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. LyleJr

    I Derivation of the Laplacian in Spherical Coordinates

    Hi all, Sorry if this is the wrong section to post this. For some time, I have wanted to derive the Laplacian in spherical coordinates for myself using what some people call the "brute force" method. I knew it would take several sheets of paper and could quickly become disorganized, so I...
  2. M

    MHB Coordinates of Points A and B

    The equation y = (x/2) + 1 is given. It forms a straight line going through the points A(x, 1.5) and B(x, 2.5). Find the x-coordinates of points A and B. Do I substitute the value of y for each point into the given equation and solve for x individually?
  3. C

    A Laplace Eq. in Cylindrical coordinates (no origin)

    Hi, I need to solve Laplace equation:##\nabla ^2 \Phi(x,r)=0## in cylindrical domain ##r_1<r<r_2##, ##0<z<+\infty##. The boundary conditions are: ## \left\{ \begin{aligned} &\Phi(0,r)=V_B \\ & -{C^{'}}_{ox} \Phi(x,r_2)=C_0 \frac{\partial \Phi(x,r)}{\partial r}\rvert_{r=r_2} \\ &\frac{\partial...
  4. Bunny-chan

    Distance between point coordinates in a straight line

    Homework Statement Let A = (1,2,5) and B = (0,1,0). Determine a point P of the line AB such that ||\vec{PB}|| = 3||\vec{PA}||. Homework EquationsThe Attempt at a Solution Initially, writing the line in parametric form\vec{AB} = B - A = (0-1,1-2,0-5) = (-1,-1,-5)\\ \\ \Rightarrow \vec{v} =...
  5. C

    Improper integral with spherical coordinates

    Homework Statement I have a question. I have a function f(x,y,z) which is a continuous positive function in D = {(x,y,z); x^2 + y^2 +z^2<=1}. And let r = sqrt(x^2 + y^2 + z^2). I have to check whether the following jntegral is convergent. x^2y^2z^2/r^(17/2) * f(x,y,z)dV. Homework Equations...
  6. Bunny-chan

    Potential gravitational energy coordinate axis

    Homework Statement I know that potential gravitational energy is relative to the reference point that I decide to choose (like in the picture below). But then if, for instance, I set my reference point in the ceiling and my vertically down y-axis to be positive. What would the potential...
  7. A

    A Schwarzschild coordinates beyond the event horizon

    We can write down the metric of the Schwarzschild black hole in Schwarzschild coordinates. Which aspect of the metric in Schwarzschild coordinates indicates that the coordinates are only valid outside the event horizon?
  8. C

    A Laplace Eq. in Cylindrical coordinates

    Hi, I need to solve Laplace equation:## \nabla ^2 \Phi(x,r)=0 ## in cylindrical domain ##0<r<r_0##, ##0<x<L## and ##0<\phi<2\pi##. The boundary conditions are the following ones: ## \left\{ \begin{aligned} &C_{di}\Phi(x,r_0)=\epsilon \frac{\partial \Phi(x,r)}{\partial r}\rvert_{r=r_0} \\...
  9. H Psi equal E Psi

    I Affine transformation and coordinates of maps

    Hi everyone! I'm having trouble with the following exercise: Let ##\mathrm {Aff}(ℝ)## be the vector space of the affine maps from ##ℝ## to ##ℝ##: $$φ_{a,b}:ℝ→ℝ$$ $$x→a x + b$$ Find the contravariant and and covariant coordinate of the map: $$φ_{1,1}:ℝ→ℝ$$ $$x→x + 1$$ with respect to the...
  10. B

    MHB Find the coordinates and the nature of any turning points (maximum and minimum)

    So I've been trying to wrap my head around this one for several hours now and it just has me stumped. I'm begging, someone, anyone, walk me through it before I swear off numbers for life hahah (Part b) Imgur: The most awesome images on the Internet Thanks in advance for any help anyone can...
  11. Mr Davis 97

    I Distinction between coordinates and vectors

    I am a little confused about the difference between between coordinates and vectors. For example, when first studying vector calculus, you learn about vector fields, which formally are maps ##f: \mathbb{R}^n \to \mathbb{R}^n##, and we say that the function associates to every point in space a...
  12. maistral

    A 2D Finite Difference formulation in polar coordinates.

