Coordinates Definition and 1000 Threads

Ive found ##\delta x/\delta r## as ##sin\theta cos\phi## ##\delta r/\delta x## as ##csc\theta sec\phi## But unsure how to do the second part? Chain rule seems to give r/x not x/r?

Let me begin by stating that I'm aware of the fact that this is a metric of de Sitter spacetime, aka I know the solution, my problem is getting there. My idea/approach so far: in the coordinates ##(u,v)## the metric is given by $$g_{\mu\nu}= \begin{pmatrix}1 & 0\\ 0 & -u^2\end{pmatrix}.$$ The...

Hi, I’ve read a set of coordinate elements: „roll, pitch and heading“ for aircrafts. What’s the difference to „roll, pitch and yaw“? Senmeis

In spherical poler coordinates the volume integral over a sphere of radius R of $$\int^R_0\vec \nabla•\frac{\hat r}{r^2}dv=\int_{surface}\frac{\hat r}{r^2}•\vec ds$$ $$=4\pi=4\pi\int_{-\inf}^{inf}\delta(r)dr$$ How can it be extended to get $$\vec \nabla•\frac{\hat r}{r^2}=4\pi\delta^3(r)??$$

B is the midpoint of AC( LIne segment) and E is the midpoint of BD( LIne segment). If A(-9, -4), C(-1,6) and E(-4,-3), find the cooridinate of D. I got really lost on that

Hello there, I'm struggling in this problem because i think i can't find the right ##\theta## or ##r## Here's my work: ##\pi/4\leq\theta\leq\pi/2## and ##0\leq r\leq 2\sin\theta## So the integral would be: ##\int_{\pi/4}^{\pi/2}\int_{0}^{2\sin\theta}\sin\theta dr d\theta## Which is equal to...

I'm watching this lecture that gives an introduction to tensors. If we have a coordinate system that's an affine transformation of the Cartesian coordinate system, then the projection of a vector ##v## (onto a particular axis) is defined as ##v_m = v.e_m## or the dot product of the vector with...

r,θ,ϕ For integration over the ##x y plane## the area element in polar coordinates is obviously ##r d \phi dr ## I can also easily see ,geometrically, how an area element on a sphere is ##r^2 sin\theta d\phi ## And I can verify these two cases with the Jacobian matrix. So that's where I'm at...

Hi! I'm studying Shankar's Principle of quantum mechanics I didn't get the last conclusion, can someone help me understand it, please. Where did the l over rho come from?

Recently, I've been studying about Lorentz boosts and found out that two perpendicular Lorentz boosts equal to a rotation after a boost. Below is an example matrix multiplication of this happening: $$ \left( \begin{array}{cccc} \frac{2}{\sqrt{3}} & 0 & -\frac{1}{\sqrt{3}} & 0 \\ 0 & 1 & 0 & 0...

The area differential ##dA## in Cartesian coordinates is ##dxdy##. The area differential ##dA## in polar coordinates is ##r dr d\theta##. How do we get from one to the other and prove that ##dxdy## is indeed equal to ##r dr d\theta##? ##dxdy=r dr d\theta## The trigonometric functions are used...

Is there a straight-forward, motivated, derivation of AdS Poincare coordinates, e.g. as given here: https://en.wikipedia.org/wiki/Anti-de_Sitter_space#Poincar.C3.A9_coordinates starting from global coordinates, as given here...

Lets say you have a vector in spherical coordinates; how do you rewrite this vector into a cartesian one and vice versa? Im fine with rewriting coordinates but vectors have got me confused. I've tried digging through info online but I couldn't find any good examples. In the following task...

A solution of equations of motion for charged particle in a uniform magnetic field are well known (##r = const##, ## \dot{\phi} = const##). But if I tring to solve this equation using only mathematical background (without physical reasoning) I can't do this due to entaglements of variables...

I have this question from Murdock's textbook called: "Perturbations: Methods and Theory": Use rescaling to solve: $\phi(x,\epsilon) = \epsilon x^2 + x+1 = 0$ and $\varphi (x,\epsilon) = \epsilon x^3+ x^2 - 4=0$. I'll write my attempt at solving these two equations, first the first polynomial...

Is the wikipedia definition of Fermi coordinates accurate? https://en.wikipedia.org/wiki/Fermi_coordinates

I considered the work done by the frictional force in an infinitesimal angular displacement: $$dW = Frd\theta = (kr\omega) rd\theta = kr^{2} \frac{d\theta}{dt} d\theta$$I now tried to integrate this quantity from pi/2 to 0, however couldn't figure out how to do this$$W =...

