Coordinates Definition and 1000 Threads

[Mentor's note: This post was moved from another thread as it raised a new question, off-topic in the originating thread] Albert Einstein in his book Relativity wrote " It is impossible to build up a system(reference body) from rigid bodies and clocks,which shall be of such a nature that...

I'm trying to derive what ##ds^2## equals to in spherical coordinates. In Euclidean space, $$ds^2= dx^2+dy^2+dz^2$$ Where ##x=r \ cos\theta \ sin\phi## , ##y=r \ sin\theta \ sin\phi## , ##z=r \ cos\phi## (I'm using ##\phi## for the polar angle) For simplicity, let ##cos...

Homework Statement Homework Equations (y-y1)/(x-x1)=mThe Attempt at a Solution I have attempted part i but I don't know how to do part ii. As point B is still part of the curve and the normal, do I still sub with the same normal eqn? :/ I have no idea how to start... Please help thanks[/B]

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...

Given the equation for a Gaussian as: ##z = f(x,y) = Ae^{[(x-x0)^2 + (y-y0)^2] /2pi*σ^2 }## , how would I go about converting this into cylindrical coordinates? The mean is non-zero, and this seems to be the biggest hurdle. I believe I read earlier that the answer is ~ ##z = f(r,θ) =...

Homework Statement Evaluate \int \int \int _R (x^2+y^2+z^2)dV where R is the cylinder 0\leq x^2+y^2\leq a^2, 0\leq z\leq h Homework Equations [/B] x = Rsin\phi cos\theta y = Rsin\phi sin\theta z = Rcos\phiThe Attempt at a Solution [/B] 2*\int_{0}^{\pi/2}d\phi \int_{0}^{2\pi}d\theta...

Homework Statement Homework EquationsThe Attempt at a Solution a) The schrödinger equation $$i \hbar \frac {\partial \Psi}{\partial t} = - \frac {\hbar^{2}}{2m} \nabla^{2} \psi + V \psi $$ For the case ##0 \le x,y,z \le a##, ##V = 0## $$i \hbar \frac {\partial \Psi}{\partial t} = - \frac...

Hi, Set up the triple integral in spherical coordinates to find the volume bounded by $$z = \sqrt{4-x^2-y^2}$$, $$z=\sqrt{1-x^2-y^2}$$, where $$x \ge 0$$ and $$y \ge 0$$. $$\int_0^{2\pi} \int_0^2 \int_{-\sqrt{4-x^2-y^2}}^{\sqrt{4-x^2-y^2}} r\ dz\ dr\ d\theta$$

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...

The angular equation: ##\frac{d}{d\theta}(\sin\theta\,\frac{d\Theta}{d\theta})=-l(l+1)\sin\theta\,\Theta## Right now, ##l## can be any number. The solutions are Legendre polynomials in the variable ##\cos\theta##: ##\Theta(\theta)=P_l(\cos\theta)##, where ##l## is a non-negative integer...

Homework Statement There is a point P(x,y) and now I rotate the x-y axis, say by θ degree. What will be the coordinates of P from this new axis. I have google but found formula for new coordinates when the points is rotated by θ degree. So I tried my own. So is there other simplified formula...

I have found via integration that the y coordinate is $$y =h/2 = 120 mm$$. The x coordinate is $$x = \frac{-4r}{3\pi} = -51.9mm$$ and the z coordinate is $$z = r - \frac{4r}{3\pi} = 69.1 mm$$. I have no answers in my textbook so can't confirm whether i am correct or not.

Homework Statement The question asks me to convert the following integral to spherical coordinates and to solve it Homework EquationsThe Attempt at a Solution just the notations θ = theta and ∅= phi dx dy dz = r2 sinθ dr dθ d∅ r2 sinθ being the jacobian and eventually solving gets me ∫ ∫ ∫...

Probably a really stupid question.. ##u=t+r+2M ln(\frac{r}{2M}-1) ## From this I get ##\frac{du}{dr}=(1-\frac{2M}{r})^{-1}## But, 1997 Sean M. Carroll lectures notes get ##\frac{du}{dr}=2(1-\frac{2M}{r})^{-1}## . (equation 7.71). No idea where this factor of 2 comes from. Thanks

Homework Statement The coordinate of an automobile in meters is x(t) = 5 + 3t + 2t2 and y(t) = 7 + 2t + t3, where t is in seconds. What is the instant acceleration of the car at time t = 2 s? ANSWERS: A. 10.2 m/s2 B. 9.5 m/s2 C. 7.9 m/s2 D. 15.0 m/s2 E. 12.6 m/s2 Homework Equations ains =...

Homework Statement ##x=r\sin\theta\cos\phi,\,\,\,\,\,y=r\sin\theta\sin\phi,\,\,\,\,\,z=r\cos\theta## ##\hat{x}=\sin\theta\cos\phi\,\hat{r}+\cos\theta\cos\phi\,\hat{\theta}-\sin\phi\,\hat{\phi}## ##\hat{y}=\sin\theta\sin\phi\,\hat{r}+\cos\theta\sin\phi\,\hat{\theta}+\cos\phi\,\hat{\phi}##...

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...

