Coordinates Definition and 1000 Threads

Homework Statement Express the quantity ∂2/∂x2+∂2/∂y2 in polar coordinates. Homework Equations x=ρcosφ y=ρsinφ ρ=sqrt(x2+y2) The Attempt at a Solution This is my first post, so I apologize for any weird looking equations, etc. I know that this is not a difficult problem, but I just cannot...

Hello, I am trying to find the magnetic field that accompanies a time dependent periodic electric field from Faraday's law. The question states that we should 'set to zero' a time dependent component of the magnetic field which is not determined by Faraday's law. I don't understand what is...

I have to evaluate $$\int^1_{-1} \int^{ \sqrt {1-x^2}}_{ - \sqrt {1-x^2}} \int^1_{-\sqrt{x^2+y^2}}dzdydx$$ using spherical coordinates. This is what I have come up with $$\int^1_{0} \int^{ 2\pi}_0 \int^{3\pi/4}_{0}r^2\sin\theta d\phi d\theta dr$$ by a combination of sketching and...

Homework Statement Change the Cartesian integral into an equivalent polar integral and then evaluate. Homework Equations x=rcosθ y=rsinθ I have: ∫∫r2cosθ dr dθ The bounds for theta would be from π/4 to π/2, but what would the bounds for r be? I only need help figuring out the bounds, not...

Homework Statement \vec J_b = 3s \hat z \int \vec J_b \, d\vec a I need to solve this integral in cylindrical coordinates. It's the bound current of an infinite cylinder, with everything done in cylindrical coordinates and s is the radius of the cylinder. The answer should end up with a phi...

Homework Statement I am currently trying to calculate the moment and products of inertia of a ring rotating about the x-axis at the moment the ring lies in the xy plane. The problem is that the notations I have from textbook are denoted for Cartesian coordinates. i.e. Ixx=∑i mi(yi2+zi2), and...

I just wanted to check that I am thinking about the coordinate transition correctly. The relativistic generalization of Euler's equation is (from Landau & Lifshitz vol. 6) ## hu^\nu \frac{\partial u_\mu}{\partial x^\nu} - \frac{\partial P}{\partial x^\mu} + u_\mu u^\nu \frac{\partial...

Homework Statement [/B] Consider ##\mathbb{R}^3## in standard Cartesian co-ordinates, and the surface ##S^2## embedded within it defined by ##(x^2+y^2+z^2)|_{S^2}=1##. A particular set of co-ords on ##S^2## are defined by ##\zeta = \frac{x}{z-1}##, ##\eta = \frac{y}{z-1}##. Express...

Homework Statement Show that ##\nabla u_i \cdot \frac{\partial \vec r}{\partial u_i} = \delta_{ij}##. (##u_i## is assumed to be a generalized coordinate.) Homework Equations Gradient in curvilinear coordinates ##\nabla \phi = \sum_{i=1}^3 \vec e_i \frac{1}{h_i} \frac{\partial \phi}{\partial...

Homework Statement The vector field ##\vec B## is given in spherical coordinates ##\vec B(r,\theta,\phi ) = \frac{B_0a}{r\sin \theta}\left( \sin \theta \hat r + \cos \theta \hat \theta + \hat \phi \right)##. Determine the line integral integral of ##\vec B## along the curve ##C## with the...

Prepare a function m-file containing a function that converts polar coordinates in two-dimensional space to rectangular (Cartesian) coordinates. Include a suitable H1 line and some additional comment lines. The input will be 2 vectors, and the output will be 2 vectors. The length of each vector...

Homework Statement Solve the Laplace equation inside a sphere, with the boundary condition: \begin{equation} u(3,\theta,\phi) = \sin(\theta) \cos(\theta)^2 \sin(\phi) \end{equation} Homework Equations \begin{equation} \sum^{\infty}_{l=0} \sum^{m}_{m=0} (A_lr^l + B_lr^{-l -1})P_l^m(\cos...

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...

Homework Statement I'm doing a question that requires me to take the dot product of 2 vectors in spherical coordinates. Both vectors have only an r component, can I just multiply the r components? Homework EquationsThe Attempt at a Solution

Homework Statement Find the volume of the solid region that lies inside the cone φ= pi/6 and inside the sphere ρ=4. Use rectangular coordinates. Homework Equations x=ρ sinφ cos θ y=ρsinφ sin θ z=ρ cos φ ρ^2=x^2+y^2+z^2 x= r cos θ y= r sin θ r^2=x^2+y^2The Attempt at a Solution at first...

