Coordinates Definition and 1000 Threads

Homework Statement Pytel Dynamics Problem 13.4 13.4 The particle passespoint O at the speed of 2.4 m/s. Between O and B, the speed changes at the rate of 2.2√v m/s2, where v is the speed in m/s. Determine the magnitude of the acceleration when the particle is (a) just to the left of pont A...

Hi, on this page: https://en.wikipedia.org/wiki/Laplace_operator#Two_dimensions the Laplacian is given for polar coordinates, however this is only for the second order derivative, also described as \delta f . Can someone point me to how to represent the first-order Laplacian operator in polar...

Homework Statement question : find the value of \iint_D \frac{x}{(x^2 + y^2)}dxdy domain : 0≤x≤1,x2≤y≤x Homework Equations The Attempt at a Solution so here, i tried to draw it first and i got that the domain is region in first quadrant bounded by y=x2 and y=x and i decided to convert...

Homework Statement A sphere of radius 6 has a cylindrical hole of radius 3 drilled into it. What is the volume of the remaining solid. The Attempt at a Solution [/B] I am able to solve this using cylindrical coordinates but I'm having trouble when I try to solve it in spherical coordinates...

When discussing about generalized coordinates, Goldstein says the following: "All sorts of quantities may be impressed to serve as generalized coordinates. Thus, the amplitudes in a Fourier expansion of vector(rj) may be used as generalized coordinates, or we may find it convenient to employ...

Hi, everyone. I had an example from my book, but I wasn't sure how they got \dfrac{1}{2}cos\theta on step 7? It seems like once they combined the constants, they ended up with just cos2\theta. Although, they have a \dfrac{1}{2} in front. Can someone help me understand where that constant came...

A force F = -K(yi + xj) (K is a positive constant) acts on a particle moving in the x-y plane. Starting from the origin, the particle is taken along the positive x-axis to the point (a, 0) and then parallel to the y-axis to the point (a, a). What is the total work done by the force F on the...

Hey! :o We have a triangle $ABC$ with $A(a_1, a_2)$, $B(b_1, b_2)$ and $C(c_1,c_2)$. I want to show that the coordinates of the centroid S is $\left (\frac{1}{3}(a_1+b_1+c_1),\frac{1}{3}(a_2+b_2+c_2) \right )$. $S$ is the intersection point of the midpoints of AB, BC and CA. We have that...

To draw oblique coordinates with the coordinates measured perpendicular to each axis would be wrong, right? I saw it done in a fairly popular book. It's usually the case that I'm the one who is wrong, but I think the book is incorrectly treating minkowski diagrams. Look at these images from...

Hello, I have 6 equations in Cartesian coordinates a) change to cylindrical coordinates b) change to spherical coordinate This book show me the answers but i don't find it If anyone can help me i will appreciate so much! Thanks for your time1) z = 2...

In Weinberg's book, it is said that a given metric ##g_{\mu \nu}## could be describing a true gravitational field or can be just the metric ##\eta_{\alpha \beta}## of special relativity written in curvilinear coordinates. Then it is said that in the latter case, there will be a set of...

In the attached pic,it is shown that the coordinates of point B are (a cos theta, a sin theta) ,but shouldn't it be (-a cos theta,a sin theta)? Can anybody please help?

Hi all, Sorry if this is the wrong section to post this. For some time, I have wanted to derive the Laplacian in spherical coordinates for myself using what some people call the "brute force" method. I knew it would take several sheets of paper and could quickly become disorganized, so I...

The equation y = (x/2) + 1 is given. It forms a straight line going through the points A(x, 1.5) and B(x, 2.5). Find the x-coordinates of points A and B. Do I substitute the value of y for each point into the given equation and solve for x individually?

Hi, I need to solve Laplace equation:##\nabla ^2 \Phi(x,r)=0## in cylindrical domain ##r_1<r<r_2##, ##0<z<+\infty##. The boundary conditions are: ## \left\{ \begin{aligned} &\Phi(0,r)=V_B \\ & -{C^{'}}_{ox} \Phi(x,r_2)=C_0 \frac{\partial \Phi(x,r)}{\partial r}\rvert_{r=r_2} \\ &\frac{\partial...

Homework Statement Let A = (1,2,5) and B = (0,1,0). Determine a point P of the line AB such that ||\vec{PB}|| = 3||\vec{PA}||. Homework EquationsThe Attempt at a Solution Initially, writing the line in parametric form\vec{AB} = B - A = (0-1,1-2,0-5) = (-1,-1,-5)\\ \\ \Rightarrow \vec{v} =...

Homework Statement I have a question. I have a function f(x,y,z) which is a continuous positive function in D = {(x,y,z); x^2 + y^2 +z^2<=1}. And let r = sqrt(x^2 + y^2 + z^2). I have to check whether the following jntegral is convergent. x^2y^2z^2/r^(17/2) * f(x,y,z)dV. Homework Equations...

