Polar coordinates Definition and 587 Threads

Hi all, I'm having some problems in grasping/properly understanding the usage of the del operator ( ##\nabla## ) in spherical co-ordinates, and I was wondering if someone could point me to some good resources on the subject, or take a bit of time to try to explain it to me. It just doesn't seem...

Hey, I know that one doesn't work with polar coordinates (t,r,θ,φ) because they don't behave well in the event horizon. But my problem is with raidal null curves, if we take ds2=0 and dφ, dθ = 0 so we have When, if I'm correct, the + sign determine that it's outgoing and the - infalling, so...

Homework Statement Evaluate r(hat and overdot), θ(hat and overdot), φ(hat and overdot) in terms of (θ , Φ) and the time derivatives of the two remaining spherical polar coordinates. Your results should depend on the spherical polar unit vectors. Homework Equations ∂/∂t= The Attempt at a...

Homework Statement A particle moves with constant speed ν around a circle of radius b, with the circle offset from the origin of coordinates by a distance b so that it is tangential to the y axis. Find the particle's velocity vector in polar coordinates. Homework Equations (dots for time...

Homework Statement Homework EquationsThe Attempt at a Solution -In order for me to figure out this problem I had to reverse the equality to isolate the Ө on the left side making the new equation 9cos(3Ө)=r^2. The first thing I’m going to do is change cos(3Ө) in terms of cos and sin. That will...

Hello :) I don't get this integral (Peskin & Schroeder P. 27 ) ##\int {{{{d^3}p} \over {{{\left( {2\pi } \right)}^3}}}{1 \over {{E_{\bf{p}}}}}{e^{i{\bf{p}} \cdot {\bf{r}}}}} = {{2\pi } \over {{{\left( {2\pi } \right)}^3}}}\int\limits_0^\infty {dp{{{p^2}} \over {2{E_{\bf{p}}}}}{{{e^{ipr}} -...

I have a 2D Gaussian: ## f(x,y) = e^{-[(x-x_o)^2 + (y-y_o)^2]/(2*{sigma}^2)}## which I converted into polar coordinates and got: ## g(r,θ) = e^{-[r^2 + r_o^2 - 2*r*r_o(cos(θ)cos(θ_o) + sin(θ)sin(θ_o))]/({2*{sigma}^2})} ## The proof for how this was done is in the attached file, and it would...

Just started self teaching myself differential geometry and tried to find the christoffel symbols of the second kind for 2d polar coordinates. I am checking to see if I did everything correctly. With a line element of: therefore the metric should be: The christoffel symbols of the second kind...

Homework Statement Let R be the triangle defined by -xtanα≤y≤xtanα and x≤1 where α is an acute angle sketch the triangle and calculate ∫∫R (x2+y2)dA using polar coordinates hint: the substitution u=tanθ may help you evaluate the integral Homework EquationsThe Attempt at a Solution so the...

Homework Statement The gradient, divergence and curl in spherical polar coordinates r, ∅, Ψ are nablaΨ = ∂Φ/∂r * er + ∂Φ/∂∅ * e∅ 1/r + ∂Φ/∂Ψ * eΨ * 1/(r*sin(∅)) nabla * a = 1/r * ∂/∂r(r2*ar) + 1/(r*sin(∅)*∂/∂∅[sin(∅)a∅] + 1/r*sin(∅) * ∂aΨ/∂Ψ nabla x a = |er r*e∅ r*sin(∅)*eΨ | |∂/∂r ∂/∂∅...

Homework Statement I have a field w=wφ(r,θ)eφ^ (e^ is supposed to be 'e hat', a unit vector) Find wφ(r,θ) given the curl is zero and find a potential for w. Homework Equations I can't type the matrix for curl in curvilinear, don't even know where to start! I've been given it in the form...

Homework Statement ∫∫dydx Where the region Ω: 1/2≤x≤1 0 ≤ y ≤ sqrt(1-x^2) Homework EquationsThe Attempt at a Solution The question asked to solve the integral using polar coordinates. The problem I have is getting r in terms of θ. I solved the integral in rectangular ordinates using a trig...

I want to solve a Laplace PDE in a polar coordinate system with finite difference method. and the boundary conditions: Here that I found in the internet: and the analytical result is: The question is how its works? Can I give an example or itd?Thanks

hi, i'm a newbie... i have this problem: i have a sphere with known and constant R (obvious), i have two point with spherical coordinates P1=(R,p_1,t_1) and P0=(R, p_0, t_0) p_x = phi x = latitude x t_x = theta x =longitude x the distance between point is D=...

Homework Statement A projectile is launched from a mountain at a given angle and velocity (which is large). Using polar coordinates find the position of the particle at time t. I'm ignoring drag (for now). Homework Equations I tried using the polar kinematic equations...

I have always been under the impression that a vector is not "fixed" in space. Given any vector, we could just move it around and it would still have the same components (in a cartesian coordinate system). What confuses me, however, is how we define the components of a vector in polar...

