) Set up the integral in polar coordinates. Homework EquationsThe Attempt at a Solution I got part a without...

I am a bit confused by the fact that in GP-coordinates https://en.wikipedia.org/wiki/Gullstrand%E2%80%93Painlev%C3%A9_coordinates the spatial part is flat. I try to imagine the following experiment: First create two rigid shells at two coordinates r1 and r2 outside of the event horizon. The...

Homework Statement Describe using spherical coordinates the solid E in the first octant that lies above the half-cone z=√(x2+y2) but inside x2+y2+z2=1. Your final answer must be written in set-builder notation. Homework Equations ρ = x2+y2+z2 x = ρsinφcosθ y = ρsinφsinθ z = ρcosφ The Attempt...

I was reading a paper that described a vector field in terms of its three components , ##A_σ,A_τ,A_φ##. with σ, τ and φ being the three bispherical coordinates. what does ##A_σ## mean? In what direction does the component point? Likewise for the other two components.

Is the position vector r=xi+yj+zk just r=rerin spherical coordinates?

Homework Statement Homework Equations x^2+y^2 = r^2 The Attempt at a Solution I think it is asking me to find the volume of the sphere, which is in the first octant (1/8 of the sphere) So I set 0<= z<= √1-r2 0 <= r <= 1 0<=θ<=π/2

Hi, I have a little doubt. I have, referred to the Sun, the cartesian positions and velocities of an asteroid (in x, y and z coordinates - 6 values). I can easely calculate the polar coordinates (longitude and latitude - along with distance). My doubt is: how do I calculate the longitude and...

We know that Schwarzschild metric describes an asymptotically flat spacetime. This means that far away from the event horizon we can safely interpret the ##r## coordinate as distance from the center. But when close enough to the event horizon the curvature becomes significant and our common...

Homework Statement Let ##x##, ##y##, and ##z## be the usual cartesian coordinates in ##\mathbb{R}^{3}## and let ##u^{1} = r##, ##u^{2} = \theta## (colatitude), and ##u^{3} = \phi## be spherical coordinates. Compute the metric tensor components for the spherical coordinates...

Homework Statement If ##\bf{v}## is a vector and ##\alpha## is a covector, compute directly in coordinates that ##\sum a_{i}^{V}v^{i}_{V}=\sum a_{i}^{U}v^{j}_{U}##. What happens if ##\bf{w}## is another vector and one considers ##\sum v^{i}w^{i}##? Homework Equations The Attempt at a...

Hello to everybody, I am solving some examples related to wave equation of shear horizontal wave in cylindrical coordinates (J.L Rose: Ultrasolic Waves in Solid Media, chapter 6), which is expressed as follows: ∇2u=1/cT2⋅∂2u/∂t2 The Laplace operator in cylindrical coordinates can can be...

Homework Statement The surface is described by the equation ## (r-2)^2 + z^2 = 1 ## in cylindrical coordinates. Assume ## r ≥ 0 ##. a) Sketch the intersection of this surface with the half plane ## θ= π/2 ## Homework Equations ## r= psin φ ## ## p^2 = r^2 + z^2 ## The Attempt at a Solution...

Homework Statement The function ##f(x)=2a^2x\left(x-a\right)^2## intersects with the line ##y=ax## at the origin, point ##Q(b,f(b))## and point ##T(c,f(c))## where ##c>b>0##. A probability density function, ##p(x)=ax-f(x)## can be formed over the domain ##[b,c]##. (a) Determine the exact...

Note: All bold and underlined variables in this post are base vectors I was reading the book 'Introduction To Mechanics' by Kleppner and Kolenkow and came across an example I don't quite understand. The example is this: a bead is moving along the spoke of a wheel at constant speed u m/s. The...

Homework Statement Show that the equation represents a sphere, and find its center and radius. 3x2+3y2+3z2 = 10+ 6y+12z Homework EquationsThe Attempt at a Solution 3x2+3y2-6y +3z2 -12z =10 My equation is how the constants in-front of the squared terms affect the sphere formula? Besides that...