So I know that ##a_t = \frac{dv}{dt}=-ks## and ##\frac{dv}{dt}=v\frac{dv}{ds}## then: $$v dv=-ks ds \rightarrow (v(s))^2=-ks^2+c$$ and using my initial conditions it follows that: $$(3.6)^2=c \approx 13$$ and $$(1.8)^2=13-5.4k \rightarrow k=1.8 \rightarrow (v(s))^2=13-1.8s$$ What bothers me is...

I was trying to construct locally Euclidean metrics. Consider the sphere with the usual coordinate system induced from spherical coordinates in ##\mathbb R^3##. Consider a point ##p## in the Equator having coordinates ##(\theta_0, \phi_0) = (\pi/2, 0)##. If you make the coordinate change ##\xi^1...

Im trying to visualize what form the light cones take in Rindler coordinates. Below is my drawing + reasoning. Is it right?

Im reading a text where the author says that the Rindler coordinates cover the first quadrant of Minkowski space and thus can be used as coordinates there. He is considering only 1 spatial dimension. I learned in high school that a quadrant is one quarter of an Euclidean plane. I looked up...

Summary: A 1963 paper by Michael Wertheim uses a Laplace transform in spherical coordinates. How is the resulting equation obtained? In 1963, Michael Wertheim published a paper (relevant page attached here), where he presented the following equation (Eq. 1): $$ y(\bar{r}) = 1 + n...

The wavefunction is Ψ(x,t) ----> Ψ(λx,t) What are the effects on <T> (av Kinetic energy) and V (potential energy) in terms of λ? From ## \frac {h^2}{2m} \frac {\partial^2\psi(x,t)}{\partial x^2} + V(x,t)\psi(x,t)=E\psi(x,t) ## if we replace x by ## \lambda x ## then it becomes ## \frac...

Hi PF! Given a matrix and vector $$ \begin{bmatrix} a & b & c\\ d & e & f \end{bmatrix},\\ \begin{bmatrix} 1\\ 2 \end{bmatrix} $$ how can I merge the two to have something like this $$ \begin{bmatrix} (1,a) & (1,b) & (1,c)\\ (2,d) & (2,e) & (2,f) \end{bmatrix} $$

Let us consider Ashtekar's definition of asymptotic flatness at null infinity: I want to see how to construct the so-called Bondi coordinates ##(u,r,x^A)## in a neighborhood of ##\mathcal{I}^+## out of this definition. In fact, a distinct approach to asymptotic flatness already starts with...

##{dx}^2+{dy}^2=3^2+3^2=18## ##{dr}^2+r^2{d\theta}^2=0^2+3^2*(\theta/2)^2\neq18## I have a feeling that what I'm doing wrong is just plugging numbers into the polar coordinate formula instead of treating it as a curve. For example, I naively plugged in 3 for r even though I know the radius...

Hi, I know I've asked this before but I didn't manage to solve the problem before. To give context I'm trying to find the angle to hit a target with given coordinates from my current location in a particular game. (I'm modding the game) I can do it with zero problems when not including air...

I am labelling this as undergraduate because I got it from an undergraduate physics book (Tipler and Mosca). The uniform semicircle has radius R and mass M. I am getting the wrong answer but I can't see where I am going wrong. Any help would be appreciated. My solution: The centre of mass...

I can see that by the tensor transformation law of the Kronecker delta that ##\frac{\partial x^a}{\partial x^b}=\delta^a_b## And thus coordinates must be independent of each other. But is there a more straightforward and fundamental reason why we don’t consider dependent coordinates? Is it...

I've been deriving ds, velocity and acceleration for an elliptic cylindrical coordinate system. When it comes to ds and velocity, its quite simple and quick. The acceleration however is tedious by my current method and I'm wondering if there is some shortcut or superior method I'm not aware...

Problem Statement: How do you calculate the rotational inertia of a hollow sphere in cartesian (x,y) coordinates? Relevant Equations: I=Mr^2 My physics teacher said its his goal to figure this out before he dies. He has personally solved all objects inertias in cartesian coordinates but can't...

