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. Gbox

    Acceleration in Plance Polar Coordinates

    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}##...
  2. M

    How to find the x coordinates of the centre of mass?

    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...
  3. W

    Double Integral in Polar Coordinates Symmetry Issue

    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...
  4. T

    Derivation of acceleration in rotating coordinates

    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...
  5. C

    Rec, Spherical and cylind coordinates.

    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?
  6. X

    Expressing A Quantity In Polar Coordinates?

    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...
  7. B

    Electromagnetic Waves in Spherical Coordinates

    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...
  8. H

    Substituting spherical coordinates to evaluate an integral

    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...
  9. M

    Double integrals: cartesian to polar coordinates

    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...
  10. M

    Surface element in cylindrical coordinates

    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...
  11. shanepitts

    Transforming Cartesian to Polar Coordinates

    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...
  12. Geofleur

    Relativistic Euler Equation in Spherical Coordinates

    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...
  13. H

    Calculating the line element after a change of coordinates

    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...
  14. I

    Gradient and curvilinear coordinates

    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...
  15. I

    Line integral in spherical coordinates

    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...
  16. I

    MATLAB Matlab polar to rectangular coordinates

    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...
  17. F

    Laplace equation in spherical coordinates

    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...
  18. Dewgale

    Usage of Del in Spherical Polar Coordinates

    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...
  19. phys-student

    Dot products in spherical or cylindrical coordinates

    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
  20. qq545282501

    Turn spherical coordinates into rectangular coordinates

    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...
  21. B

    Why does not dipole moment depend on coordinates system?

    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
  22. SarahAlbert

    Laplace in Spherical and Cylindrical Coordinates

    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...
  23. T

    Is there an error in my coordinate system transformation and vector matching?

    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>...
  24. W

    Volume integral in cylindrical coordinates

    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...
  25. A

    MHB Finding Conversion Formula for P & Q Coordinates

    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...
  26. F

    Find a normal vector to a unit sphere using cartesian coordinates

    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...
  27. T

    Force and Potential Energy Coordinates

    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}##...
  28. azizlwl

    How to find the vector between two points given in spherical coordinates?

    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?
  29. C

    Show that wave function in coordinates x,y is normalized

    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...
  30. Harel

    A problem with polar coordinates and black hole

    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...
  31. S

    Distance function in Riemannian normal coordinates

    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}...
  32. B

    Spherical coordinates path integral and stokes theorem

    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...
  33. nettle404

    Deriving the heat equation in cylindrical coordinates

    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)...
  34. MexChemE

    Shell balances in cylindrical coordinates

    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...
  35. A

    Integrate a vector field in spherical coordinates

    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...
  36. E

    Bases and Coordinates: B1 and B2 for [R][/3] - Homework Statement

    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...
  37. S

    Generalized Coordinates - Landau & Lifshitz

    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...
  38. B

    Time derivatives in Spherical Polar Coordinates

    Homework Statement Evaluate r(hat and overdot), θ(hat and overdot), φ(hat and overdot) in terms of (θ , Φ) and the time derivatives of the two remaining spherical polar coordinates. Your results should depend on the spherical polar unit vectors. Homework Equations ∂/∂t= The Attempt at a...
  39. C

    Can You Help With Finite Element Analysis in Cylindrical Coordinates?

    I am trying to numerically calculate the electric potential inside a truncated cone using the finite element method (FEM). The cone is embedded in cylindrical coordinates (r,phi,z). I am assuming phi-independence on the potential, therefore the problem is essentially 2D; I am working only with...
  40. B

    Solving PDE for 2nd Order Conic Equation

    I want to understand the solutions for the PDE of 2nd order \begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}:\begin{bmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{bmatrix} + \begin{bmatrix} b_1\\ b_2 \end{bmatrix}\cdot \begin{bmatrix} f_x\\ f_y \end{bmatrix} +cf=0 But...
  41. duran9987

    Motion On An Off Center Circle In Polar Coordinates

    Homework Statement A particle moves with constant speed ν around a circle of radius b, with the circle offset from the origin of coordinates by a distance b so that it is tangential to the y axis. Find the particle's velocity vector in polar coordinates. Homework Equations (dots for time...
  42. Ken G

    A Measuring 4-Vectors: Is It Possible?

    We know that 4-vectors are invariants, in the sense that they have the same meaning in all reference frames/coordinate systems. We know they transform by the Lorentz transformation in SR, and have an invariant Minkowski norm (let's not bring in GR at this point unless it becomes necessary). It...
  43. W

    Dot product for vectors in spherical coordinates

    Hi all. I'm struggling with taking dot products between vectors in spherical coordinates. I just cannot figure out how to take the dot product between two arbitrary spherical-coordinate vectors ##\bf{v_1}## centered in ##(r_1,\theta_1,\phi_1)## and ##\bf{v_2}## centered in...
  44. S

    Finding the mass of a solid, using Spherical Coordinates.

    Homework Statement Find the mass of the solid bounded from above z = √(25 - x2-y2) and below from z = 4, if its density is δ = k(x^2 + y^2 + z^2)^(-1/2). Homework Equations m = ∫∫∫δdV The Attempt at a Solution The plane z = 4 is transformed into ρcosφ = 4, that is, ρ = 4secφ. And x^2 +...
  45. PWiz

    Meaning of r in Schwarzchild coordinates

    I'm trying to understand *quote unquote thread title* by performing some simple (heuristic) analysis on my own. Before beginning, I'd like to present what I've been given to understand here at PF: -r is not the distance from the center of a spherical shell to an arbitrary spatial coordinate on...
  46. J

    Vector coordinates and its points

    Is there a way to know the points if I only have the vector coordinates and I can't use the origin as one of the points? For example, if I have vec(PQ) <-1,4,-5> . Is there a way to know the points of this vector?
  47. M

    "Semi" Synchronous coordinates

    I understand that one can always construct a set Synchronous coordinates (or Gaussian normal coordinates) on a neighborhood of a point in spacetime. My question is: Does one can construct a metric with only $g_{0i}=0$ such that $dS^2=g_{00}dt^2 + g_{ij}dx^{i} dx^{j}$ (where $i=1,,,,D$ and...
  48. P

    Changing to polar coordinates in an exponential

    Hello :) I don't get this integral (Peskin & Schroeder P. 27 ) ##\int {{{{d^3}p} \over {{{\left( {2\pi } \right)}^3}}}{1 \over {{E_{\bf{p}}}}}{e^{i{\bf{p}} \cdot {\bf{r}}}}} = {{2\pi } \over {{{\left( {2\pi } \right)}^3}}}\int\limits_0^\infty {dp{{{p^2}} \over {2{E_{\bf{p}}}}}{{{e^{ipr}} -...
  49. P

    Why is ##x = r \sin{\phi} \cos{\theta}## in spherical coordinates?

    My question is really about converting between spherical coordinates and cartesian coordinates. Suppose that ##\phi## and ##\theta## are defined as follows: ##\phi## is the angle between the position vector of a point and the ##z##-axis. ##\theta## is the angle between the projection of that...
  50. M

    Integrating Gaussian in polar coordinates problem

    I have a 2D Gaussian: ## f(x,y) = e^{-[(x-x_o)^2 + (y-y_o)^2]/(2*{sigma}^2)}## which I converted into polar coordinates and got: ## g(r,θ) = e^{-[r^2 + r_o^2 - 2*r*r_o(cos(θ)cos(θ_o) + sin(θ)sin(θ_o))]/({2*{sigma}^2})} ## The proof for how this was done is in the attached file, and it would...
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