What is Coordinates: Definition and 1000 Discussions

In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.

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  1. sergiokapone

    Equation of motion in polar coordinates for charged particle

    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...
  2. A

    MHB Rescaled Coordinates in a polynomial equation.

    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...
  3. K

    I Wiki definition of Fermi coordinates

    Is the wikipedia definition of Fermi coordinates accurate? https://en.wikipedia.org/wiki/Fermi_coordinates
  4. jhongg7

    Equilibrium equations - spherical coordinates

  5. E

    Can't work out integral in polar 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 =...
  6. Santilopez10

    Intrinsic coordinates kinematics problem

    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...
  7. K

    I Locally Cartesian Coordinates on the Sphere

    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...
  8. K

    I Light Cone in Rindler Coordinates: Visualization & Reasoning

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

    I Rindler Coordinates & Quadrants: Resolving an Issue

    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...
  10. L

    A Laplace transform in spherical coordinates

    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...
  11. Baibhab Bose

    Effects of KE & PE of a Harmonic Oscillator under Re-scaling of coordinates

    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...
  12. M

    Mathematica Adding coordinates to a matrix

    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 $$ \begin{bmatrix} (1,a) & (1,b) & (1,c)\\ (2,d) & (2,e) & (2,f) \end{bmatrix} $$
  13. L

    A Construct BMS Coordinates near Null Infinity

    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...
  14. SamRoss

    Trying to use polar coordinates to find the distance between two points

    ##{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...
  15. T

    Angle required to hit coordinates including air resistance

    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...
  16. W

    I Centre of mass of a semicircle using polar coordinates

    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...
  17. T

    I Why Are Coordinates Independent in GR? - Exploring the Motivation

    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...
  18. mishima

    I Elliptic Cylinder Coordinates, Acceleration Derivation....options?

    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...
  19. colemc20

    Hollow Sphere Inertia in Cartesian Coordinates

    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...
  20. Z

    I Derivation of Divergence in Cartesian Coordinates

    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: $$\nabla\cdot\mathbf{F} = \lim_{V\rightarrow 0}\frac{1}{V}\int_S\mathbf{F\cdot n}da$$...
  21. A

    I Fixing Rindler Coordinates when a=0

    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...
  22. Luke Tan

    I Transforming Vector Fields between Cylindrical Coordinates

    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...
  23. majormuss

    Visualizing & Solving a 2D Laplace Eq problem (Polar Coordinates)

  24. A

    Doubt about a unit vector in toroidal coordinates

    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...
  25. SamRoss

    I Is there an algebraic derivation of the area element in polar coordinates?

    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 ##...
  26. J

    MHB Polar Coordinates Intersection

    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.
  27. O

    I Radius vector in cylindrical coordinates

    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...
  28. S

    Fortran Generate a circle in FORTRAN having polar coordinates

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

    I Compute Physical Coordinates of an Event in a Curved Spacetime

    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...
  30. J

    MHB Integration in Polar Coordinates (Fubini/Tonelli)

    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...
  31. Pencilvester

    I Parameterized surfaces from coordinates

    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...
  32. jleon008

    Heliocentric Coordinates for a 2 body problem and the Hamiltonian

    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...
  33. A

    Solving a Cipher: Celestial coordinates and precession

    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...
  34. A

    B Is there a standard mapping of celestial coordinates to geo-coordinates

    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...
  35. C

    I Is x{hat} a unit vector and why is theta a vector?

    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}...
  36. K

    I Finding Most General Form of Rindler Coordinates

    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 =...
  37. L

    I Finding the coordinates of a point on a sphere

    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...
  38. stephchia

    Finding the linear mapping between homogeneous coordinates

    Homework Statement If I have an affine camera with a projection relationship governed by: \begin{equation} \begin{bmatrix} x & y \end{bmatrix}^T = A \begin{bmatrix} X & Y & Z \end{bmatrix}^T + b \end{equation} where A is a 2x3 matrix and b is a 2x1 vector. How can I form a matrix...
  39. PhysicS FAN

    Graphs, functions, and coordinates

    Homework Statement If a staight ε: y=(-λ+μ)x +2λ -μ , (where μ and λ are real numbers) passes through point A(0,1) and is parallel to an other straight lin. ζ: y= -2x + 2008 find λ and μ Homework EquationsThe Attempt at a Solution It is clear that when x=0 we know that 2λ-μ=1 which is one of...
  40. CivilSigma

    What are the Benefits of Using Modal Coordinates in Structural Dynamics?

