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. I

    Gaussian Coordinates Explained: Resources & Definition

    Hey, Whenever I read Relativity I get stuck around Gaussian coordinates, i can't seem to find much out about them, do they have another name? does anyone know any good resources explaining them? or am I just in over my head? thanks guys,
  2. F

    Area under the curve using polar coordinates - help

    Hi, I have a pretty simple question but I'm not certain I know how to phrase it properly. I will try. When we are integrating using cartesian coordinates to find the area under a curve, area under the x-axis is negative and area above the x-axis is positive. This makes sense when I...
  3. U

    Transforming Coordinates: Exploring Non-Perpendicular Unit Vectors

    Homework Statement The x-y coordinates are being transformed into the u-v coordinates. Based on the diagram, u lies along x while v makes an angle α with x.The Attempt at a Solution The answer defined u and v weirdly.. Shouldn't x = u and y = v sin α ??
  4. W

    Horizontal + Celestial Coordinates to Geographic Coordinates

    Hey everyone, I cannot seem to figure this out and I'm having a hard time finding any guides online for this stuff. All I can find are calculators. I was wondering if it would be possible to calculate my Geographic Coordinates on Earth if I had the Horizontal and Celestial coordinates of a...
  5. T

    Cylindrical coordinates question

    Homework Statement https://dl.dropbox.com/u/64325990/cylindrical.PNG The Attempt at a Solution Okay so I found r = 2.24 and z = -3. However I am stuck at finding theta. I think I just don't understand what the question means when it says "In addition, the line defined by theta = 0 in...
  6. V

    Outgoing Eddington-Finkelstein coordinates

    I'm not sure what's going on in outgoing Eddington-Finkelstein coordinates for a Schwarzschild black hole. Future-directed timelike curves can be followed from inside the event horizon to outside it (page 185/186 of Sean Carroll's online GR notes...
  7. R

    Converting Rotation matrix to operate on fractional coordinates

    Hi I have an orthogonalized rotation matrix -0.500000 -0.866025 0.000000 0.866025 -0.500000 0.000000 0.000000 0.000000 1.000000 for the following unit cell: a b c alpha beta gamma space group 131.760 131.760 120.910...
  8. H

    When plotting graphs in polar coordinates, how does one know when to

    When plotting graphs in polar coordinates, how does one know when to make the graph sharp (at θ=0) (as in for the graph for r=1-cosθ) as opposed to a dimple (r=3/2 + cos θ) ?
  9. V

    Locally inertial coordinates on geodesics

    It's a standard fact of GR that at a given point in space-time, we can construct a coordinate system such that the metric tensor takes the form of Minkowski spacetime and its first derivatives vanish. Equivalently, we can make the Christoffel symbols vanish at point. Moreover, the fact that, in...
  10. mnb96

    Lie groups actions and curvilinear coordinates question

    Hello, let's suppose I have the following system of curvilinear coordinates in ℝ2: x(u,v) = u y(u,v) = v + e^u where one arbitrary coordinate line C_\lambda(u)=u \mathbf{e_1} + (\lambda+e^u) \mathbf{e_2} represents the orbit of some point in ℝ2 under the action of a Lie group. Now I consider...
  11. D

    Change to polar coordinates integration Problem

    Homework Statement Integrate y/(x^2+y^2) for x^2+y^2<1 and y> 1/2 ; use change of variables to polar coordinates Homework Equations THe above The Attempt at a Solution the variables transform as y=rsinz x=rcosz, where z is an angle between pi/6 and 5*pi/6 = which is the...
  12. H

    Parametric Surfaces: rectangular and polar coordinates

    Homework Statement I'm not grasping how to convert a surface with known rectangular graph to a parametric surface (using some polar techniques, I assume). I would appreciate it if someone could clarify the conversion process. One of the examples is as follows: A sphere...
  13. G

    Mathematica [Mathematica] Solving Heat Equation in Spherical Coordinates

    Hello Folks, I have this equation to solve (expressed in LaTeX): \frac{\partial{h}}{\partial t} = \frac{1}{n} \left[ \frac{1}{r^2 \sin^2{\phi}} \frac{\partial}{\partial \theta} \left( K \frac{\partial h}{\partial \theta} \right) + \frac{1}{r^2 \sin \phi} \frac{\partial}{\partial \phi}...
  14. G

