What is Coordinate: Definition and 908 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. G

    Spacetime represented in coordinate geometry

    if the 3 spatial dimensions are represented on a 3 axis graph. would time be a straight line through the origin? if not could your direct me to some reading that i may find answers? I'm uncertain whether space-time can be represented using euclidean geometry.
  2. pellman

    Coordinate basis vs local frame?

    The wikipedia article on connection forms refers to a local frame. What is the relationship between local frames and coordinate bases? Are they the same thing? Is one a subset of the other? The connection form article uses general notation e_\alpha for the basis elements instead of the...
  3. B

    Magnitude and coordinate direction angles of the resultant force.

    Problem is here http://engineeringhomework.net/statics/FG02_32-03UNP62.JPG Determine the magnitude and coordinate direction angles of the resultant force. I don't know where to start on this one.
  4. C

    Del operator with coordinate transformations

    How can you express the del operator after a change of variables? For example, if I want to use cylindrical coordinates for a fluids problem, what is the del operator in terms of the new coordinates? And how do you derive it for any other arbitrary coordinate transforms?
  5. L

    Block on Plane with Friction in strange coordinate system

    Homework Statement A block of mass m slides down a plane inclined at an angle theta with initial velocity v down the slope and with friction coefficient mu. Find T, the time in which the block comes to a rest due to friction. Use coordinates with y vertical and x horizontal. Homework Equations...
  6. G

    Calculate a Triple Integral using the cylindrical coordinate system

    I don't understand what's wrong with it:
  7. I

    Covariant derivative in spherical coordinate

    I am confused with the spherical coordinate. Say, in 2D, the polar coordinate (r, \theta) The mathworld website says that http://mathworld.wolfram.com/SphericalCoordinates.html D_k A_j = \frac{1}{g_{kk}} \frac{\partial A_j}{\partial x_k} - \Gamma^i_{ij}A_i I don't know why we...
  8. P

    Polar Coordinate Symmetry with Double Angles

    Homework Statement Given the equation r²=25sin2Θ Asked to find symmetry with respect to line Θ = pi/2 Homework Equations w.r.t. Θ = pi/2: (r,Θ) - (r, pi-Θ) and (r, Θ) - (-r,-Θ) The Attempt at a Solution For the first case, I plugged in (pi-Θ) for Θ, but I'm confused about what to do...
  9. Jonnyb42

    Coordinate Systems: GR vs Newtonian

    I just want to ask a simple question: Is it true that Newtonian/Classical Mechanics does not hold true for all coordinate systems, while General Relativity does?
  10. A

    Coordinate transformation and metric tensor

    General four-dimensional (symmetric) metric tensor has 10 algebraic independent components. But transformation of coordinates allows choose four components of metric tensor almost arbitrarily. My question is how much freedom is in choose this components? Do exist for most general metric...
  11. E

    Maxwell stress tensor coordinate system

    Hello, I am trying to understand the Maxwell Stress Tensor. Specifically, I would like to know if it is coordinate-system dependent (and if so, what the expressions are for the stress tensor in cylindrical and spherical coordinates). Griffiths gives the definition of the maxwell stress tensor...
  12. Z

    Coordinate Geometry- distance between two points

    Hi Homework Statement Two points X(a,b) and Y(b,a) Prove that the distance = sqrt2 (a-b) Homework Equations d = sqrt(x2-x1)^2 + (y2-y1)^2) The Attempt at a Solution I have drilled it down to (sqrt 2b^2+2a^2-4ba) but unable to drill it down to sqrt2 (a-b) Help...
  13. P

    Coordinate System Transformations

    Lets say I have Coordinate Frame's A and B. and... I have the coordinates of the 3 principle axes of B in terms of Frame A, So for a simple example, a rotation of +pi/2 about the z axis of A would yield the following mapping of the xyz axes of B in terms of Frame A: XA -> -YB YA -> XB ZA ->...
  14. S

    Computeing the coordinate vector

    hello, am confuse with this problem. I have B = {U1,U2} and B' = {u'1,u'2} u1 = [2,2], u2 =[4,-1] u'1 = [1,3], u'2 = [1,1] Now I have found the transition matrix from B' to B which is 13/10 -1/2 -2/5 0 Now, the question that am having trouble with is: Compute the coordinate...
  15. T

    Is this the correct method for finding a coordinate vector?