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...

We can write down the metric of the Schwarzschild black hole in Schwarzschild coordinates. Which aspect of the metric in Schwarzschild coordinates indicates that the coordinates are only valid outside the event horizon?

Hi, I need to solve Laplace equation:## \nabla ^2 \Phi(x,r)=0 ## in cylindrical domain ##0<r<r_0##, ##0<x<L## and ##0<\phi<2\pi##. The boundary conditions are the following ones: ## \left\{ \begin{aligned} &C_{di}\Phi(x,r_0)=\epsilon \frac{\partial \Phi(x,r)}{\partial r}\rvert_{r=r_0} \\...

Hi everyone! I'm having trouble with the following exercise: Let ##\mathrm {Aff}(ℝ)## be the vector space of the affine maps from ##ℝ## to ##ℝ##: $$φ_{a,b}:ℝ→ℝ$$ $$x→a x + b$$ Find the contravariant and and covariant coordinate of the map: $$φ_{1,1}:ℝ→ℝ$$ $$x→x + 1$$ with respect to the...

So I've been trying to wrap my head around this one for several hours now and it just has me stumped. I'm begging, someone, anyone, walk me through it before I swear off numbers for life hahah (Part b) Imgur: The most awesome images on the Internet Thanks in advance for any help anyone can...

I am a little confused about the difference between between coordinates and vectors. For example, when first studying vector calculus, you learn about vector fields, which formally are maps ##f: \mathbb{R}^n \to \mathbb{R}^n##, and we say that the function associates to every point in space a...

So I have this PDE: d2T/dr2 + 1/r dT/dr + d2T/dθ2 = 0. How do I implement dT/dr || [r = 0] = 0? Also, what should I do about 1/r? This is actually the first time I am going to attack FDF in polar/cylindrical coordinates. I can finite-difference the base equation fairly decently; I am just...

The 2D Laplacian in polar coordinates has the form of $$ \frac{1}{r}(ru_r)_r +\frac{1}{r^2}u_{\theta \theta} =0 $$ By separation of variables, we can write the ## \theta## part as $$ \Theta'' (\theta) = \lambda \Theta (\theta)$$ Now, the book said because we need to satisfy the condition ##...

Relations between vectors in cylindrical and Cartesian coordinate systems are given by \vec{e}_{\rho}=\cos \varphi \vec{e}_x+\sin \varphi \vec{e}_y \vec{e}_{\varphi}=-\sin \varphi \vec{e}_x+\cos \varphi \vec{e}_y \vec{e}_z=\vec{e}_z We can write this in form \begin{bmatrix}...

Evaluate the double integral by converting to polar coordinates. Let S S be the double integral symbol S S xy dydx Inner limits: 0 to sqrt{2x - x^2} Outer limits: 0 to 2 The answer is 2/3. I know that x = rcosϴ and y = rsinϴ. S S rcosϴ*rsinϴ r drdϴ. S S (r^3)cosϴ*sinϴ drdϴ. I am stuck...

I have a question regarding writing a shell balance for a cylindrical system with transport in one direction (in any area of transport phenomena). When we set up the conservation equation(say steady state), we multiply the flux and the area at the surfaces of our control volume and plug them...

Evaluate the iterated integral by converting to polar coordinates. Let S S = double integral symbol S S y dx dy The outer integral is from 0 to a. The inner integral is from 0 to sqrt{a^2 - y^2}. I started by letting y = r sin ϴ S S r sinϴ dxdy. I then let dxdy = r dr d ϴ S S r sin ϴ rdr...

I have been reading Thomas Hartman's lecture notes (http://www.hartmanhep.net/topics2015/) on Quantum Gravity and Black Holes. -------------------------------------------------------------------------------------------------------------------------------- In page 97, he derives (9.4), which is...

Homework Statement If (r, θ) are the polar coordinates of a point then describe the region defined by the restrictions -1 < r < 0, π/2 < θ < 3π/2 Homework Equations No clue The Attempt at a Solution I tried drawing the curve in a polar grid by starting at π/2 and finishing at 3π/2. I was...

In an inertial coordinate system in two-dimensional Minkowski spacetime, the metric takes the form $$(ds)^{2} = - (dt)^{2} + (dx)^{2},$$ and in an accelerating coordinate system in two-dimensional Minkowski spacetime, the metric takes the form $$(ds)^{2} = - R^{2}(d\eta)^{2} + (dR)^{2}.$$ The...

