Coordinates Definition and 1000 Threads

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. J

    R, dr and d²r and curvilinear coordinates

    Hellow everybody! If ##d\vec{r}## can be written in terms of curvilinear coordinates as ##d\vec{r} = h_1 dq_1 \hat{q_1} + h_2 dq_2 \hat{q_2} + h_2 dq_2 \hat{q_2}## so, how is the vectors ##d^2\vec{r}## and ##\vec{r}## in terms of curvilinear coordinates? Thanks!
  2. B

    Moving boundary diffusion equation (transformation of coordinates)

    I'm trying to implement a numerical code for the diffusion equation with moving boundaries. I have no problems with the numerical implementation, but with the transformation of coordinates. In spherical coordinates, the diffusion equation is \frac{\partial c}{\partial t} = D...
  3. Lebombo

    Find the corresponding rectangular coordinates for the point.

    Homework Statement Find the corresponding rectangular coordinates for the point. (-2, \frac{5\pi}{3}) x = -2cos(\frac{5\pi}{3}) x = -2cos(\frac{2\pi}{3}) x = -2* \frac{-1}{2} = 1 y = -2sin(\frac{5\pi}{3}) y = -2sin(\frac{2\pi}{3}) y = -2*\frac{\sqrt{3}}{2} =...
  4. C

    How do you find the coordinates flipped for Trig question 11 part c?

    For question 11 , how do you do part c? I know that (cos theta corresponds to x value and sin theta corresponds to y value. Using that I found the angle to be 318 degrees for part a. For part c, how would you start that? The answer is the coordinates flipped , x and y values with a positive...
  5. B

    MHB How to calculate center coordinates of two reverse arcs in 3D space

    Hi, Given 3D points P1(200,60,140), P2(300,120,110), P3(3,0,-1), P4(-100,0,-1) and the radius of arc C1MP3 is equal to radius of arc C2MP1. How do I calculate coordinates x, y, z of points C1 and C2? Points C1 and C2 are centers of two reverse arcs which are tangent to each other at point...
  6. B

    How to calculate center coordinates of two reverse arcs in 3D space

    Hi, Given 3D points P1(200,60,140), P2(300,120,110), P3(3,0,-1), P4(-100,0,-1) and the radius of arc C1MP3 is equal to radius of arc C2MP1. How do I calculate coordinates x, y, z of points C1 and C2? See this image. Points C1 and C2 are centers of two reverse arcs which are tangent to...
  7. J

    Linear system in polar coordinates

    Hellow! I have searched for some theory about linear system in polar coordinates, unfortunately, I not found anything... exist some theory, some book, anything about this topic for study? Thanks!
  8. E

    Coordinates and casimir effect

    Hello, I am reading this paper on the casimir effect and I am failing to understand where the 1/(2âˆ)^2 comes in and how the polar coordinates are converted to Cartesian. The equations are (3.23) and (3.24). http://aphyr.com/data/journals/113/comps.pdf Thank you!
  9. DrClaude

    Hamilton-Jacobi equation in spherical coordinates

    I was looking at the Wikipedia entry on the Hamilton-Jacobi equation, and was confounded by the equation at the beginning of the section on spherical coordinates: http://en.wikipedia.org/wiki/Hamilton–Jacobi_equation#Spherical_coordinates Shouldn't the Hamiltonian simply be $$ H =...
  10. M

    Representing displacement vectors in cylindrical coordinates

    Hello, In Cartesian coordinates, if we have a point P(x1,y1,z1) and another point Q(x,y,z) we can easily find the displacement vector by just subtracting components (unit vectors are not changing directions) and dotting with the unit products. In fact we can relate any point with a position...
  11. I

    Double integral: Cartesian to Polar coordinates

    Homework Statement ∫∫√(x^2+y^2)dxdy with 0<=y<=1 and -SQRT(y-y^2)<=x<=0 Homework Equations x=rcos(theta) y=rsin(theta) The Attempt at a Solution 0.5<=r=1, we get r=0.5 from -SQRT(y-y^2)<=x by completing the square on the LHS then, 0<=theta<=pi But, when I calculated the...
  12. M

    Time-independence of original coordinates in canonical transform

    I am going through my professors notes on generating functions. I come across the following equation: \frac{\partial}{\partial t} \frac{\partial F}{\partial \xi^k} = \frac{\partial}{\partial t} \left( \gamma_{il} \frac{\partial \eta^i}{\partial \xi^k}\eta^l - \gamma_{kl}\xi^l \right ). Here...
  13. Y

    Differentiation in spherical coordinates.