Homework Statement This is problem 6.5 in Griffiths EM. I can't understand why dipole moment does not depend on coordinate systems. Homework EquationsThe Attempt at a Solution

Homework Statement I'm suppose to verify the given Laplace in (a) Cartesian (b) Sperical and (c) Cylindrical coordinates. (a) was easy enough but I need to know if I'm doing (b) and (c) correctly. I don't need a solution, I simply need to know if the my Spherical formula is correct, my...

So I am going through the exam guide for my exam tomorrow and there is a second problem that stump me. We transform the cartesian axis to <1/√3,1/√3,1√3> and <1/√2,0,-1/√2> which are orthogonal and we find the third axis by taking the cross product which gives <-881/2158,881/1079,-881/2158>...

Homework Statement OK, I thought once I knew what the question was asking I'd be able to do it. I was wrong! Consider the volume V inside the cylinder x2 +y2 = 4R2 and between z = (x2 + 3y2)/R and the (x,y) plane, where x, y, z are Cartesian coordinates and R is a constant. Write down a triple...

Points on the same line have two different coordinate systems: P and Q. The corresponding coordinates are denoted by small letters p and q. The two systems are related by a conversion formula q=sp+t. The point with P-coordinate -52 has Q-coordinate 634. The point with P-coordinate -4 has...

Homework Statement Consider a unit sphere centered at the origin. In terms of the Cartesian unit vectors i, j and k, find the unit normal vector on the surface Homework Equations A dot B = AB cos(theta) A cross B = AB (normal vector) sin(theta) Unit sphere radius = 1 The Attempt at a...

Homework Statement Evaluate the force corresponding to the potential energy function ##V (r) = \frac{cz}{r^3}##, where ##c## is a constant. Write your answer in vector notation, and also in spherical polars, and verify that it satisfies ##∇∧F = 0##. Homework Equations ##F(x)=-\frac{dU}{dx}##...

Homework Statement Find the vector directed from (10,3π/4,π/6) to (5, π/4,π), where the endpoints are given in spherical coordinates. Ans -9.660ax, - 3ay. + 10.61az Homework Equations az=rCosΦ The Attempt at a Solution az=10Cos(π/6) +5Cos(π) =13.6 My answer differs. Where did i go wrong?

Homework Statement A particle is described by the state of the following wave function. wavefunction(x,y) = 30/[(a^5)(b^5)]^1/2 * x(a-x) * b(b-y) Homework Equations integral from 0 to i of x^n * (1-x)^m dx = (n!m!)/(n+m+1)! The Attempt at a Solution I know that normalizing means taking the...

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...

Hi, I read somewhere the geodesic distance between an arbitrary point ##x## and the base point ##x_0## in normal coordinates is just the Euclidean distance. Why?! That's the part I don't understand. I know that one can write g_{\mu \nu} = \delta_{\mu \nu} - \frac{1}{6} (R_{\mu \rho \nu \sigma}...

Homework Statement Homework Equations The path integral equation, Stokes Theorem, the curl The Attempt at a Solution [/B] sorry to put it in like this but it seemed easier than typing it all out. I have a couple of questions regarding this problem that I hope can be answered. First...

Homework Statement Consider heat flow in a long circular cylinder where the temperature depends only on t and on the distance r to the axis of the cylinder. Here r=\sqrt{x^2+y^2} is the cylindrical coordinate. From the three-dimensional heat equation derive the equation U_t=k(U_{rr}+2U_r/r)...

Hello, PF! I have some doubts about setting up shell balances in a cylindrical geometry. Consider a fluid flowing down a vertical pipe. In order to perform the momentum balance, we take a cylindrical (annular) shell of length L and width Δr. The analysis of such system can be found in chapter 2...

I have the following integral: ## \oint_{S}^{ } f(\theta,\phi) \hat \phi \; ds ##Where s is a sphere of radius R.so ds = ##R^2 Sin(\theta) d\theta d\phi ## Where ds is a scalar surface element. If I was working in Cartesian Coordinates I know the unit vector can be pulled out of integral and...

