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. W

    Stokes's theorem in spherical coordinates

    Problem: Say we have a vector function ##\vec{F} (\vec{r})=\hat{\phi}##. a. Calculate ##\oint_C \vec{F} \cdot d\vec{\ell}##, where C is the circle of radius R in the xy plane centered at the origin b. Calculate ##\int_H \nabla \times \vec{F} \cdot d\vec{a}##, where H is the hemisphere...
  2. A

    Gnuplot in Polar Coordinates

    I am brand new to Gnuplot and am having a problem trying to figure out how to graph in Polar Coordinates for a school assignment. What bothers me is we didn't go over other coordinate systems like Polar or Parametric at all for Gnuplot, and the internet tutorials I find seem to assume some basic...
  3. W

    Deriving a forumla for the gradient in cylindrical coordinates

    Problem: Starting from the gradient of a scalar function T(x,y,z) in cartesian coordinates find the formula for the gradient of T(s,ϕ,z) in cylindrical coordinates. Solution (so far): I know that the gradient is given by \nabla T = \frac{\partial T}{\partial x}\hat{x}+\frac{\partial...
  4. W

    Rewrite Indefinite Integral in Terms of Elliptic Coordinates

    Problem: Rewrite the indefinite integral ## \iint\limits_R\, (x+y) dx \ dy ## in terms of elliptic coordinates ##(u,v)##, where ## x=acosh(u)cos(v) ## and ## y=asinh(u)sin(v) ##. Attempt at a Solution: So would it be something like, ## \iint\limits_R\, (x+y) dx \ dy =...
  5. H

    Derive the divergence formula for spherical coordinates

    Homework Statement The formula for divergence in the spherical coordinate system can be defined as follows: \nabla\bullet\vec{f} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 f_r) + \frac{1}{r sinθ} \frac{\partial}{\partial θ} (f_θ sinθ) + \frac{1}{r sinθ}\frac{\partial f_\phi}{\partial...
  6. PsychonautQQ

    Finding area between two curves Polar Coordinates

    Homework Statement Find the area inside the circle r = 3sinθ and outside the carotid r = 1 + sinθ The Attempt at a Solution Alright so I graphed it and found that they intersect at ∏/6 and 5∏/6. I can't think of a good way to approach the problem. The carotid has some of it's area...
  7. PsychonautQQ

    Graphing r = 1 - cos(theta) (polar coordinates

    Homework Statement Okay the graph SHOULD look like this. http://jwilson.coe.uga.edu/EMAT6680Fa11/Chun/11/21.png I can't make sense of this at all. It looks so weird. Why does it bend around the y-axis in such an asymmetric way? I just graphed r = sin(θ) with ease by making a table of r vs θ...
  8. PsychonautQQ

    Finding center of circle with Polar Coordinates

    Homework Statement r=7sin(∅) find the center of the circle in Cartesian coordinates and the radius of the circle The Attempt at a Solution My math teacher is impossible to understand >.< and then the stupid homework is online and crap blah this class but I REALLY want to understand the material...
  9. PsychonautQQ

    Polar Coordinates inverse Radius

    Homework Statement I have to turn this homework in online... I just want someone to check my work Convert from Cartesian coordinates to Polar coordinates (-1,-sqrt(3)) if r > 0 and if r < 0. Homework Equations The Attempt at a Solution if r > 0 then I believe the answer is...
  10. PsychonautQQ

    Polar Coordinates Tangent line

    Homework Statement I don't know how to make theta so ∅ = theta. find the slope of the tangent line at r = sin(6∅) when ∅ = pi/12 Homework Equations y=rsin(6∅) x=rcos(6∅) r=sin(6∅) tangent line equation y-y' = m(x-x') m = dy/dx The Attempt at a Solution when ∅ = pi/12 then...
  11. B

    Is a (hyper)sphere a (hyper)plane in spherical coordinates?

    Hi, can I say that a sphere is a plane, because in spherical coordinates, I can simply express it as <(r, \theta, \varphi)^T, (1, 0, 0)^T> = R? It does sound too easy to me. I'm asking because I'm thinking about whether it is valid to generalize results from the John-Radon transform (over...
  12. Y

    Partial derivative in Spherical Coordinates

    Is partial derivative of ##u(x,y,z)## equals to \frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z} Is partial derivative of ##u(r,\theta,\phi)## in Spherical Coordinates equals to \frac{\partial u}{\partial r}+\frac{\partial u}{\partial...
  13. andrewkirk

    Can contraction of a tensor be defined without using coordinates?