Homework Statement A vase is filled to the top with water of uniform density f = 1. The side profile of the barrel is given by the surface of revolution obtained by revolving the graph of g(z) = 2 + cos(z) over the z-axis, and bounded by 0 ≤ z ≤ π. Find the mass of the vase. Homework...

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

Homework Statement My textbook states that when ##\theta = c ## where c is the constant angle with respect to the x-axis, the graph is a "half-plane". However, when ##\phi = c ## it is a half-cone. The only difference I see is that ##\phi## is the angle with respect to the z-axis, rather than...

Homework Statement Evaluate ## \int \int \int_E {x}dV ## where E is enclosed by the planes ##z=0## and ##z=x+y+5## and by the cylinders ##x^2+y^2=4## and ##x^2+y^2=9##. Homework Equations ## \int \int \int_E {f(cos(\theta),sin(\theta),z)}dzdrd \theta ## How do I type limits in for...

[Mentors note: this thread was split off from an older discussion of Rindler coordinates] Can somebody help me understand why acceleration along the hyperbola is constant? To be more precise: assume (X,T) is the inertial coordinates and (x,t) the corresponding Rindler transformations...

We went over this concept quite fast in class and there is one thing that confused me: When transforming from a set of ##q_i## and ##p_i##to ##Q_i## and ##P_i##, if one checks that the transormations are canonical the new Hamiltonian ##K(Q_i, P_i)## obeys exactly the same equations.This has...

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...

Homework Statement [/B] Matrix A = 1 1 0 1 Matrix B = a b c d Find coordinates on a, b, c, d such that AB = BA. Homework EquationsThe Attempt at a Solution I calculated AB and BA with simple matrix multiplication, but am not sure where to go from here. AB = a + c a + b...

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...

Hey! :o Using spherical coordinates and the orthonormal system of vectors $\overrightarrow{e}_{\rho}, \overrightarrow{e}_{\theta}, \overrightarrow{e}_{\phi}$ describe each of the $\overrightarrow{e}_{\rho}$, $\overrightarrow{e}_{\theta}$ and $\overrightarrow{e}_{\phi}$ as a function of...

By 'proper frame' of observer O, I mean any reference frame (coordinate system) in which (Condition A:) The worldline of O is always at the spatial origin for every time coordinate. Clearly such a frame is not unique because spatial rotations do not invalidate (A). What I am interested in is...

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##...

Hi everyone, I am new to the physics forums and I need your help :) I understand that depending on the symmetry of the problem, it may be easier to change the coordinate system you are using. My question is, how would I convert the electric field due to a point charge at the origin, from...

Hey! :o Using cylindrical coordinates and the orthonormal system of vectors $\overrightarrow{e}_r, \overrightarrow{e}_{\theta}, \overrightarrow{e}_z$ describe each of the $\overrightarrow{e}_r$, $\overrightarrow{e}_{\theta}$ and $\overrightarrow{e}_z$ as a function of $\overrightarrow{i}...

Hey! :o We are given the following point in cylindrical coordinates. We have to write in orthogonal and spherical coordinates. The point is $\left (2, \frac{\pi}{2}, -4\right )$. First of all, do orthogonal coordinates mean cartesian coordinates?? (Wondering) The cylindrical coordinates...

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...

I want to prove that the barycentric coordinates of a point $P$ inside the triangle with vertices in $(1,0,0), (0,1,0), (0,0,1)$ are distances from $P$ to the sides of the triangle. Let's denote the triangle by $ABC, \ A = (1,0,0), B=(0,1,0), C= (0,0,1)$. We consider triangles $ABP, \ BCP, \...

Homework Statement The question and my attempt are attached as pics Homework EquationsThe Attempt at a Solution I can't seem to find r¨ and θ¨. Assuming I already got r˙ and θ˙ (the answers are written after the question). The idea I tried was to get the acceleration equation in cylindrical...

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...

Let A(a, b, c) and A'(a′,b′,c′) be two distinct points in R3. Let f from [0 , 1] to R3 be defined by f(t) = (1 -t) A + t A'. Instead of calling the component functions of f ,(f1, f2, f3) let us simply write f = (x, y, z). Express x; y; z in terms of the coordinates of A and A, and t. I thought...

I can start explaining the problem but a more quicker way would be to open this link: http://onlinelibrary.wiley.com/doi/10.1002/9783527627486.app2/pdf and check the paragraph resulting in expression (B.5). Note that I don't really care about the kinetic energy they talk about in this link...

Homework Statement Working in Cartesian coordinates (x,y,z) and given that the function g is independent of x, find the functions f and g such that: v=coszi+f(x,y,z)j+g(y,z)k is a Beltrami field. Homework Equations From wolfram alpha a Beltrami field is defined as v x (curl v)=0 The Attempt...

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...

After setting up Laplace's equation in spherical coordinates and separating the variables, it is not clear to me why the constants are put in the form of l(l+1) and why m runs from -l to l. Could anyone please help me ununderstand, or better yet, point me to a source that explains the entire...

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...

Hi, I am going through the derivation of an instanton solution (n=1) in Srednicki Chp. 93. Specifically, I went through eqn.s 93.29-93.38. However the sign of the Levi-Civita Symbol is bugging me: It says that in 4D Euclidean space, \epsilon^{1234}=+1 in Cartesian coordinates implies...

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...