    Developing Inner Product in Polar Coordinates via metric

    Hey all, I've never taken a formal class on tensor analysis, but I've been trying to learn a few things about it. I was looking at the metric tensor in curvilinear coordinates. This Wikipedia article claims that you can formulate a dot product in curvilinear coordinates through the following...
  15. P

    Polar Coordinates: Understanding Negative Distance r

    Hi, I am learning about Polar Coordinates and how they can be written in several equivalent ways. I understand how you can add 360 to angles and use negative angles to represent the same point. However, I have a very hard time understanding how you can write the same point but with a...
  16. R

    Vectors/Tensors-spherical coordinates. z component of force of fluid on a sphere

    i am a chemical engineer but this is fluid mechanics stuff so i figured you physics geniuses would know this stuff so to find the z component of force exerted by fluid on the surface of the sphere they find the normal force acting on a surface element of the sphere, integrated over the entire...
  17. S

    Minkowski metric - to sperical coordinates transformation

    I need to transform cartesian coordinates to spherical ones for Minkowski metric. Taking: (x0, x1, x2, x3) = (t, r, α, β) And than write down all Christoffel symbols for it. I really have no clue, but from other examples I've seen i should use chain rule in first and symmetry of...
  18. N

    Electromagnetic Field Tensor in Curvilinear Coordinates

    How to express electromagnetic field tensor in curvilinear coordinates, that is given a curvilinear coordinates (t,\alpha,\beta,\gamma) with metric tensor as follows: n_{\mu \nu }= \left[ \begin{array}{cccc}h_0^2& 0 & 0 & 0 \\ 0 & -h_1^2 & 0 & 0 \\ 0 & 0 & -h_2^2 & 0 \\ 0 & 0 & 0 & -h_3^2...
  19. G

    Metric tensor in spherical coordinates

    Hi all, In flat space-time the metric is ds^2=-dt^2+dr^2+r^2\Omega^2 The Schwarzschild metric is ds^2=-(1-\frac{2MG}{r})dt^2+\frac{dr^2}{(1-\frac{2MG}{r})}+r^2d\Omega^2 Very far from the planet, assuming it is symmetrical and non-spinning, the Schwarzschild metric reduces to the...
  20. P

    D Alembert's Principle: Dependence of kinetic energy on generalized coordinates.

    Hey! I was reading Goldestein's book on classical mechanics and I came across this (Page 20 3rd Edition): "Note that in a system of Cartesian coordinates the partial derivative of T with respect to qj vanishes. Thus, speaking in the language of differential geometry, this term arises...
  21. S

    Coordinates in GR: An Introduction &amp; Question

    Hello everyone! I'm new on the forum (been browsing threads for some time though) and this post is both an introduction of myself and a first question. I have a huge interest for physics but my working knowledge (having studied it in school, oh well some 15 years ago) is limited to classical...
  22. ShayanJ

    Generalized coordinates in Lagrangian mechanics

    In some texts about Lagrangian mechanics,its written that the generalized coordinates need not be length and angles(as is usual in coordinate systems)but they also can be quantities with other dimensions,say,energy,length^2 or even dimensionless. I want to know how will be the Lagrange's...
  23. P

    Sketch the Curve in Polar Coordinates

    Homework Statement Sketch the curve r = 1 + 2cosθ in polar coordinates. Homework Equations None that I can think of, it's graphing. The Attempt at a Solution What I was trying to was use the method of finding cartesian coordinates and plugging different values of θ into the equation to...
  24. G

    Integrating the metric in 3-D Spherical coordinates

    Guys, I read that integrating the ds gives the arc length along the curved manifold. So in this case, I have a unit sphere and its metric is ds^2=dθ^2+sin(θ)^2*dψ^2. So how to integrate it? What is the solution for S? Note, it also is known as ds^2=dΩ^2 Thanks!
  25. F

    Proof - Express in Clyndrical Coordinates

    Proof -- Express in Clyndrical Coordinates Homework Statement Show that when you express ds^2 = dx^2 + dy^2 +dz^2 in cylindrical coordinates, you get ds^2 = dr^2 + r^2d^2 + dz^2. Homework Equations x=rcosθ y=rsinθ z=z The Attempt at a Solution EDIT// I was really over thinking...
  26. T

    Triple integral for cone in cylindrical coordinates.