Homework Statement The curve ##C## has polar equation ## r\theta =1 ## for ## 0<\theta<2\pi## Use the fact that ## \lim_{\theta \rightarrow 0}\frac{sin \theta }{\theta }=1## to show the line ## y=1## is an asymptote to ## C##.The Attempt at a Solution **Attempt** $$\ r\theta =1$$ $$\...

Homework Statement My end goal is to plot null geodesics around a black hole with realistic representations within the horizon (r<2GM, with c=1) using Mathematica. I've done this for outside the horizon using normal Schwarzschild coordinates and gained equation (1) below, and then used this...

Homework Statement The coordinates of the parallelogram ABCD are: A (-2; 1) B (5; 2) C (6; 5) D (-1; 4) We also know that the diagonals intercept in the middle of each other (so if the diagonals are AC and BD, and the intercept in point M, then AM = MC, and BM = MD). Not sure if this...

How do i get an idea, or a 'feel' of the components of the acceleration in polar coordinates which constitute the component in the eθ direction? from what i know, a= (r¨−rθ˙^2) er + (rθ¨+ 2r˙θ˙) eθ ; (where er and eθ are unit vectors in the radial direction and the direction of increase of the...

I have a series of data points for X and Y points on a graph. The data is quite random and I am trying to work out a trend line so I can then form an equation for the line. How would I go about working out the equation for the data below. (0, 580) (6.7, 495) (13.4, 445) (18.7, 365) (22.8, 350)...

Hi there, I've been trying to solve the following problem, which I found looks pretty basic, but actually got me really confused about the definition of angular momentum. Problem The trajectory of a point mass m is described by the following equations, in spherical coordinates: r(t) = r_0 +...

Homework Statement I am trying to calculate the laplacian in polar coordinates but I failed.Please see the attached Homework Equations The Attempt at a Solution My solution to this was uploaded in the attached.I was wondering what's wrong with the purple brackets since they shouldn't exist(...

Homework Statement F(x,y,z) = xzi Homework Equations N/A The Attempt at a Solution I just said that x = rcos(θ) so F(r,θ,z) = rcos(θ)z. Is this correct? Beaucse I am also asked to find curl of F in Cartesian coordinates and compare to curl of F in cylindrical coordinates. For Curl of F in...

What are the coordinates on the 3D Bloch ball of a qubit's mixed state of the form: ##\rho=p_{00}|0\rangle \langle 0|+p_{01}|0\rangle \langle 1|+p_{10}|1\rangle \langle 0|+p_{11}|1\rangle \langle 1|##

Homework Statement Find the coordinates of each member of set S relative to B. B = {1, cos(x), cos2(x), cos3(x), cos4(x), cos5(x)} S = {1, cos(x), cos(2x), cos(3x), cos(4x), cos(5x)} I am to do this using Mathematica software. Each spanning equation will need to be sampled at six separate...

<Mentor note: moved from a technical forum and therefore without template>So I´m trying to understand how to use the equation for finding the gradient in spherical coordinates, just going from cartesian to spherical seemed crazy. Now I´m at a point where I want to try out what I have read and I...

Homework Statement Here are the problem statement and the solution. I'm stuck at where the book suggests the formulas for the x and y coordinations (highlighted in yellow) of the third charge. Any explanations or proof on how they came to the conclusion for the third charge coordinations would...

Homework Statement It's just an example in the textbook. A vector in cylindrical coordinates. A=arAr+aΦAΦ+azAz to be expressed in cartesian coordinates. Start with the Ax component: Ax=A⋅ax=Arar⋅ax+AΦaΦ⋅ax ar⋅ax=cosΦ aΦ⋅ax=-sinΦ Ax=ArcosΦ - AΦsinΦ Looking at a figure of the unit vectors I...

Homework Statement Homework EquationsThe Attempt at a Solution the answer given is the same but without the negative sign, I don't understand because the crossproduct of unit vectors when using a Cartesian coordinates of the directions given by the right-hand rule? Is the positive z...

Suppose we wish to use Cartesian coordinates for points on the surface of a sphere. Then all derivatives of the metric would vanish and so the Riemann curvature tensor would vanish. But it would give us a wrong result, namely that the space is not curved. So it means that if we want to get...

Homework Statement hi I have read a lot on homogenous coordinates and I feel like I now have a solid foundation. However none of the videos or books I have read give an explicit reason as to why translation with the extra dimension works(i.e. it does not result in scaling). Here 's what I...

Homework Statement The angular velocity vector of a rigid object rotating about the z-axis is given by ω = ω z-hat. At any point in the rotating object, the linear velocity vector is given by v = ω X r, where r is the position vector to that point. a.) Assuming that ω is constant, evaluate v...

The coordinates of A and B are A(-1, 2) and B(5, -3). If B is the midpoint of line segment AC, what are the coordinates of C? I know this question is connected to the midpoint formula. If so, how do I use the formula to find the x and y coordinates of C?