Spherical coordinates Definition and 351 Threads

Now, this is kind of embarrassing, but I've been trying to do this for too long now and failed: I want to construct an atlas for ##S^2##, but I want to use spherical coordinates rather than stereographic projection. Of course the first chart image is simply ##\theta \in (0, \pi), \varphi \in...

Homework Statement So I'm doing a question from one of my past exams as attached, there are no copy right issues with this document that I know of and have asked my lecturer who wrote the exam and he said I am welcome to upload it. The question is 1)b)iv), my attempt is attached. I end up with...

Homework Statement The problem and its solution are attached as TheProblemAndSolution.jpg. Homework Equations V(D) = ∫∫∫_D ρ^2 sinθ dρ dϕ dθ The Attempt at a Solution How exactly does the solution get cos α = 1/√(3)? Also, when the solution goes from the step with two integrals to the step...

Homework Statement I have a PDE and I need to solve it in spherical domain: $$\frac{\partial F(r,t)}{\partial t}=\alpha \frac{1}{r^2} \frac{\partial}{\partial r} r^2 \frac{\partial F(r,t)}{\partial r} +g(r,t) $$ I have BC's: $$ \frac{\partial F}{\partial dr} = 0, r =0$$ $$ \frac{\partial...

Homework Statement There is a sphere of magnetic material in a uniform magnetic field \vec H_0=H_0\vec a_z, and after some calculations I got the magnetic moment vector \vec M_0=M_0\vec a_z, where M_0 is something that isn't important to my question. I am unsure if this magnetic moment vector...

Homework Statement The position-space representation of the radial component of the momentum operator is given by ## p_r \rightarrow \frac{\hbar}{i}\left ( \frac{\partial }{\partial r} + \frac{1}{r}\right ) ## Show that for its expectation value to be real:## \left \langle \psi|p_r|\psi \right...

I am accustomed to ##x=rcos(\theta)sin(\phi)## ##y=rsin(\theta)sin(\phi)## ##z=rcos(\phi)## ##-\pi<\theta<\pi##, ##-\pi/2 < \phi < \pi/2## but see some people using these instead ##x=rcos(\theta)cos(\phi)## ##y=rsin(\theta)cos(\phi)## ##z=rsin(\phi)## ##-\pi<\theta<\pi##, ##-\pi/2 < \phi <...

Homework Statement Integrate $$\int_V \delta^3(\vec r)~ d\tau$$ over all of space by using V as a sphere of radius r centered at the origin, by having r go to infinity. Homework EquationsThe Attempt at a Solution This integral actually came up in a homework problem for my E&M class and I'm...

Is A x B = | i j k | also true for Spherical Coordinates? | r1 theta1 phi1 | | r2 theta2 phi2 | Or I have to convert them to Cartesian Coordinates and do the cross product and then...

Homework Statement I'm feeling a bit ambivalent about my interpretation of spherical coordinates and I'd appreciate it if someone could clarify things for me! In particular, I'd like to know whether or not my derivation of the coordinates is legitimate. Homework Equations Considering...

Homework Statement Hi! This is not really a problem. I'm just confused on how to express the charge distribution of a set of point charges in spherical coordinates. From our discussion, ρ(\vec{r})=\sum\limits_{i=1}^N q_i δ(\vec{r}-\vec{r}') where \vec{r} is the position of the point where...

Homework Statement In spherical coordinates (ρ,θ,ø); I understood the ranges of ρ, and θ. But ø, still eludes my understanding. Why is ø only from 0 to π, why not 0 to 2π??

Dear all, As I was reading my book. It said that the line element of a particular coordinate system (spherical) in R^{3} is so and so. Then it said that the metric is flat. I don't get how the metric is flat in spherical coordinate. Could someone shed some light on this please? Thanks

I just derived the 3-D Cristoffel symbol of the 2nd kind for spherical coordinates. I don't think I made any careless mistakes, but once again, I just want to verify that I am correct and I can't find any place on line that will give me the components of the symbol so I can check myself. Here...

I recently derived a matrix which I believe to be the metric tensor in spherical polar coordinates in 3-D. Here were the components of the tensor that I derived. I will show my work afterwards: g11 = sin2(ø) + cos2(θ) g12 = -rsin(θ)cos(θ) g13 = rsin(ø)cos(ø) g21 = -rsin(θ)cos(θ)...

Hi all, I'm not sure how to get the boundaries in terms of both the spherical and cylindrical coordinates for this question. Here are the boundaries we were given in the solution. How was \frac{\pi}{4} for φ and \frac{1}{\sqrt{2}} for r obtained? Thanks!

1. . An electric dipole located at the origin in free space has a moment p = 3ax −2ay +az nC·m. Find V at r = 2.5, θ =30◦, φ =40◦. I find it difficult to solve when its in spherical co-ordinates.2.Relevent Eq V =P.(r-r')/( 4∏ε|r−r'|2)(|r-r'|)I am confused how to find a unit vector on spherical...