    Homework Statement Find limits of integration for volume of upside down cone with vertex on origin and base at z=1/sqrt(2). Angle at vertex is pi/2. Do this in cylindrical coordinates. Homework Equations None. The Attempt at a Solution My inner integral conflicts with the books...
  27. T

    Position of the Sun and Moon in ECEF coordinates

    Hi! I was just wondering if anyone knows where I can find a software/algorithm that can give me the postion of the sun and the moon in ECEF coordinates? If not, do you have any clues to how I might start building one? I am an engeneering student, so I don't have a lot of knowlegde about...
  28. A

    Why use primed coordinates for this

    Griffiths notation kind of bothers me. Can anyone explain why he uses primed coordinates in the attached picture. Wouldn't dl, da, dτ do just as well? Cheers :)
  29. T

    Triple Integrals: Spherical Coordinates - Finding the Bounds for ρ

    Homework Statement Find the volume of the solid that lies above the cone z = root(x2 + y2) and below the sphere x2 + y2 + x2 = z. Homework Equations x2 + y2 + x2 = ρ2 The Attempt at a Solution The main issue I have with this question is finding what the boundary of integration is for ρ. I...
  30. Dale

    Null coordinates in flat spacetime

    There are some "standard" coordinate systems in flat spacetime, such as Minkowski (inertial), Rindler (uniform acceleration), and Born (rotation). Is there a "standard" coordinate system in flat spacetime which has at least one null coordinate?
  31. E

    Finding polar coordinates of polar points

    Homework Statement Plot the Following points(given in polar coordinates). Find all the polar coordinates of each point. a. (2, pi/2) b. (2,0) c. (-2, pi/2) d. (-2,0) Homework Equations none The Attempt at a Solution I have plotted it on a graph but could someone explain to me...
  32. A

    Jacobian Matrix for Polar Coordinates

    Hi, I need some help understanding the solution to a problem. Equations: x = r.cos(θ) y = r.sin(θ) r = x2 + y2 theta = arctan(y/x)Question: Determine the Jacobian Matrix for (x,y)T and for (r, θ)T SOLUTION: I understand and can compute by myself the Jacobian for (x,y)T, but the solution to...
  33. V

    Coordinates of a point outside a plane

    Hello, If you can get me a hint for solving this matter it would be much appreciated. I have the 3D coordinates of three points on a plane A, B, C. There's another point G and we know AG, BG, CG. My problem is to find the coordinates of point G:cry: Thanks in advance!
  34. Z

    Integral in cylindrical coordinates

    I recently did an integral of the form: ∫∫1/ρ dρρdθ the extra ρ between dρ and dθ is the cost of switching to cylindrical coordinates. Now I want to know, do you carry out the integration in ρ, keeping the ρ outside the integration (since it's technically a scaling factor that belongs to...
  35. E

    Converting to Spherical Coordinates then integrating? Am I doing this right?

    Converting to Spherical Coordinates...then integrating? Am I doing this right? Homework Statement Consider the integral ∫∫∫(x2z + y2z + z3) dz dy dx, where the left-most integral is from -2 to 2, the second -√(4-x2) to √(4-x2) and the right-most integral is from 2-√(4-x2-y2) to...
  36. maistral

    Gaussian integral to polar coordinates - limit help?

    I'm trying my very best to understand it, but really, I just couldn't get it. I read four books now, and some 6 pdf files and they don't give me a clear cut answer :( Alright, so this integral; ∫e-x2dx from -∞ to ∞, when converted to polar integral, limits become from 0 to 2∏ for the outer...
  37. R

    Converting cartesian to polar coordinates in multiple integrals

    Homework Statement Do you see how y gets converted to csc? I don't get that. I would y would be converted to sin in polar coordinates.
  38. F

    Kinematics Vectors and cartesian coordinates. Plane with wind blowing.