Hello guys, How can i check if coordinantes A,B,C and D are in the same plane? 3D space(x,y,z) Can i take the cross product: AB x AC and check if its perpendicular to for example DC x DB. and then check if the crossproducts are parallell? but i guess this can give me two parallell vectors in...

can some one help me with part b finding the co-ordinates of p. i tried this by letting sin 2x=1/2 but when i work out x i do not get the right answer. the right answer is (17pi/12, 1/2)

Homework Statement What is the magnitude of the velocity vector if ##\vec{v} = 4 \hat{r} + 6 \hat{\theta}## Homework EquationsThe Attempt at a Solution I know how do do this in Cartesian coordinates (use the Pythagorean theorem), but not so sure how to do it in polar coordinates.

Hi, In an article on theoretical fluid dynamics I recently came across the following equation: $$M_i = \sqrt{g} \rho v_i$$ where ##M_i## denotes momentum density, ##v_i## velocity, ##\rho## the mass density and g is the determinant of the metric tensor. It is probably quite obvious, but I do...

I have read that in polar coordinates, we can form the position vector, velocity, and acceleration, just as with Cartesian coordinates. The position vector in Cartesian coordinates is ##\vec{r} = r_x \hat{i} + r_y \hat{j}##. And any choice of ##r_x## and ##r_y## maps the vector to a position in...

I am trying to convert this polar equation to Cartesian coordinates. r = 8 cos theta I type the equation into wolfram alpha and it gives me a graph, but no Cartesian points. If somebody could help me find the cartesian points, I would appreciate it. Thank you.

I have been reading these notes on Rindler coordinates for an accelerated observer. In Rindler coordinates, the hyperbolic motion of the observer is expressed through the coordinate transformation $$t=a^{-1}e^{a{{\xi}}}\sinh a{\eta}\\ {}x=a^{-1}e^{a{{\xi}}}\cosh a{\eta}.$$On a space-time...

1. The question The position of a particle is given by r(t) = acos(wt) i + bsin(wt) j. Assume a and b are both positive and a > b. The plane polar coordinates of a particle at a time t equal to 1/8 of the time period T will be given by _ Homework Equations r(t) = acos(wt) i + bsin(wt) j. The...

GH is a straight line. the coordinates of G are (-2,8) the midpoint of GH is (5,-3) work out the coordinates of H

I am trying to find and solve the geodesics equation for polar coordinates. If I start by the definition of Christoffel symbols with the following expressions : $$de_{i}=w_{i}^{j}\,de_{j}=\Gamma_{ik}^{j}du^{k}\,de_{j}$$ with $$u^{k}$$ is the k-th component of polar coordinates ($$1$$ is for...

I am currently reading Goldstein's Classical mechanics and come on to this problem. Let q1,q2,...,qn be generalized coordinates of a holonomic system and T its kinetic energy. qk correspondes to a translation of the entire system and qj a rotation of the entire system around some axis, then...

I'd like to expand a 3D scalar function I'm working with, ##f(r,\theta,\phi)##, in an orthogonal spherical 3D basis set. For the angular component I intend to use spherical harmonics, but what should I do for the radial direction? Close to zero, ##f(r)\propto r##, and above a fuzzy threshold...

Why isn't ##\frac{\partial L}{\partial t}\frac{\partial t}{\partial \dot{q_m}}## included in (5.41), given that ##L## could depend on ##t## explicitly?

Hello, I calculated the Vector Laplacian of a uniform vector field in Cartesian and in Cylindrical coordinates. I found different results. I can't see why. In Cartesian coordinates the vector field is: (vx,vy,vz)=(1,0,0). Its Laplacian is: (0,0,0) . That's the result I expected. In...

I am hoping someone can clarify some confusion I have. It is my understanding that there is no such thing as absolute velocity or acceleration in GR. If one observer is moving near the speed of light and the other is stationary each observer will see the other as in motion. But if they each...