Hey pf! I was thinking about how div(curl(f)) = 0 for any vector field f. However, is this true for div and curl in spherical coordinates? It doesn't seem to be. If not, what needs to happen for this to be true in spherical coordinates?? Thanks all!

Homework Statement Write a triple integral in spherical coordinates that represents the volume of the part of the sphere X^2+Y^2+Z^2=16 that lies in the first octant(where x,y, and z are coordinates are all greater than or equal to zero) Homework Equations So i know this is in...

I am finding the electric field from a spherical shell at a point on the z-axis outside the shell. The shell is centered at the origin,and I am only allowed to use coulomb's law. I want to find dE in spherical coordinates first then transform it to Cartesian before integrating to get E. So I...

Homework Statement Let V be the volume of the solid enclosed by the sphere x^2 + y^2 + z^2 - 2z = 0 , and the hemisphere x^2 + y^2 + z^2 = 9 , z ≥ 0. Find VHomework Equations Using spherical coordinates: x^2 + y^2 + z^2 = ρ^2 z = ρcos(ø) The Attempt at a Solution So I changed both of them to...

I've got a Green's function in which all the impulses are on the line from the north pole to the origin (polar angle θ=0) and terminating with a point impulse at the north pole. I've found its gradient at a field point, and I want to rotate everything to a new coordinate system with the source...

is it logical to ask this question in Spherical coordinates: Using the differential length dl , find the length where r=1 0<Θ<∏/4 ∏/2< θ <∏/4 where Θ is the azimuthal angle. What I mean by ∏/2< θ <∏/4 is that the line is a "diagonal" line which has an ascention of ∏/4 from the xy...

Hello, I am looking into finding geodesic distances for an ellipsoid. I will designate two points then find the distance between them. This will be my geodesic distance. I have put together a schematic (attached) for reference. Ultimately I need to know the distance D as shown on the...

Note: physics conventions, \theta is measured from z-axis We have a vector operator \vec{L} = -i \vec{r} \times \vec{\nabla} = -i\left(\hat{\phi} \frac{\partial}{\partial \theta} - \hat{\theta} \frac{1}{\sin\theta} \frac{\partial}{\partial \phi} \right) And apparently \vec{L}\cdot\vec{L}=...

Homework Statement Find the surface area of the Earth (as a fraction of the total surface of the earth) that lies above 50 degrees latitude North. Homework Equations $$A = \int_R\sqrt{|\det(g)|}d\theta d\phi$$ The Attempt at a Solution Hence I get $$\int_0^{360}...

I was looking at the Wikipedia entry on the Hamilton-Jacobi equation, and was confounded by the equation at the beginning of the section on spherical coordinates: http://en.wikipedia.org/wiki/Hamilton–Jacobi_equation#Spherical_coordinates Shouldn't the Hamiltonian simply be $$ H =...

1) If u(r,\theta,\phi)=\frac{1}{r}, is \frac{\partial{u}}{\partial {\theta}}=\frac{\partial{u}}{\partial {\phi}}=0 because u is independent of \theta and \;\phi? 2) If u(r,\theta,\phi)=\frac{1}{r}, is: \nabla^2u(r,\theta,\phi)=\frac{\partial^2{u}}{\partial...

So, I was curious about this and found more or less what I was looking for here: http://electron9.phys.utk.edu/vectors/3dcoordinates.htm Except, something is bothering me about those equations. At the very bottom, the equation for Theta in a spherical coordinate system; shouldn't it be...

Homework Statement Evaluate the iterated integral ∫ (from 0 to 1) ∫ [from -sqrt(1-x^2) to sqrt(1-x^2) ] ∫ (from 0 to 2-x^2-y^2) the function given as √(x^2 + y^2) dz dy dx The Attempt at a Solution I changed the coordinates and I got the new limits as ∫(from 0 to pi) ∫(from...

I recently had to do an integral like the one in the thread below: http://math.stackexchange.com/questions/142235/three-dimensional-fourier-transform-of-radial-function-without-bessel-and-neuman The problem I had was also evaluating the product and I am quite sure that the answer in the thread...

I'm not sure whether this falls in the homework category, or the standard calculus section, so apologies in advance if this doesn't fall in the right category. Homework Statement Evaluate ∫∫∫e^[(x^2 + y^2 + z^2)^3/2]dV, where the region is the unit ball x^2 + y^2 + z^2 ≤ 1. (or any...

Homework Statement Say I am given a spherically symmetric potential function V(r), written in terms of r and a bunch of other constants, and say it is just a polynomial of some type with r as the variable, \frac{q}{4\pi\varepsilon_o}P(r), and we are inside the sphere of radius R, so r<R…...

Q: Consider the solid that lies above the cone z=√(3x^2+3y^2) and below the sphere X^2+y^2+Z^2=36. It looks somewhat like an ice cream cone. Use spherical coordinates to write inequalities that describe this solid. What I tried to do: I started by graphing this on a 3D graph at...