    Homework Statement An airplane flies at an air speed of 300 miles per hour, in the direction toward southwest. There is a head wind of 75 mi/hr in the direction toward due east. (A) Determine the ground speed. (B) Determine the direction of motion of the plane, expressed as an angle...
  39. C

    Find volume of solid elliptic paraboloid using polar coordinates

    Homework Statement a elliptic paraboloid is x^2/a^2+y^2/b^2<=(h-z)/h, 0<=z<=h. Its apex occurs at the point (0,0,h). Suppose a>=b. Calculate the volume of that part of the paraboloid that lies above the disc x^2+y^2<=b^2.:confused: 2. The attempt at a solution We normally do the...
  40. E

    Derivation of heat transfer equation for spherical coordinates

    Homework Statement where λ= thermal conductivity \dot{q}= dissipation rate per volume Homework Equations qx=-kA\frac{dT}{dx} The Attempt at a Solution I don't know where to start from to be honest, so any help would be greatly appreciated
  41. H

    Rotating the coordinates to coincide the principal axes

    Dear all, We can rotate the local coordinates of the element so that the stress tensor becomes diagonal. The new coordinate system would be the principal stress axes of which are in fact the eignevectors of the stress tensor. Once we have the eigenvectors ( which are generally orthogonal)...
  42. Y

    Computing a surface integral with polar coordinates

    Homework Statement Show that ##\iint_{S}(x^2 + y^2)d\sigma = \frac{9\pi}{4}## where ##S = \{(x,y,z): x > 0, y > 0, 3 > z > 0, z^2 = 3(x^2 + y^2)\}## Homework Equations ##\iint_{S}f(x,y,z)d\sigma = \iint_{R}f(r(x,y))\sqrt{[r_x(x,y)]^2 + [r_y(x,y)]^2 + 1}## where ##r : R → ℝ^3, R \in ℝ^2##...
  43. M

    Double integral with polar coordinates

    Homework Statement It is given a set defined as: 0≤x≤1, 0≤y≤1-x. With x,y in ℝ. f(x,y)=1 (plane parallel to Oxy plane) They ask you to express the integral ∫∫Setf(x,y)dxdy in polar coordinates and calculate it. Homework Equations x=rcosθ y=rsenθ r=√x2+y2 The Attempt at a...
  44. M

    Evaluate the triple integral (with spherical coordinates)

    Homework Statement Firstly sorry for my bad english,i have a one question for you(İ try it but i didn't solve it ) Homework Equations The Attempt at a Solution i know problem will be solved spherical coordinates but i don't know how i get angles (interval) theta and fi ...
  45. E

    Area of overlapping polar coordinates

    Homework Statement find the over lapping area of the following equations r=3sin(x) r=1+sin(x)Homework Equations area =1/2 ∫ f(x)^2 dxThe Attempt at a Solution first off I started by finding the intersecting angle by: 3sin(x)=1+sin(x) 2sin(x)=1 sin(x)=1/2 x=pi/6 and the peak is at pi/2 so I...
  46. M

    Exploring the Mystery of 3D Coordinates: A= (X, Y, Z, 1)

    hello EveryBody, In the 3D Coordinates I always find 4 parameters instead of 3. A = (X, Y, Z, 1) I wonder why? thank you.
  47. R

    Coordinates and change of base

    Homework Statement The Attempt at a Solution I don't understand where 2v1 + 3v2 and 4v1 - 3v2 came from.
  48. A

    Finding xy coordinates of obtuse and acute triangle

    This might seem easy, but I am sort of rusty on the math since i haven't taken a math course in a while. Homework Statement A 2 meter long bar lies in the xy plane with one end at the origin. find position at the xy plane of the other? end point of the bar if the angle the bar makes with...
  49. R

    Express the given vector in terms of its coordinates

    Homework Statement Express the given vector in terms of its coordinates: The vector from the origin to the end point of the vector from (-3,7,2) in the direction and with the length of u = (2, -3, 4) The Attempt at a Solution I don't even know the algorithm for solving this...
  50. C

    Polar Coordinates - Areas

    I am asked to consider the following graph: r2=a+sin(θ), where a=2 I have a picture of this plot, which I have attached, We are asked to find the area of the upper 'cresent' of the curve, contained at the top How would I go about calculating that? I've found that if I plot...
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