1. Homework Statement I am trying to solve a triple integral using cylindrical coordinates. This is what I have to far . But I think I have choosen the limits wrong. Homework EquationsThe Attempt at a Solution [/B]

Homework Statement To integrate a function (the function itself is not important) over the region Q. Q is bounded by the sphere x²+y²+z²=2 (ρ=sqrt2) and the cylinder x²+y²=1 (ρ=cscφ). To avoid any confusion, for the coordinates (ρ,φ,θ), θ is essentially the same θ from polar coordinates in 2...

From the Eulerian form of the continuity equation, where x is the Eulerian coordinate: \frac {\partial \rho}{ \partial t } + u \frac {\partial \rho}{\partial x} + \rho \frac { \partial u}{\partial x} = 0 The incremental change in mass is, where m is the Lagrangian coordinate: dm = \rho dx...

So I'm reading the Schaum's outlines while trying to prepare for a big test I have in September. And I'm trying to understand something here that maybe someone can offer some clarification and guidance. So, using Coulomb's Law, we can find the electric field as follows: \begin{equation} dE...

Yes, that is a serious title for the thread. Could someone please define GENERALIZED COORDINATES? In other words (and with a thread title like that, I damn well better be sure there are other words ) I understand variational methods, Lagrange, Hamilton, (and all that). I understand the...

Ok, the reason for my absence is I have been working on a project of building a multitasking Robot. Since the connection of Robots to Physics never entered my mind I assumed I have nothing of value to add to any discussion on the PhysicsForums. If anyone is interested in the problem that I am...

According to this video the length of basis is r. It grows as we further from the origin . Why?

hi, I really wonder what the difference between curvilinear coordinates in a Euclidean space and embedding a curved space into Euclidean space is ? They resemble to each other for me, so Could you explain or spell out the difference? Thanks in advance...

First, I'd like to say I apologize if my formatting is off! I am trying to figure out how to do all of this on here, so please bear with me! So I was watching this video on spherical coordinates via a rotation matrix: and in the end, he gets: x = \rho * sin(\theta) * sin(\phi) y = \rho*...

Homework Statement Hi everybody! I would like to discuss with you a problem that I am wondering if I understand it correctly: Find expressions for the cartesian components and for the magnitude of the angular momentum of a particle in cylindrical coordinates ##(r,\varphi,z)##. Homework...

hi, I am just curious about, and really wonder if there is a proof which demonstrates that curvature tensor is 0 in all flat space coordinates. Nevertheless, I have seen the proofs related to curvature tensor in Cartesian coordinates and polar coordinates, but have not been able to see that zero...

Homework Statement I need to find the work done by the force field: $$\vec{F}=(5x-8y\sqrt{x^2+y^2})\vec{i}+(4x+10y\sqrt{x^2+y^2})\vec{j}+z\vec{k}$$ moving a particle from a to b along a path given by: $$\vec{r}=\frac{1}{2}\cos(t)\vec{i}+\frac{1}{2}\sin(t)\vec{j}+4\arctan(t)\vec{k}$$ The Attempt...

Hi all, i am having a problem with question 3, as its not clear if i should use the Z value for the camera as 15 or 25 m or... Could you suggest me. Cheers The goal of this project is to obtain some understanding of the camera’s motion in space. Also, based on the camera’s motion, we will...

There have been many questions on this forum about celestial mechanics in general, and concerning position and velocity in an orbit in particular. So I offer this post as a summary and reference. Here's a method for finding heliocentric position and sun-relative velocity in ecliptic coordinates...

Homework Statement Use spherical coordinates to find the volume of the solid enclosed between the spheres $$x^2+y^2+z^2=4$$ and $$x^2+y^2+z^2=4z$$ Homework Equations $$z=\rho cos\phi$$ $$\rho^2=x^2+y^2+z^2$$ $$dxdydz = \rho^2sin\phi d\rho d\phi d\theta$$ The Attempt at a Solution The first...