    All but one of the tensor operations can be defined without reference to either coordinates or a basis. This can be done for instance by defining a ##(^m_n)## tensor over vector space ##V## as a multi-linear function from ##V^m(V^*)^n## to the background field ##F##. This allows us to define...
  14. T

    Reducing a PDE to an ODE Using a Change of Coordinates

    I've been studying Walter A. Strauss' Partial Differential Equations, 2nd edition in an attempt to prepare for my upcoming class on Partial Differential Equations but this problem has me stumped. I feel like it should be fairly simple, but I just can't get it. 10. Solve ##u_{x} + u_{y} + u =...
  15. karush

    MHB *3 coordinates of parallelogram STUV

    (a) $\vec{ST} = \pmatrix{9 \\ 9}$ so $V=(5,15)-(9,9)=(-4,6)$ (b) $UV = \pmatrix{-4,6}-\lambda \pmatrix{9,9}$ (c) eq of line $UV$ is $y=x+10$ so from position vector $\pmatrix{1 \\11}$ we have $11=1+10$ didn't know how to find the value of $\lambda$ (d) ?
  16. L

    Rotation in spherical coordinates

    Hi guys, This isn't really a homework problem but I just need a bit of help grasping rotations in spherical coordinates. My main question is, Is it possible to rotate a vector r about the y-axis by an angle β if r is expressed in spherical coordinates and you don't want to convert r...
  17. U

    Jacobian in spherical coordinates?

    Hi, Started to learn about Jacobians recently and found something I do not understand. Say there is a vector field F(r, phi, theta), and I want to find the flux across the surface of a sphere. eg: ∫∫F⋅dA Do I need to use the Jacobian if the function is already in spherical...
  18. E

    Differentiation spherical coordinates

    Hi ! I'm trying to inverse a mass matrix so I need to do something like this \dfrac{q}{\partial \mathbf{r}} where \cos q = \dfrac{\mathbf{r}\cdot \hat{\mathbf{k}}}{r} However, when \mathbf{r} = \hat{\mathbf{k}} \text{ or } -\hat{\mathbf{k}} I have problems. ¿What can I do...
  19. karush

    MHB IBV4 Quadrilateral OABC: O(0, 0), A(5, 1), B(10, 5), C(2, 7)

    wasn't sure about $\overrightarrow{AC}$
  20. M

    Dipole of Magnetic field in polar coordinates

    Homework Statement Hi everybody... i have a bad problem with my brain: starting from the Vectorial form of the magnetic dipole: \vec{B}(\vec{r}) =\frac{\mu_0}{4 \pi} \frac{3 \vec{r} ( \vec{r} \cdot \vec{m}) - r^2 \vec{m}}{r^5} Homework Equations i want to derive the spherical...
  21. M

    Find the shortest path between two points in polar coordinates

    Homework Statement Find the shortest distance between two points using polar coordinates, ie, using them as a line element: ds^2 = dr^2 + r^2 dθ^2Homework Equations For an integral I = ∫f Euler-Lagrange Eq must hold df/dθ - d/dr(df/dθ') = 0 The Attempt at a Solution f = ds = √(1 + (r *...
  22. 5

    Calculus problem involving finding coordinates

    Homework Statement Using calculus, find the coordinates of the point on the line y =-2x+5, which is closest to the origin, and the corresponding value of D Homework Equations y = -2x +5 The Attempt at a Solution I know I need to find a line that is perpendicular to the line of...
  23. Y

    Divergence in spherical coordinates.

    I want to verify: \vec A=\hat R \frac{k}{R^2}\;\hbox{ where k is a constant.} \nabla\cdot\vec A=\frac{1}{R^2}\frac{\partial (R^2A_R)}{\partial R}+\frac{1}{R\sin\theta}\frac{\partial (A_{\theta}\sin\theta)}{\partial \theta}+\frac{1}{R\sin\theta}\frac{\partial A_{\phi}}{\partial \phi}...
  24. W

    Compensation factor for converting dy dx to cylindrical coordinates?