    1) If u(r,\theta,\phi)=\frac{1}{r}, is \frac{\partial{u}}{\partial {\theta}}=\frac{\partial{u}}{\partial {\phi}}=0 because u is independent of \theta and \;\phi? 2) If u(r,\theta,\phi)=\frac{1}{r}, is: \nabla^2u(r,\theta,\phi)=\frac{\partial^2{u}}{\partial...
  14. M

    How cyclic coordinates affect the dimension of the cotangent manifold

    Our professor's notes say that "In general, in Hamiltonian dynamics a constant of motion will reduce the dimension of the phase space by two dimensions, not just one as it does in Lagrangian dynamics." To demonstrate this, he uses the central force Hamiltonian...
  15. M

    Polar coordinates to set up and evaluate double integral

    Homework Statement Use polar coordinates to set up and evaluate the double integral f(x,y) = e-(x2+y2)/2 R: x2+y2≤25, x≥0 The Attempt at a Solution First I just want to make sure I'm understanding this my double integral would be ∫^{\pi/2}_{-\pi/2} because x≥0 ∫^{5}_{0}...
  16. mnb96

    Curvilinear coordinates on tangent spaces

    Hello, if we consider a diffeomorphism f:M-->N between two manifolds, we can easily obtain a basis for the tangent space of N at p from the differential of f. I was wondering, why should we always express tangent vectors as linear combinations of tangent basis vectors? Could it be useful in...
  17. I

    Double Integrals in Polar Coordinates

    Homework Statement Use polar coordinates to find the volume of the given solid. Enclosed by the hyperboloid -x2 - y2 + z2 = 1 and the plane z = 2 Homework Equations r2 = x2 + y2, x = rcosθ, y = rsinθ ∫∫f(x,y)dA = ∫∫f(rcosθ,rsinθ)rdrdθ The Attempt at a Solution -x2 - y2 + 4...
  18. MattRob

    Spherical Coordinates: Understanding Theta Equation

    So, I was curious about this and found more or less what I was looking for here: http://electron9.phys.utk.edu/vectors/3dcoordinates.htm Except, something is bothering me about those equations. At the very bottom, the equation for Theta in a spherical coordinate system; shouldn't it be...
  19. S

    What is the equation for the given curve in polar coordinates?

    Homework Statement x = eKcos(k) y=eKsin(k) -∞ < K < ∞ Find an equation in polar coordinates for the above curve The Attempt at a Solution I am not fully clear as to what the question is asking. If its asking for (r,k), where K is normally a theta value then it would be...
  20. A

    Evaluating triple integral with spherical coordinates

    Homework Statement Evaluate the iterated integral ∫ (from 0 to 1) ∫ [from -sqrt(1-x^2) to sqrt(1-x^2) ] ∫ (from 0 to 2-x^2-y^2) the function given as √(x^2 + y^2) dz dy dx The Attempt at a Solution I changed the coordinates and I got the new limits as ∫(from 0 to pi) ∫(from...
  21. A

    Setting up triple integrals in different coordinates

    Homework Statement Assume that f(x,y,z) is a continuous function. Let U be the region inside the cone z=√x^2+y^2 for 2≤x≤7. Set up the intregal ∫f(x,y,z)dV over U using cartesian, spherical, and cylindrical coordinates. Homework Equations CYLINDRICAL COORDINATES x=rcosθ y=rsinθ z=z...
  22. A

    Integral in spherical coordinates

    I recently had to do an integral like the one in the thread below: http://math.stackexchange.com/questions/142235/three-dimensional-fourier-transform-of-radial-function-without-bessel-and-neuman The problem I had was also evaluating the product and I am quite sure that the answer in the thread...
  23. karush

    MHB Convert r = 5sin(2θ) to rectangular coordinates

    convert r=5\sin{2\theta} to rectangular coordinates the ans to this is $\left(x^2+y^2\right)^{3/2}=10xy$ however... multiply both sides by $r$ to get $r^2=5\cdot r \cdot \sin{2\theta}$ then substitute $r^2$ with $x^2+y^2$ and $\sin{2\theta}$ with $2\sin\theta\cos\theta$ and divide each side...
  24. M

    I don't understand the ranges of the angles in spherical coordinates

    I'm not sure whether this falls in the homework category, or the standard calculus section, so apologies in advance if this doesn't fall in the right category. Homework Statement Evaluate ∫∫∫e^[(x^2 + y^2 + z^2)^3/2]dV, where the region is the unit ball x^2 + y^2 + z^2 ≤ 1. (or any...
  25. 1