Hello MHB, So when I change to space polar I Dont understand how facit got \frac{\pi}{4} \leq \theta \leq \frac{\pi}{2} Regards, |\pi\rangle \int\int\int_D(x^2y^2z)dxdydz where D is D={(x,y,z);0\leq z \leq \sqrt{x^2+y^2}, x^2+y^2+z^2 \leq 1}

Homework Statement Exercise 1.3 on uploaded Problem Sheet. Homework Equations Shown in Exercise 1.3 on Problem Sheet The Attempt at a Solution Uploaded working: I have found the inverse of the Transformation Matrix from Cartesian to Spherical Coordinates by transposing...

Homework Statement In some region of space, the electric field is \vec{E} =k r^2 \hat{r} , in spherical coordinates, where k is a constant. (a) Use Gauss' law (differential form) to find the charge density \rho (\vec{r}) . (b) Use Gauss' law (integral form) to find the total charge...

Homework Statement OK, we've been asked to derive the equations of motion in spherical coordinates. According to the assignment, we should end up with this: $$ \bf \vec{v} \rm = \frac{d \bf \vec{r} \rm}{dt} = \dot{r} \bf \hat{r} \rm + r \dot{\theta}\hat{\boldsymbol \theta} \rm + r...

Problem: For the vector function \vec{F}(\vec{r})=\frac{r\hat{r}}{(r^2+{\epsilon}^2)^{3/2}} a. Calculate the divergence of ##\vec{F}(\vec{r})##, and sketch a plot of the divergence as a function ##r##, for ##\epsilon##<<1, ##\epsilon##≈1 , and ##\epsilon##>>1. b. Calculate the flux of...

Problem: Say we have a vector function ##\vec{F} (\vec{r})=\hat{\phi}##. a. Calculate ##\oint_C \vec{F} \cdot d\vec{\ell}##, where C is the circle of radius R in the xy plane centered at the origin b. Calculate ##\int_H \nabla \times \vec{F} \cdot d\vec{a}##, where H is the hemisphere...

Homework Statement The formula for divergence in the spherical coordinate system can be defined as follows: \nabla\bullet\vec{f} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 f_r) + \frac{1}{r sinθ} \frac{\partial}{\partial θ} (f_θ sinθ) + \frac{1}{r sinθ}\frac{\partial f_\phi}{\partial...

Hi, can I say that a sphere is a plane, because in spherical coordinates, I can simply express it as <(r, \theta, \varphi)^T, (1, 0, 0)^T> = R? It does sound too easy to me. I'm asking because I'm thinking about whether it is valid to generalize results from the John-Radon transform (over...

Is partial derivative of ##u(x,y,z)## equals to \frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z} Is partial derivative of ##u(r,\theta,\phi)## in Spherical Coordinates equals to \frac{\partial u}{\partial r}+\frac{\partial u}{\partial...

Hi guys, This isn't really a homework problem but I just need a bit of help grasping rotations in spherical coordinates. My main question is, Is it possible to rotate a vector r about the y-axis by an angle β if r is expressed in spherical coordinates and you don't want to convert r...

Hi, Started to learn about Jacobians recently and found something I do not understand. Say there is a vector field F(r, phi, theta), and I want to find the flux across the surface of a sphere. eg: ∫∫F⋅dA Do I need to use the Jacobian if the function is already in spherical...

Hi ! I'm trying to inverse a mass matrix so I need to do something like this \dfrac{q}{\partial \mathbf{r}} where \cos q = \dfrac{\mathbf{r}\cdot \hat{\mathbf{k}}}{r} However, when \mathbf{r} = \hat{\mathbf{k}} \text{ or } -\hat{\mathbf{k}} I have problems. ¿What can I do...

I want to verify: \vec A=\hat R \frac{k}{R^2}\;\hbox{ where k is a constant.} \nabla\cdot\vec A=\frac{1}{R^2}\frac{\partial (R^2A_R)}{\partial R}+\frac{1}{R\sin\theta}\frac{\partial (A_{\theta}\sin\theta)}{\partial \theta}+\frac{1}{R\sin\theta}\frac{\partial A_{\phi}}{\partial \phi}...

Homework Statement Calculate the moment of inertia of a uniformly distributed sphere about an axis through its center. Homework Equations I know that $$I= \frac{2}{5} M R^{2},$$ where ##M## is the mass and ##R## is the radius of the sphere. However, for some reason, when I do this...

http://en.wikipedia.org/wiki/Gradient#Cylindrical_and_spherical_coordinates which formula do we apply to get the gradient in spherical coordinates?

Hi Say I have a point on a unit sphere, given by the spherical coordinate $(r=1, \theta, \phi)$. Is this point equivalent to the point that one can obtain by $(x,y,z)=(1,0,0)$ around the $y$-axis by an angle $\pi/2-\theta$ and around the $z$-axis by the angle $\phi$? I'm not sure this is...