I'm in the middle of the Great Courses Multivariable Calculus course. A double integral example involves a quarter circle, in the first quadrant, of radius 2. In Cartesian coordinates, the integrand is y dx dy and the outer integral goes from 0 to 2 and the inner from 0 to sqrt(4-y^2). In...

The magnitude of the parallel component of the time derivative of a vector ##\vec{A}## is given by: $$|\frac{d\vec{A}_{\parallel}}{dt}| = |\frac{dA}{dt}|$$ Where ##A## is the magnitude of the vector. Can we write the actual derivative in vector form as ##\frac{dA}{dt} \hat{A}##? Notice how I...

The position of a point in cartesian coordinates is given by: $$\vec{r} = x \hat{\imath} + y \hat{\jmath}$$ In polar coordinates, it is given by: $$\vec{r} = r \hat{r}$$ Now, ##x = r \cos{θ}## and ##y = r \sin{θ}## assuming ##θ## is measured counterclockwise from the ##x##-axis. Equating the two...

Homework Statement I am trying to convert the following vector at (1, 1, 0) to cylindrical polar coordinates, and show that in both forms it has the same direction and magnitude: ##4xy\hat{x}+2x^2\hat{y}+3z^2\hat{z}## Homework Equations ##\rho^2=x^2+y^2## ##tan \phi = \frac{y}{x}## ##z=z##...

Homework Statement Write the chain rule for the following composition using a tree diagram: z =g(x,y) where x=x(r,theta) and y=y(r,theta). Write formulas for the partial derivatives dz/dr and dz/dtheta. Use them to answer: Find first partial derivatives of the function z=e^x+yx^2, in polar...

It it bad practice to consider values of ##r## that are greater than or equal to zero, while ignoring negative values? Do I lose any information in my analysis of motion? I understand what values of ##r < 0## represent, and I'm willing to use them in a pure mathematics context. In classical...

For a dipole, if there is point subtending an angle ##\theta## at the centre of dipole and at a distance ##r## from centre of dipole, then the electric field at that point can be broken into 2 components. One along the line joining the point and centre of dipole and point given by...

Homework Statement Given that: theta_dot = 6 rad/sec m_A = 0.8kg u_k = 0.40 The problem also mentions that movement is at a constant angular rate so I think that means: r_doubleDot = 0 theta_doubleDot = 0 Lastly, at an instant: r_dot = 800mm/sec = 0.8m/sec 2. Homework Equations...

Homework Statement Express f(x,y) = \frac{1}{\sqrt{x^{2} + y^{2}}}\frac{y}{\sqrt{x^{2} + y^{2}}}e^{-2\sqrt{x^2 + y^2}} in terms of the polar coordinates \rho and \phi and then evaluate the integral of f(x,y) over a circle of radius 1 centered at the origin. Homework Equations y = \rho...

The usual change of variables in this case (mentioned in the title of this topic) is this: ##x = rcos(t)## ##y = rsin(t)## When I rewrite (say my integral) in polar coordinates I have to change ##dxdy## to ##rdrdt## My question is why can't I just compute dx and dy the usual way (the already...

Hi, I have a problem with the following explanation of velocity in plane polar coordinates. I don't understand why the magnitude of Δer is approximately equal to Δθ. Thanks

Homework Statement I didn't know if this was considered "advanced" physics, but it's an intermediate classical mechanics course so I'll just post my question here. Basically, if you have a cardioid ##r(\theta)=k(1+\cos(\theta))##, you can show that the ##\dot{\theta}=\frac{v}{\sqrt{2kr}}##...

I've attached the solution to this post. The question is essentially just asking to find the area in one loop for r = cos[3(theta)]. This seems like a fairly simple question (and answer). I've solved and understand the general integration, but I am just a little uncertain on why exactly...

I am reading Spivak's Differential Geometry Vol. 1. I am stuck for some days in chapter 8 about integrating forms on manifolds. Maybe someone can clear my doubt. First, I will 'type' what the corollary says: My doubt is regarding this affirmation: The book it says is easy to see. Well...

Homework Statement r=2cos(theta) I want to find the area using polar integration. Homework Equations area=(1/2)r^2 from 0-pi The Attempt at a Solution When I plug everything in I get 2pi as the answer. I'm in multivariable calculus so this is very frustrating. What am I doing wrong, I don't...

I've been using MATLAB (ode45) to simulate the mechanics of a rocket under the forces of gravity, drag, and internal thrust.y I've recently refactored my simulation to include 2d space, orientation of the rocket, etc. (So I can try to make it orbit, finding optimal ascent profiles, etc)...

Homework Statement Prove: $$\frac{d\hat{r}}{dt} = \dot{\phi} \hat{\phi }$$ and $$\frac{d\hat{\phi}}{dt} = -\dot{\phi} \hat{r }$$Homework EquationsThe Attempt at a Solution I solved this for an Analytical Mechanics assignment a month ago, and completely forgot how it goes.. $$\hat{r} ⊥...