    Converting to Polar Coordinates

    Homework Statement Convert ∫ from 0 to 3/√2 ∫ from y to √(9-y^2) of xydxdy to polar form. Homework Equations x2+y2=r2 The Attempt at a Solution I found the equation x2+y^2=9 from the upper range of the second integral. So r=3. Therefore r ranges from 0 to 3. The integrand is...
  26. Ed Aboud

    Double integral, cylindrical coordinates

    Homework Statement The problem states: Use cylindrical coordinates to evaluate \iiint_V \sqrt{x^2 +y^2 +z^2} \,dx\,dy\,dz where V is the region bounded by the plane z = 3 and the cone z = \sqrt{x^2 + y^2} Homework Equations x = r cos( \theta ) y = r sin( \theta ) z =...
  27. A

    Deformation gradient f(3,3) vs Coordinates

    Dear, I have a task to model the behaviour of certain interphase material. Let's say that functions which describe the change of material parameters are known. i.g. change of the Young's modulus as function of distance from neighbouring material (or (0,0) origin) - PAR=PAR(x)...
  28. J

    What are the polar coordinates of (1,-2) and how do you find them?

    Homework Statement Convert (1,-2) to polar coordinates find one representation with r >0 and one with r <0. Also 0<= theta <= 2 PI Homework Equations I used tantheta = y /x , and x^2 +y^2 = r^2 The Attempt at a Solution I got (sqrt(5) , arctan(-2)) , (-sqrt(5) , arctan(-2) + pi...
  29. I

    Expressing the limits of integration for radius in polar coordinates

    i'm trying to integrate some some function bounded by the x-y domain of x2+y2=6y which is a circle on the x-y plane shifted upward where the outer part of the circle is 6. i'm trying to integrate a double integral.. ∫∫f(x)rdrdθ i don't know how to express my limits of integration for r...
  30. T

    Double integral over a region needing polar coordinates.

    1. Evaluate the double integral ∫∫arctan(y/x) dA by converting to polar coordinates over the Region R= { (x,y) | 1≤x^2+y^2≤4 , 0≤y≤x } My attempt at solving Converting to polar using x=rcosθ and y=rsinθ I get ∫∫arctan(tan(θ))r drdθ I understand that I have to integrate first with respect...
  31. C

    Integrals in cylindrical coordinates.

    Integrate the function f(x,y,z)=−7x+2y over the solid given by the "slice" of an ice-cream cone in the first octant bounded by the planes x=0 and y=sqrt(263/137)x and contained in a sphere centered at the origin with radius 25 and a cone opening upwards from the origin with top radius 20. I...
  32. P

    Volume in cylindrical coordinates

    Homework Statement Find the volume using cylindrical coordinates bounded by: x2+y2+z2=2 and z = x2+y2 Homework Equations Converting to cylindrical coordinates: z = √2-r2 and z = r2 The Attempt at a Solution I figured z would go from r2 to √2-r2 r from 0 to √2 and θ...
  33. PsychonautQQ

    Finding volume in Polar Coordinates

    Homework Statement Find the volume of the wedge-shaped region contained in the cylinder x^2+y^2=9 bounded by the plane z=x and below by the xy planeHomework Equations The Attempt at a Solution So it seems a common theme for me I have a hard time finding the limits of integration for the dθ term...
  34. PsychonautQQ

    Evaluating an Integral in Polar Coordinates

    Homework Statement Evalutate the double integral sin(x^2+y^2)dA between the region 1≥x^2+y^2≥49 The Attempt at a Solution so r^2 = x^2 + y^2 dA = rdrdθ so I can turn this into double integral sin(r^2)rdrdθ where the inner integral integrated with respect to dr goes from 1 to 7...
  35. Z

    Approximating speed from past several lat/lon coordinates

    Suppose you are observing the movement of an object on the Earth's surface. At any given moment, you know its current position (in lat/lon coordinates) and three prior positions. Each prior position is separated in time from the one after it by a small but variable number of seconds (say several...
  36. B

    Geometry problem - calculating curve coordinates from versines

    Hi, I was wondering if anyone can help me. I don’t have a homework problem, but a problem I have encountered at work. I am a mechanical engineer working in the railway industry and I am struggling with a problem of reconstructing the vertical geometry of a rail in terms of height and...
  37. B

    Lagrange's Equation Generalized Coordinates

    Hello, I am currently reading about the topic alluded to in the topic of this thread. In Taylor's Classical Mechanics, the author appears to be making a requirement about any arbitrary coordinate system you employ in solving some particular problem. He says, "Instead of the Cartesian...
  38. J