Homework Statement Let B1={([u][/1]),([u][/2]),([u][/3])}={(1,1,1),(0,2,-1),(1,0,2)} and B2={([v][/1]),([v][/2]),([v][/3])}={(1,0,1),(1,-1,2),(0,2,1)} a) Show that B1 is a basis for [R][/3] b) Find the coordinates of w=(2,3,1) relative to B1 c)Given that B2 is a basis for [R[/3], find...

If suppose only if the velocities are determined for all N particles can the system be completely determined, can we not extend and say that only if acceleration for all particles are provided can the system be completely determined? For instance can there not be two systems of N particles with...

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...

I am trying to numerically calculate the electric potential inside a truncated cone using the finite element method (FEM). The cone is embedded in cylindrical coordinates (r,phi,z). I am assuming phi-independence on the potential, therefore the problem is essentially 2D; I am working only with...

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...

We know that 4-vectors are invariants, in the sense that they have the same meaning in all reference frames/coordinate systems. We know they transform by the Lorentz transformation in SR, and have an invariant Minkowski norm (let's not bring in GR at this point unless it becomes necessary). It...

Hi all. I'm struggling with taking dot products between vectors in spherical coordinates. I just cannot figure out how to take the dot product between two arbitrary spherical-coordinate vectors ##\bf{v_1}## centered in ##(r_1,\theta_1,\phi_1)## and ##\bf{v_2}## centered in...

Homework Statement Find the mass of the solid bounded from above z = √(25 - x2-y2) and below from z = 4, if its density is δ = k(x^2 + y^2 + z^2)^(-1/2). Homework Equations m = ∫∫∫δdV The Attempt at a Solution The plane z = 4 is transformed into ρcosφ = 4, that is, ρ = 4secφ. And x^2 +...

I'm trying to understand *quote unquote thread title* by performing some simple (heuristic) analysis on my own. Before beginning, I'd like to present what I've been given to understand here at PF: -r is not the distance from the center of a spherical shell to an arbitrary spatial coordinate on...

Is there a way to know the points if I only have the vector coordinates and I can't use the origin as one of the points? For example, if I have vec(PQ) <-1,4,-5> . Is there a way to know the points of this vector?

I understand that one can always construct a set Synchronous coordinates (or Gaussian normal coordinates) on a neighborhood of a point in spacetime. My question is: Does one can construct a metric with only $g_{0i}=0$ such that $dS^2=g_{00}dt^2 + g_{ij}dx^{i} dx^{j}$ (where $i=1,,,,D$ and...

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}} -...

My question is really about converting between spherical coordinates and cartesian coordinates. Suppose that ##\phi## and ##\theta## are defined as follows: ##\phi## is the angle between the position vector of a point and the ##z##-axis. ##\theta## is the angle between the projection of that...

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...

I have just been asked why we use curvilinear coordinate systems in general relativity. I replied that, from a heuristic point of view, space and time are relative, such that the way in which you measure them is dependent on the reference frame that you observe them in. This implies that...

Sorry again for all these ongoing question as I try to fix my math deficiencies. (Back to working on differential forms.) So... I understand that the equation of steepest ascent/descent in Cartesian coordinates is written: dxi/dt = ∂f/∂xi And I can follow the "physical interpretations" of...

Referring to the problem in the attachment, the author mentions that if we consider the coordinate system attached to the bicycle and the bicycle accelerates or decelerates, the flow past the bicycle becomes unsteady. For an unsteady flow, we know that nothing changes at a given location on a...

In the Einstein Field Equations: Rμν - 1/2gμνR + Λgμν = 8πG/c^4 × Tμν, which tensor will describe the coordinates for the curvature of spacetime? The equations above describe the curvature of spacetime as it relates to mass and energy, but if I were to want to graph the curvature of spacetime...

Given an energy functional $ E=\int_{0}^{\infty} \,dr.r\left[\frac{1}{2}\left(\d{\phi}{r}\right)^2 - S.\phi\right] $ I am told that discretizing on a lattice ri=ih (h=lattice size, i is i axis) leads to : $ 2{r}_{i}{\phi}_{i} - {r}_{i+\frac{1}{2}}{\phi}_{i+1} - {r}_{i-\frac{1}{2}}{\phi}_{i-1}...

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...