    So I have this PDE: d2T/dr2 + 1/r dT/dr + d2T/dθ2 = 0. How do I implement dT/dr || [r = 0] = 0? Also, what should I do about 1/r? This is actually the first time I am going to attack FDF in polar/cylindrical coordinates. I can finite-difference the base equation fairly decently; I am just...
  13. Mayan Fung

    I 2D Laplacian in polar coordinates

    The 2D Laplacian in polar coordinates has the form of $$ \frac{1}{r}(ru_r)_r +\frac{1}{r^2}u_{\theta \theta} =0 $$ By separation of variables, we can write the ## \theta## part as $$ \Theta'' (\theta) = \lambda \Theta (\theta)$$ Now, the book said because we need to satisfy the condition ##...
  14. L

    A Relation between Vector Norms in Cylindrical and Cartesian Coordinates

    Relations between vectors in cylindrical and Cartesian coordinate systems are given by \vec{e}_{\rho}=\cos \varphi \vec{e}_x+\sin \varphi \vec{e}_y \vec{e}_{\varphi}=-\sin \varphi \vec{e}_x+\cos \varphi \vec{e}_y \vec{e}_z=\vec{e}_z We can write this in form \begin{bmatrix}...
  15. harpazo

    MHB Double Integrals in Polar Coordinates

    Evaluate the double integral by converting to polar coordinates. Let S S be the double integral symbol S S xy dydx Inner limits: 0 to sqrt{2x - x^2} Outer limits: 0 to 2 The answer is 2/3. I know that x = rcosϴ and y = rsinϴ. S S rcosϴ*rsinϴ r drdϴ. S S (r^3)cosϴ*sinϴ drdϴ. I am stuck...
  16. M

    Shell balances in cylindrical coordinates

    I have a question regarding writing a shell balance for a cylindrical system with transport in one direction (in any area of transport phenomena). When we set up the conservation equation(say steady state), we multiply the flux and the area at the surfaces of our control volume and plug them...
  17. harpazo

    MHB Double Integrals in Polar Coordinates

    Evaluate the iterated integral by converting to polar coordinates. Let S S = double integral symbol S S y dx dy The outer integral is from 0 to a. The inner integral is from 0 to sqrt{a^2 - y^2}. I started by letting y = r sin ϴ S S r sinϴ dxdy. I then let dxdy = r dr d ϴ S S r sin ϴ rdr...
  18. S

    A Deriving the Poincare patch from global coordinates in AdS##_{3}##

    I have been reading Thomas Hartman's lecture notes (http://www.hartmanhep.net/topics2015/) on Quantum Gravity and Black Holes. -------------------------------------------------------------------------------------------------------------------------------- In page 97, he derives (9.4), which is...
  19. B

    Help describing a region in polar coordinates

    Homework Statement If (r, θ) are the polar coordinates of a point then describe the region defined by the restrictions -1 < r < 0, π/2 < θ < 3π/2 Homework Equations No clue The Attempt at a Solution I tried drawing the curve in a polar grid by starting at π/2 and finishing at 3π/2. I was...
  20. S

    I Rindler coordinates in Minkowski spacetime

    In an inertial coordinate system in two-dimensional Minkowski spacetime, the metric takes the form $$(ds)^{2} = - (dt)^{2} + (dx)^{2},$$ and in an accelerating coordinate system in two-dimensional Minkowski spacetime, the metric takes the form $$(ds)^{2} = - R^{2}(d\eta)^{2} + (dR)^{2}.$$ The...
  21. F

    Asymptote of a curve in polar coordinates

    Homework Statement The curve ##C## has polar equation ## r\theta =1 ## for ## 0<\theta<2\pi## Use the fact that ## \lim_{\theta \rightarrow 0}\frac{sin \theta }{\theta }=1## to show the line ## y=1## is an asymptote to ## C##.The Attempt at a Solution **Attempt** $$\ r\theta =1$$ $$\...
  22. P

    Non-radial null geodesics in Eddington-Finkelstein coordinates

    Homework Statement My end goal is to plot null geodesics around a black hole with realistic representations within the horizon (r<2GM, with c=1) using Mathematica. I've done this for outside the horizon using normal Schwarzschild coordinates and gained equation (1) below, and then used this...
  23. A

    Parallelogram area (coordinates)

    Homework Statement The coordinates of the parallelogram ABCD are: A (-2; 1) B (5; 2) C (6; 5) D (-1; 4) We also know that the diagonals intercept in the middle of each other (so if the diagonals are AC and BD, and the intercept in point M, then AM = MC, and BM = MD). Not sure if this...
  24. Mind----Blown

    Significance of terms of acceleration in polar coordinates

    How do i get an idea, or a 'feel' of the components of the acceleration in polar coordinates which constitute the component in the eθ direction? from what i know, a= (r¨−rθ˙^2) er + (rθ¨+ 2r˙θ˙) eθ ; (where er and eθ are unit vectors in the radial direction and the direction of increase of the...
  25. S