    Homework Statement In structural dynamics of multiple degrees of freedom structures, the solution of the following PDE varies with the respect of the applied load, however in numerous literature I have read, the solution is a combination of modal coordinates and modal shapes: $$m \ddot v + c...
  41. F

    I Polar coordinates and unit vectors

    Hello, I get that both polar unit vectors, ##\hat{r}## and ##\hat{\theta}##, are unit vectors whose directions varies from point to point in the plane. In polar coordinates, the location of an arbitrary point ##P## on the plane is solely given in terms of one of the unit vector, the vector...
  42. Z

    Kinematics in Cylindrical Coordinates

    Homework Statement A small bead of mass m slides on a frictionless cylinder of radius R which lies with its cylindrical axis horizontal. At t = 0 , when the bead is at (R,0), vz = 0 and the bead has an initial angular momentum Lo < mR sqrt(Rg) about the axis of the cylinder where g is the...
  43. F

    Intrinsic coordinates and an intrinsic description of motion

    Hello, For 2D motion, I understand that velocity, position and acceleration of a point object can be described using the fixed basis vector ##\hat {i}## and ##\hat {j}## and the rectangular coordinates ##x(t)## and ##y(t)## which which are functions of time ##t##. Another option is to use...
  44. Mason Smith

    Cylindrical coordinates: unit vectors and time derivatives

    Homework Statement Homework EquationsThe Attempt at a Solution I have found expressions for the unit vectors for cylindrical coordinates in terms of unit vectors in rectangular coordinates. I have also found the time derivatives of the unit vectors in cylindrical coordinates. However, I am...
  45. T

    B Angle required to hit coordinates x, y, z with air ressitance

    The formula for the angle required for you to launch a projectile with a given velocity, gravity, distance and height difference is, taking g as gravity, v as total velocity, x as total distance on the horizontal plane and y as how high the target is above you (Negative value means the target is...
  46. MattIverson

    What are phase space coordinates and how do you plot them?

    Homework Statement I have phase space coordinates (x0,y0,z0,vx,vy,vz)=(1,0,0,0,1,0). I need to analytically show that these phase space coordinates correspond to a circular orbit. Homework Equations r=sqrt(x^2+y^2+z^2) maybe? The Attempt at a Solution My core problem here is maybe that I...
  47. Q

    Cylindrical Coordinates: Line Integral Of Electrostatic Field

    Homework Statement An electrostatic field ## \mathbf{E}## in a particular region is expressed in cylindrical coordinates ## ( r, \theta, z)## as $$ \mathbf{E} = \frac{\sin{\theta}}{r^{2}} \mathbf{e}_{r} - \frac{\cos{\theta}}{r^{2}} \mathbf{e}_{\theta} $$ Where ##\mathbf{e}_{r}##...
  48. Arman777

    I Confusion about the math of the Comoving Coordinates

    From the FLRW metric Proper distance can be derived like this, $$ds^2=-c^2dt^2+a^2(t)[dr^2+S_k(r)^2d\Omega^2]$$ Let us fixed the time at ##t=t_0## for the measurement and assume that the object has only radial component, then the metric equation turns out to be, $$ds^2=a^2(t_0)dr^2$$...
  49. M

    Surface area of a shifted sphere in spherical coordinates

    Homework Statement find the surface area of a sphere shifted R in the z direction using spherical coordinate system. Homework Equations $$S= \int\int \rho^2 sin(\theta) d\theta d\phi$$ $$x^2+y^2+(z-R)^2=R^2$$ The Attempt at a Solution I tried to use the sphere equation mentioned above and...
  50. M

    I Converting from spherical to cylindrical coordinates

    I have the coordinates of a hurricane at a particular point defined on the surface of a sphere i.e. longitude and latitude. Now I want to transform these coordinates into a axisymmetric representation cylindrical coordinate i.e. radial and azimuth angle. Is there a way to do the mathematical...
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