    Homework Statement What is the compensation factor for converting dy dx to cylindrical coordinates? Homework Equations None that I know of besides the bottom ones as part of the attempt The Attempt at a Solution So I know that the conversion formulas for going from Cartesian (x,y,z)...
  25. Ackbach

    MoI of a Sphere using Spherical Coordinates

    Homework Statement Calculate the moment of inertia of a uniformly distributed sphere about an axis through its center. Homework Equations I know that $$I= \frac{2}{5} M R^{2},$$ where ##M## is the mass and ##R## is the radius of the sphere. However, for some reason, when I do this...
  26. I

    How to deduct the gradient in spherical coordinates?

    http://en.wikipedia.org/wiki/Gradient#Cylindrical_and_spherical_coordinates which formula do we apply to get the gradient in spherical coordinates?
  27. J

    Triple integral problem: cylindrical coordinates

    Homework Statement I have a graph 1/x^2=y^2+z^2 where z=rsin(θ) and y=rcos(θ) where 0≤r≤1 and 0≤θ≤2∏ on the zy-plane The end result is attached (sorry, I'm not aware of how to use Latex :[ ) I can kind of understand how they determined the first bounds for the integral: the lowest x...
  28. D

    Kernal density estimate in polar coordinates.

    Hi, I have a data set containing values for power and direction. I would like to produce a probability density estimate. The data can have multiple sources so I want to use a nonparametric method. I work in python which has a method for kernal density estimation (KDE), which I think should be...
  29. S

    Fortran [Fortran90] fdtd in polar coordinates, got infinity output

    hi all, attached here is my code for 2d fdtd in polar coordinates, from 'numerical electromagnetic: the fdtd method (umran s inan, pg 94-96) written in fortran90. I have try a few approach I could think about to troubleshoot this code but the output is still infinity. Anybody here can give me...
  30. J

    Help understanding measured coordinates of an electron, etc. Examples?

    I am trying to read into quantum mechanics and am reading a lot of rules that do not cite evidence and while it is probably just the books I am reading, I was wondering if anyone could post some links to experiments that verify some of this. First of all, this book "Quantum Mechanics -...
  31. N

    Obtaining spherical coordinates by rotations

    Hi Say I have a point on a unit sphere, given by the spherical coordinate $(r=1, \theta, \phi)$. Is this point equivalent to the point that one can obtain by $(x,y,z)=(1,0,0)$ around the $y$-axis by an angle $\pi/2-\theta$ and around the $z$-axis by the angle $\phi$? I'm not sure this is...
  32. Fantini

    MHB Solve Diff Eq: Change of Coordinates to Eliminate Squared Terms

    Here is the question: Consider the differential equation $$x' = a_1 x + a_2 x^2 + a_3 x^3 + \cdots,$$ with $a_1 \neq 0$. Show that there exists a $C^2$ change of coordinates of the form $x = y + \alpha y^2$ that rewrites the equation (locally around $x=0$) as $$y' = a_1 y + b_3 y^3 +...
  33. I

    Double Integrals with Polar Coordinates

    Homework Statement Use polar coordinates to find the volume of the solid bounded by the paraboloid z = 47 - 5x2 - 5y2 and the plane z = 2. Homework Equations x2 + y2 = r2 x = rcosθ y = rsinθ The Attempt at a Solution I substituted the z = 2 into the equation given, 2 = 47 -...
  34. B

    Finding The Coordinates of The Center Of Curvature

    Homework Statement Let C be a curve given by y = f(x). Let K be the curvature (K \ne 0) and let z = \frac{1+ f'(x_0)^2}{f''(x_0)}. Show that the coordinates ( \alpha , \beta ) of the center of curvature at P are ( \alpha , \beta ) = (x_0 -f'(x_0)z , y_0 + z) Homework Equations The...
  35. T

    Help With: Area (Polar Coordinates), Confusing Integral

    Find the area of the following region: Inside: r2 = 6 cos 2θ Outside: r = √3 Here's how I've set up the integral. I have to be making a mistake somewhere in the set up, but I can't figure it out. r1 = √3 r2 = (\sqrt{6 cos 2θ}) \frac{Area}{4}= \frac{1}{2}\int\ (\sqrt{6 cos 2θ})^{2}...
  36. S

    Projective coordinates vs vectors

    There is a technical distinction between a vector and the coordinates of a vector. Are projective (also called "affine") coordinates the coordinates of vectors? I'm thinking of how translation is accomplished by matrix multiplication. For example the point (x,y) in 2-D is given coordinates...
  37. O

    Differential Geometry - Finding Flat Coordinates

    Homework Statement Hello, I posted a similar question in the physics section but no one was able to help, I am first going to include a link to the older problem where I was attempting to find the ,(Finding the local flat space of the Poincare half disk metric), and explain what is different...
  38. IridescentRain