    I Working out the equation for coordinates on a graph

    I have a series of data points for X and Y points on a graph. The data is quite random and I am trying to work out a trend line so I can then form an equation for the line. How would I go about working out the equation for the data below. (0, 580) (6.7, 495) (13.4, 445) (18.7, 365) (22.8, 350)...
  26. welssen

    Determine the angular momentum in polar coordinates

    Hi there, I've been trying to solve the following problem, which I found looks pretty basic, but actually got me really confused about the definition of angular momentum. Problem The trajectory of a point mass m is described by the following equations, in spherical coordinates: r(t) = r_0 +...
  27. D

    Laplacian of polar coordinates

    Homework Statement I am trying to calculate the laplacian in polar coordinates but I failed.Please see the attached Homework Equations The Attempt at a Solution My solution to this was uploaded in the attached.I was wondering what's wrong with the purple brackets since they shouldn't exist(...
  28. B

    Recast a given vector field F in cylindrical coordinates

    Homework Statement F(x,y,z) = xzi Homework Equations N/A The Attempt at a Solution I just said that x = rcos(θ) so F(r,θ,z) = rcos(θ)z. Is this correct? Beaucse I am also asked to find curl of F in Cartesian coordinates and compare to curl of F in cylindrical coordinates. For Curl of F in...
  29. M

    I Qubit mixed state density matrix coordinates on a Bloch ball

    What are the coordinates on the 3D Bloch ball of a qubit's mixed state of the form: ##\rho=p_{00}|0\rangle \langle 0|+p_{01}|0\rangle \langle 1|+p_{10}|1\rangle \langle 0|+p_{11}|1\rangle \langle 1|##
  30. C

    Linear Algebra - Finding coordinates of a set

    Homework Statement Find the coordinates of each member of set S relative to B. B = {1, cos(x), cos2(x), cos3(x), cos4(x), cos5(x)} S = {1, cos(x), cos(2x), cos(3x), cos(4x), cos(5x)} I am to do this using Mathematica software. Each spanning equation will need to be sampled at six separate...
  31. K

    Equation for finding the gradient in spherical coordinates

    <Mentor note: moved from a technical forum and therefore without template>So I´m trying to understand how to use the equation for finding the gradient in spherical coordinates, just going from cartesian to spherical seemed crazy. Now I´m at a point where I want to try out what I have read and I...
  32. AdrianMachin

    Finding third charge coordinates in an equilibrium position

    Homework Statement Here are the problem statement and the solution. I'm stuck at where the book suggests the formulas for the x and y coordinations (highlighted in yellow) of the third charge. Any explanations or proof on how they came to the conclusion for the third charge coordinations would...
  33. F

    Conversion vectors in cylindrical to cartesian coordinates

    Homework Statement It's just an example in the textbook. A vector in cylindrical coordinates. A=arAr+aΦAΦ+azAz to be expressed in cartesian coordinates. Start with the Ax component: Ax=A⋅ax=Arar⋅ax+AΦaΦ⋅ax ar⋅ax=cosΦ aΦ⋅ax=-sinΦ Ax=ArcosΦ - AΦsinΦ Looking at a figure of the unit vectors I...
  34. The black vegetable

    Cartesian Coordinates and Cross Product of Vectors for Magnetic Field Direction?

    Homework Statement Homework EquationsThe Attempt at a Solution the answer given is the same but without the negative sign, I don't understand because the crossproduct of unit vectors when using a Cartesian coordinates of the directions given by the right-hand rule? Is the positive z...
  35. davidge

    I Riemann tensor in 3d Cartesian coordinates

    Suppose we wish to use Cartesian coordinates for points on the surface of a sphere. Then all derivatives of the metric would vanish and so the Riemann curvature tensor would vanish. But it would give us a wrong result, namely that the space is not curved. So it means that if we want to get...
  36. R

    Why does translation work with the extra dimension (Homogeneous coordinates.)

    Homework Statement hi I have read a lot on homogenous coordinates and I feel like I now have a solid foundation. However none of the videos or books I have read give an explicit reason as to why translation with the extra dimension works(i.e. it does not result in scaling). Here 's what I...
  37. J

    Finding the curl of velocity in spherical coordinates

    Homework Statement The angular velocity vector of a rigid object rotating about the z-axis is given by ω = ω z-hat. At any point in the rotating object, the linear velocity vector is given by v = ω X r, where r is the position vector to that point. a.) Assuming that ω is constant, evaluate v...
  38. M

    MHB How do I use the midpoint formula to find the coordinates of point C?