1. Problem Consider a particle that feels an angular force only of the form: F_θ = 3mr'θ'. Show that r' = ± (Ar^4 + B)^(1/2), where A and B are constants of integration, determined by the initial conditions. Also, show that if the particle starts with θ' ≠ 0 and r' > 0, it reaches r = ∞ in a...

Homework Statement A block of mass M is on a frictionless table that has a hole a distance S from the block. The block is attached to a massless string that goes through the hole. A force F is applied to the string and the block is given an angular velocity w0 , with the hole as the origin, so...

Homework Statement The problem and its solution are attached as TheProblemAndTheSolution.jpg. If you don't want to view the attached image, the cartesian-coordinate version that the problem wants me to convert to a polar-coordinate version is the following (let "int" = "integral").: int int (1...

Homework Statement Use polar coordinates to find the volume of the solid inside the hemisphere z=sqrt(16-x^2-y^2) and inside the cylinder x^2+y^2-4x=0 Homework Equations z=sqrt(16-x2-y2) x2+y2-4x=0 x=rcos(Θ) y=rsin(Θ) z=√(16-r2) The Attempt at a Solution ∫∫ r√(16-r2) dr dΘ The problem is...

Hi, Say I have a variable 'x' which has the polar value 10@-75°, would '-x' be -10@+75° or 10@+75° as I am a touch confused as to which bit I have to invert

Homework Statement I'm currently trying to follow a derivation done by Shankar in his "Basic Training in Mathematics" textbook. The derivation is on pages 343-344 and it is based on the solution to the two dimensional heat equation in polar coordinates, and I'm not sure how he gets from one...

Given the force (derived from a potential in planar polar coordinates) F(p,w) = 2p+sin(w)e_p+cos(w)e_w Where e_p and e_w are unit vectors How do I calculate the line integral over a circumference that is defined as: p = 2 0 ≤ w ≤ 2pi Using the definition of a line integral \int_0^{2pi} \...

I have a course next semester on Classical Mechanics (mostly Lagrangian problems), for a second time. I'm ok for the theoretical preparation, but I'm trying to work ahead on problems and exercises, which was badly explained and without much of any resources. So, one of the sources to exercise on...

Homework Statement evaluate the double integral of cosh(6x^2+10y^2) dxdy by making the change of variables x=rcos(theta)/sqrt(3) and y=rsin(theta)/sqrt(5) let D be the region enclosed by the ellipse 3x^2+5y^2=1 and the line x=0 where x>0. Homework EquationsThe Attempt at a Solution first I...

Homework Statement The projectile A is being tracked by the radar at O. At a given instant, the radar readings are θ = 30degrees, R = 2000m, dR/dt = 200 m/s, and d^2R/dt^2 = 20 m/s^2. Determine the speed of the projectile at that instant. THE ANSWER AT THE BACK IS 299.7m/s [PLEASE SEE...

Homework Statement As a part of my self study I am trying to find the covariant basis vectors in the spherical polar coordinates. Since I have never done anything like this before I would appreciate if someone could tell me whether I am on the rigth track. Homework Equations...

The torque contribution due to the uniaxial anisotropy is given by the equation below \frac{\Gamma}{l_m K} = (2 \sin\theta \cos\theta)[\sin\phi e_x - \cos\phi e_y] (3) This contribution can be taken in the LLG equation to derive the LLG equation in polar coordinates \frac{\partial...

Homework Statement My answer seems to differ from the books answer, so I'm wondering where something has gone wrong. Find the volume inside the prism bounded by the planes ##y = x##, ##y = 0##, ##x = \frac{a}{\sqrt{2}}##, ##z = 0## and the cone ##az = h\sqrt{x^2 + y^2}##. Homework...

I'm reading Leonard Susskind's The Theoretical Minimum Vol. 1. 1. The problem: I'm on the section in which he asks the readers to derive the Lagrangian for a particle on a rotating carousel in polar coordinates. 2. Relevant ideas: The same Lagrangian in Cartesian coordinates is given as...

Alright, so I was reading up on tensors and such with non-Cartesian coordinate systems all day but now I'm a bit tired an confused so you'll have to forgive me if it's a stupid question. So to express the dot product in some coordinate system, it's: g(\vec{A}\,,\vec{B})=A^aB^bg_{ab} And, if...

Homework Statement Find the Volume of the solid that the cylinder ##r = acos\theta## cuts out of the sphere of radius a centered at the origin.Homework Equations The Attempt at a Solution I have defined the polar region as follows, $$D = \{ (r,\theta) | -\pi/2 ≤ \theta ≤ \pi/2 , 0 ≤ r...

Hello, I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows: 2W = σijεij Where σ and ε are symmetric rank 2 tensors. For cartesian coordinates it is really easy because the metric is just the identity matrix, hence: 2W = σxxεxx +...