    Solution to the scalar wave equation in cylindrical coordinates

    Hello. I don't know how to prove that a certain function is a solution to the scalar wave equation in cylindrical coordinates. The scalar wave equation is \left(\nabla^2+k^2\right)\,\phi(\vec{r})=0,which in cylindrical coordinates is...
  39. K

    The length of a path on a sphere (in spherical coordinates)

    So, I'm to show that in spherical coordinates, the length of a given path on a sphere of radius R is given by: L= R\int_{\theta_1}^{\theta_2} \sqrt{1+\sin^2(\theta) \phi'^2(\theta)}d\theta, where it is assumed \phi(\theta), and start coordinates are (\theta_1,\phi_1) and (\theta_2, \phi_2)...
  40. Y

    Coordinates transformation by rotating at the origin.

    I want to transform from rectangular coordinates ( xyz) to (x",y",z") rectangular coordinates that is rotated at the origin as show in the attachment. Then I want to transform (x",y",z") rectangular coordinates into Spherical coordinates. Attach is the method I use, I want to verify I am doing...
  41. C

    MHB Integral - cylindrical coordinates

    Hello, my best problem is about find the integration limits. in cylindrical coordinates- where V is limited by the cylinder y^2+z^2=9 and the planes x = 0, y = 3x and z = 0 in the first octant.
  42. Fernando Revilla

    MHB Integration in polar coordinates

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
  43. N

    Synchronous Coordinates transformation

    Given a specific metric, is there a easy way to transform it in Synchronous coordinates? For example having dsigma2 = (1+z)^2 dt^2 - ds^2 - s^2 dphi^2 - dz^2 , I was able to do some substitutions, but I had to stop at the differential equations presented in the attachement.
  44. Hepth

    Massive Vector Polarizations in Spherical Coordinates

    I can't seem to find one, but does anyone have a reference to the fourvector polarizations for a massive vector particle in spherical coordinates where a momentum is defined as p = \{E, |\vec{p}| \sin \theta \sin \phi, |\vec{p}|\sin \theta \cos \phi , |\vec{p}| \cos \theta\} theta goes...
  45. G

    Comsol 3.5x: Obtaining boundary coordinates from irregular geom obj

    I don't have access to Comsol 4.x. I imported a 3D mesh generated from point cloud data and generated a geometry. (A hollow almost-ellipsoid.) I solve my system on the surface/boundary alone; there is no volumetric data. I need to extract 1D data from the surface/boundary at points other...
  46. M

    Kruzkal Coordinates Inside Horizon: Defining u', v

    We start by defining two coordinates ##u=t+r^*## and ##v=t-r^*##. Then we define another two coordinates ##u'=e^{u/4GM}## and ##v'=-e^{-v/4GM}##. But from what I have understood this is true for ##r>2GM##. How do we define ##u'## and ##v'## for ##r<2GM##? I think it's ##u'=e^{u/4GM}## and...
  47. A

    Change of variables from one set of coordinates to another in Fourier

    ... ... I am curious to know why we have to multiply with e^{-j\omega t} in Fourier transform? What is the purpose of this? I have heard somewhere that the transform is merely a change of variables from one set of coordinates to another. I would like to know more about this. Can you help me?
  48. V

    Velocity Vector in Polar Coordinates (Kleppner p.30)

    In polar coordinates we have \vec{r} = r \hat{r} \Rightarrow \vec{v} = \frac{d}{dt}({r \hat{r}}) = \dot{r}\hat{r} + r \frac{d \hat{r}}{dt} . In the book Introduction to Mechanics, K & K says the right term is the component of velocity directed radially outward. (Surely a typo, as the left...
  49. S

    How to translate from polar to cartesian coordinates:

    How to translate r = 2 /(2 - cos(theta)) to cartesian coordinates: so far: r = 2 /(2 - cos(theta)) r = 2 /(2 - cos(theta)) |* (2 - cos(theta)) both sides r (2 - cos(theta))= 2 2*r - rcos(theta) = 2 | know x = rcos(theta) 2*r - x...
  50. J

    Line integral around a circle in polar coordinates

    I know that \oint_{C}\mathrm{d}\vec{l} = 0, for any closed curve C. But when i try to calculate the integral around the unit circle in polar coordinates, I get a result different from zero. Here is my approach : \oint_{C}\mathrm{d}\vec{l} = \int_{0}^{2\pi}\hat{\phi}\mathrm{d}\phi =...
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