    Spherical cylindrical and rectangular coordinates

    Homework Statement Suppose that in spherical coordinates the surface S is given by the equation rho * sin(phi)= 2 * cos(theta). Find an equation for the surface in cylindrical and rectangular coordinates. Describe the surface- what kind of surface is S? Homework Equations The...
  39. M

    Graphing with polar coordinates Problem

    Homework Statement Draw the graph of r = 1/2 + cos(theta) Homework Equations The equation is itself given in the question. It is a Limacon. The Attempt at a Solution Step-1 ---> Max. value of r is 1/2 + 1 = 3/2 [ at cos (0) ] Min. value of r is 1/2 - 1...
  40. A

    Potential, field, Laplacian and Spherical Coordinates

    Homework Statement Say I am given a spherically symmetric potential function V(r), written in terms of r and a bunch of other constants, and say it is just a polynomial of some type with r as the variable, \frac{q}{4\pi\varepsilon_o}P(r), and we are inside the sphere of radius R, so r<R…...
  41. P

    Parameterize a geodesic using one of the coordinates

    I've been working on a problem where I have to find the geodesics for a given Riemannian Manifold. To present my doubt, I tried to find a simpler example that would demonstrate my uncertainty but the one I found, and shall present bellow, has actually a simplification that my problem doesn't, so...
  42. W

    Describing a Solid Ice Cream Cone with Spherical Coordinates

    Q: Consider the solid that lies above the cone z=√(3x^2+3y^2) and below the sphere X^2+y^2+Z^2=36. It looks somewhat like an ice cream cone. Use spherical coordinates to write inequalities that describe this solid. What I tried to do: I started by graphing this on a 3D graph at...
  43. Petrus

    MHB Triple integral, spherical coordinates

    Hello MHB, So when I change to space polar I Dont understand how facit got \frac{\pi}{4} \leq \theta \leq \frac{\pi}{2} Regards, |\pi\rangle \int\int\int_D(x^2y^2z)dxdydz where D is D={(x,y,z);0\leq z \leq \sqrt{x^2+y^2}, x^2+y^2+z^2 \leq 1}
  44. Q

    A GR question about null surfaces, vectors and coordinates

    I wondered anyone can explain the significance of the above as applied to metrics in the context of general relativity. This came up when the video lecturer in GR mentioned that r for example, was null or this or that vector or surface was null, say in the context of the eddington finkelstein...
  45. V

    Nabla Operator in Spherical Coordinates

    Homework Statement Exercise 1.3 on uploaded Problem Sheet. Homework Equations Shown in Exercise 1.3 on Problem Sheet The Attempt at a Solution Uploaded working: I have found the inverse of the Transformation Matrix from Cartesian to Spherical Coordinates by transposing...
  46. E

    Angular Momentum In Polar Coordinates

    Homework Statement Consider a planet orbiting the fixed sun. Take the plane of the planet's orbit to be the xy-plane, with the sun at the origin, and label the planet's position by polar coordinates (r, \theta). (a) Show that the planet's angular momentum has magnitude L = mr^2 \omega, where...
  47. B

    Lagrangian, Hamiltonian coordinates

    Dear All, To give a background about myself in Classical Mechanics, I know to solve problems using Newton's laws, momentum principle, etc. I din't have a exposure to Lagrangian and Hamiltonian until recently. So I tried to read about it and I found that I was pretty weak in coordinate...
  48. I

    Re: Entropy - Actually a question about working in Polar Coordinates

    show that \frac{d\hat{r}}{dt}=\hat{θ}\dot{θ} also, \frac{d\hat{θ}}{dt}=-\dot{θ}r i've tried finding the relationship between r and theta via turning it into Cartesian coord.s, and I've tried the S=theta r but still no luck. S=theta r dS/dt=d(theta)/dt r which is similar to the RHS...
  49. Einj

    Laplacian in toroidal coordinates

    Hi everyone, I would like to write the Laplacian operator in toroidal coordinate given by: $$ \begin{cases} x=(R+r\cos\phi)\cos\theta \\ y=(R+r\cos\phi)\sin\theta \\ z=r\sin\phi \end{cases} $$ where r and R are fixed. How do I do? More generally how do I find the Laplacian under a...
  50. N

    Forces, displacement, and coordinates of a particle

    Homework Statement Two forces, vector F 1 = (4 i hat bold + 6 j hat bold) N and vector F 2 = (4 i hat bold + 8 j hat bold) N, act on a particle of mass 1.90 kg that is initially at rest at coordinates (+1.95 m, -3.95 m). A) What are the components of the particle's velocity at t = 10.3...
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