In section 1-5 of the third edition of Foundations of Electromagnetic Theory by Reitz, Milford and Christy, the authors give a coordinate-system-independent definition of the divergence of a vector field: $$\nabla\cdot\mathbf{F} = \lim_{V\rightarrow 0}\frac{1}{V}\int_S\mathbf{F\cdot n}da$$...

Rindler coordinates are nice, but they fall apart when a=0, where ##T=\frac{sinh(at)}{a}=\frac{0}{0}##. Is there a good way to fix that? Intuitively I'd want to do out the taylor expansion, divide by a, then collapse it back to... something... $$T=\frac {\sinh(at)} {a}=\frac{ \sum_{n=0}^\infty...

In dealing with rotating objects, I have found the need to be able to transform a vector field from cylindrical coordinate systems with one set of coordinate axes to another set. For eg i'd like to transform a vector field from being measured in a set of cylindrical coordinates with origin at...

The system considers a torus that has a wire wrapped around it, through which a current flows. In this way, a field originates in the phi direction. The direction of current is "theta" in the spherical coordinate system but in toroidal system, in several book shows that the electrical current...

There is a simple geometric derivation of the area element ## r dr d\theta## in polar coordinates such as in the following link: http://citadel.sjfc.edu/faculty/kgreen/vector/Block3/jacob/node4.html Is there an algebraic derivation as well beginning with Cartesian coordinates and using ##...

Determine the polar coordinates of the two points at which the polar curves r=5sin(theta) and r=5cos(theta) intersect. Restrict your answers to r >= 0 and 0 <= theta < 2pi.

I am starting to learn classical physics for my own. One exercise was, to calculate the vector r (see picture: 1.47 b). The vector r is r=z*z+p*p. I don’t understand this solution. My problem is: in a vector space with n dimensions there are n basis vectors. In the case of cylindrical...

Say "I have grid in polar coordinates (r, theta). How do I plot it in tecplot. Tecplot plots it in cartesian coordinates."

Hi everyone, this is my first post on PhysicsForums. Thank you so much in advance for your help! My question is the following. Let us suppose we have an event A in a curved spacetime which, for definiteness, is the spacetime curved by the bodies of the solar system. Adopting a coordinate system...

Let $S^{n-1} = \left\{ x \in R^2 : \left| x \right| = 1 \right\}$ and for any Borel set $E \in S^{n-1}$ set $E* = \left\{ r \theta : 0 < r < 1, \theta \in E \right\}$. Define the measure $\sigma$ on $S^{n-1}$ by $\sigma(E) = n \left| E* \right|$. With this definition the surface area...

For all parameterized (hyper)surfaces that form smooth manifolds of dimension ##n-1## embedded in Euclidean ##\mathbb {R}^n##, will there always exist a coordinate system ##\partial_{\bar \mu}## on ##\mathbb {R}^n## that yields the same manifold when the right coordinate (say ##\partial_1##) is...

It seems as though in general, the equations of motion are described with two equations which result from the definition of a Hamiltonian problem, where the problems are of the form: $$\dot p=-H_q(p,q), \dot q=H_p(p,q)$$ It is a little confusing to me how the equations of motion go from two...

You guys were really helpful last time I came to you. Let's hope you can do it again. I have a sort of weird question, in a weird context. It's pretty complex, which is why I'm asking for help from experts. Let me explain. (Better grab a beverage, this will take awhile.) . I am not even an...

Sorry, I'm not an astronomer. This question relates to the book "S." by Doug Dorst. I understand that the celestial coordinates have a zero-point at the vernal equinox. (0h, 0m, 0s RA, 0⁰, 0", 0' Dec.) I also understand that it's possible to map these coordinates to spherical, or...

I have a physics test tommorow, and my physics professor said this homework was important. However I have been having difficulty interpreting it. Can someone help me with this. Is x{hat} a unit vector. The part that says +x{hat} axis seems to imply that, but then where does the length for r{hat}...

I'm searching, but so far I have not found a derivation of the coordinates shown by wikipedia in the very beggining of https://en.wikipedia.org/wiki/Rindler_coordinates#Characteristics_of_the_Rindler_frame. It seems obvious from the relation ##X^2 - T^2 = 1 / a^2##, (##c = 1##), that ##X =...

I have three points: A, B and C, which are all on the surface of the same sphere. I need to find the xyz coordinates of C. What I know: - the radius of the sphere - the origin of the sphere - the xyz coordinates of A and B - the arc distance from A to C and from B to C - the angle between AB and...