    The coordinates of A and B are A(-1, 2) and B(5, -3). If B is the midpoint of line segment AC, what are the coordinates of C? I know this question is connected to the midpoint formula. If so, how do I use the formula to find the x and y coordinates of C?
  39. Dopplershift

    Need Help With Gradient (Spherical Coordinates)

    Homework Statement Find te gradient of the following function f(r) = rcos(##\theta##) in spherical coordinates. Homework Equations \begin{equation} \nabla f = \frac{\partial f}{\partial r} \hat{r} + (\frac{1}{r}) \frac{\partial f}{\partial \theta} \hat{\theta} + \frac{1}{rsin\theta}...
  40. jlmccart03

    Find the electric field at x and y coordinates?

    Homework Statement A 60 μC point charge is at the origin. Find the electric field at the point x1 = 50 cm , y1 = 0. Homework Equations Coulombs Law: F = (kQ1Q2)/r2 Electric Field: E = kQ/r2 k = 8.99*109 C The Attempt at a Solution So taking Coulombs Law and deriving the Electric Field...
  41. D

    A Change of coordinates in quantum phase space

    Hello! I was reading a paper on formulation of QM in phase space (https://arxiv.org/abs/physics/0405029) and I have some doubts related to chapter 5. It seems to me that there is a transformation to modified polar coordinates (instead of radius there is u which is square of radius multiplied by...
  42. P

    MHB Mass of a Torus in Spherical Coordinates

    consider a torus whose equation in terms of spherical coordinates(r,\theta,\phi) is r=2sin\phi for 0\le\phi\le2\Pi. determine the mass of the region bounded by the torus if the density is given by \rho=\phi.
  43. Dilemma

    Finding the volume - Polar coordinates

    Hello everyone, 1. Homework Statement Question : Find the volume of the region which remains inside the cyclinder x 2 + y 2 = 2y, and is bounded from above by the paraboloid surface x 2 + y 2 + z = 1 and from below by the plane z = 0 Homework Equations The Attempt at a Solution This looks...
  44. FallenApple

    Correct operation for coordinates of hoop particle system?

    Homework Statement All in the pic below. Part of the solution presented. Didn't present the whole thing as that would clutter the page. I just want to know how they set up the x coordinate for the particle. Homework Equations This problem is just about using the lagragian. So my issue is...
  45. maxhersch

    Double Integration in Polar Coordinates

    Homework Statement Integrate by changing to polar coordinates: ## \int_{0}^6 \int_{0}^\sqrt{36-x^2} tan^{-1} \left( \frac y x \right) \, dy \, dx ## Homework Equations ## x = r \cos \left( \theta \right) ## ## y = r \sin \left( \theta \right) ## The Attempt at a Solution So this is a...
  46. L

    A Laplacian cylindrical coordinates

    Laplacian in cylindrical coordinates is defined by \Delta=\frac{\partial^2}{\partial \rho^2}+\frac{1}{\rho}\frac{\partial}{\partial \rho}+\frac{1}{\rho^2}\frac{\partial^2}{\partial \varphi^2}+\frac{\partial^2}{\partial z^2} I am confused. I I have spherical symmetric function f(r) then \Delta...
  47. Hannibal247

    One-dimensional steady state conduction in Cylindrical coordinates

    Hello, Im having some issues with my task. 1. Homework Statement The heat generation rate of a cylindrical fuel (D=0.2 m and 1 m long) is 160 kW. The thermal conductivity of the fuel is 100 W/mK and its surface temperature is maintained at 283 K. Determine the temperature at the axis...
  48. W

    Derivative in spherical coordinates

    Homework Statement -here is the problem statement -here is a bit of their answer Homework Equations Chain rule, partial derivative in spherical coord. The Attempt at a Solution I tried dragging out the constant and partial derivate with respect to t but still I can't reach their df/dt and...
  49. J

    A Need a Proof that Action-Angle Coordinates are Periodic

    Can someone point me to a proof that Action-Angle coordinates in Hamilton-Jacobi Theory must be periodic. I have looked all over and no one seems to prove it, they just assume it. Thanks.
  50. T

    I Converting galactical coordinates to horizontal/geographical coordinates

    Hello, I am doing an astrophysics project about Orion belt constelation and the pyramids of Giza. I am interested in the myth that the three pyramids is an exact copy of Orion belt. How should I convert galactical latitude/longtitude to linear/geographical coordinates so I could compare the...
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