Coordinates Definition and 1000 Threads

I know that an arc length in polar coordinates can be computed by integrating $$\int ds$$ using the formula ##ds=\sqrt{\rho^2 + \frac{dr}{d\theta}^2}d\theta##. But, seeing that ##s=\rho\theta## and ##ds = \rho d\theta##, why is it wrong to calculate arc lengths with this expression for ##ds##?

The following integral arises in the calculation of the new density of a non-uniform elastic medium under stress: ∫dx ρ(r,θ)δ(x+u(x)-x') where ρ is a known mass density and u = ru_r+θu_θ a known vector function of spherical coordinates (r,θ) (no azimuthal dependence). How should the Dirac...

Homework Statement Provide an expression in Cartesian coordinates for a plane wave of amplitude 1 [V/m] and wavelength 700 nm propagating in u = cosθx + sinθy direction, where x and y are unit vectors along the x and y-axis and θ is the measured angle from the x axis. Homework Equations...

Homework Statement Homework EquationsThe Attempt at a Solution I have stared at this for hours and don't know where to start. I think I need to get r in terms of t but I don't really know how with the information given. I just need a good hint to get started.

Homework Statement Sketch each of the following vector fields. E_5 = \hat \phi r E_6 = \hat r \sin(\phi) I wish to determine the \hat x and \hat y components for the vector fields so that I can plot them using the quiver function in MATLAB. Homework Equations A cylindrical coordinate...

Homework Statement We need to find the shortest distance between two given cities. For this I'll use Bangkok, Thailand (13°N, 100°E) and Havana, Cuba (23°N, 82°W ). Earth is assumed to be perfectly spherical with a radius of 6.4x106m. These aren't the places we were given but the coordinates...

I have seen both rk and qj both used to represent generalized coordinates in the Lagrange equations. Are these both the same things? Does it matter which you use? Thanks!

Hello people! I have ended up to this integral ##\int_{φ=0}^{2π} \int_{θ=0}^π \sin θ \ \cos θ~Y_{00}^*~Y_{00}~dθ \, dφ## while I was solving a problem. I know that in spherical coordinates when ##\vec r → -\vec r## : 1) The magnitude of ##\vec r## does not change : ##r' → r## 2) The angles...

Hi this isn't my homework, but it is taken from a worksheet for a Maths course(trying to refresh my rusty math), so I hope it fits in here. 1. Homework Statement two cylindrical polar vectors with same origin: P(2,55°,3); Q(4,25°,6) units in m Homework Equations a) Express in cartesian...

What is the physical interpretation of Cartesian coordinates in GR? Say, e.g., a system centered at the center of a spherical mass. What are x,y, and z physically, i.e., how are they measured?

I'm stuck on a seemingly simple 2D electrostatics problem. The problem is as follows: A parabolic interface ($$x(y)=cy^2$$) separates two regions of different conductivities, with a uniform electric field at infinity aligned with the x-axis. I write the Laplace operator in parabolic...

I am looking at this derivation of velocity in spherical polar coordinates and I am confused by the definition of r, theta and phi. http://www.usna.edu/Users/math/rmm/SphericalCoordinates.pdf I thought phi was the co latitude in the r,θ,∅ system and not the latitude. Of course the two are...

First post ! I hope that my question will not make some long time physicists laugh. It is about geometrical quantization and the phase space in which we use : z=1/sqrt(2)(q+ip) My question is simple what is this 1/sqrt(2) ? And what is it is interpretation ? Thank you !

I am looking to understand more about ##a=(\ddot{r}-r(\ddot{\theta})^2)\hat{r}+(r\ddot{\theta}+2\dot{r}\dot{\theta})\hat{\theta}## I understand the terms ##\ddot{r}## and ##r\ddot{\theta}## ,but why ##-r(\ddot{\theta})^2## has opposite direction to ##\hat{r}## and why ##2\dot{r}\dot{\theta}##...

Homework Statement A sphere of mass M and Radius R had two spheres of R/4 removed. the centres of cavities are R/4 and 3R/4 from the centre of the original sphere (at x=0). what is the x coordinate of the centre of mass of this object? there is a drawing next to the question literally showing...

Homework Statement Find the volume of the solid lying inside both the sphere x^2 + y^2 + z^2 = 4a^2 and the cylinder x^2 + y^2 = 2ay above the xy plane. Homework Equations Polar coordinates: r^2 = x^2 + y^2 x = r\cos(\theta) y = r\sin(\theta) The Attempt at a Solution So I tried this...

I was just trying to write out the derivation for an object's trajectory from an inertial coordinate system if the object is rotating in another coordinate system (e.g. finding Coriolis, centrifugal acceleration). I seem to have gotten something close to what I was looking for, but after...

Homework Statement Homework EquationsThe Attempt at a Solutionhere is the setup for each, can someone check if they are correct before I evaluate the volume?

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...