    Homework Statement [PLAIN]http://img25.imageshack.us/img25/3409/linearf.jpg Homework Equations The Attempt at a Solution Is this the right method to do this type of question (as I haven't seen an example of finding a coordinate vector before)? Do I solve the following system of...
  16. D

    General coordinate transformations for tensors

    Homework Statement Write down the transformation laws under general coordinate transformations for a tensor of type (0,1) and a tensor of type (2,1) respectively The Attempt at a Solution I seem to have two transformation formulas but they could in fact just be the same thing. I'll just do...
  17. K

    Working with different coordinate systems

    Does anyone know of a good book for relearning and working with different cooridinate systems like polar cylindricaly spherical the typicall engineering stuff...
  18. J

    Coordinate singularities and coordinate transformations

    I have a metric of the form ds^2 = (1-r^2)dt^2 -\frac{1}{1-r^2}dr^2-r^2 d\theta^2 - r^2 sin^2\theta d\phi^2 A singularity exists at r=\pm 1 . By calculating R^{abcd}R_{abcd} i found out that this singularity is a coordinate singularity. I found the geodesic equations for radial photons...
  19. P

    What are valid coordinate transforms (diffeomorphisms)?

    The thread bcrowell had on time reversal in GR got me thinking about this. Some limitations are obvious: mapping two events onto one, discontinuity,... I will use x*, t*, etc. to refere to tansformed coordinates (primes always confuse me with derivatives). Similarly, the transform x* = t, t*...
  20. K

    How to derive the spherical coordinate form for Laplacian

    Homework Statement \Delta f = \frac{1}{{r^2 }}\frac{\partial }{{\partial r}}\left( {r^2 \frac{{\partial f}}{{\partial r}}} \right) + \frac{1}{{r^2 \sin \phi }}\frac{\partial }{{\partial \phi }}\left( {\sin \phi \frac{{\partial f}}{{\partial \phi }}} \right) + \frac{1}{{r^2 \sin ^2 \phi...
  21. L

    Coordinate free definition of hessian

    Let M be a manifold and f: M \rightarrow \mathbb{R} be a smooth function such that df=0 at some point p \in M. Let \{ x^\mu \} be a coordinate chart defined in a neighbourhood of p. Define F_{\mu \nu} = \frac{ \partial f}{ \partial x^\mu \partial x^\nu } By considering the transoformation...
  22. T

    Coordinate Geometry: Find A & B on 5x-12y=6 & 3x-4y=-2

    Homework Statement The points A and B are located on the line 5x-12y=6.The perpendicular distances of A and B respectively from the line 3x-4y=-2 are both 4 units. A lies on the same side of 3x-4y=-2 as the origin ,and B lies on the other side. Find the coordinates of A and B. Homework...
  23. T

    Vector coordinate transformation: Help?

    Homework Statement How does \delta_{b}C^{d} transform? Also compute \delta^{'}_{b} C^{'d}The Attempt at a Solution \delta_{b} C^{d} = \frac{dC^{d}}{dX^{b}} ?I think I am supposed to prove that its a scalar, but I really have no starting point. Any extensive help would be really great.
  24. P

    MTW Formula 25.38 Coordinate Time

    In "Gravitation" - Misner, Thorne, Wheeler, the second part of formula 25.38 (first part describes the proper time left till the singularity) seems to suggest it describes some kind of coordinate time left. But the question is what exactly? Here is the formula: t={\it 2M}\, \left( -2/3\...
  25. M

    What Is the X Coordinate of the Particle?

    Homework Statement The graph below represents the time versus velocity of a particle. The x coordinate of the particle at 30 seconds is 20 feet. What is the x coordinate of the particle at 50 seconds?Homework Equations These are the equations I think could be used to solve this problem; the...
  26. P

    Calculate coordinate in 3D triangle.

    Hi, I'm looking for an efficient way of achieving the following, but having trouble thinking it through: Let's say I have a triangle in three dimensions with vertices A, B and C. Let's say I also have a three dimensional coordinate D. I know the x and z component values for D and I am...
  27. M

    Proof: Coordinate Rotation Around (0,0)

    Homework Statement Prove that the coordinates of the point (x',y') where the counter-clockwise rotation through the angle @ around (0,0) brings the given point (x,y) are x' = xcos@ - ysin@ y' = xsin@ + ycos@ Hint: show that for the points (x,y) = (1,0) and (x,y) = (0,1) directly, and...
  28. A

    Spherical coordinate derivatives

    1. Find the derivatives of the spherical coordinates in terms of df/dx, df/dy, and df/dz. 2. f(x,y,z) x=rcos\thetasin\varphi y=rsin\varthetacos\varphi z=rcos\varphi 3. The Attempt at a Solution [/b] I took the derivatives of the three equations and I got...
  29. B

    Having trouble with this Moments about the coordinate axes problem

    Having trouble with this "Moments about the coordinate axes" problem! When a force F is applied to the handle of the valve shown, its moments about the x and z axes are, respectively, M_x= -77 lb ft and M_z= -81 lb ft. For d=27 in., determine the moment M_y of F about the y axis...
  30. T

    Calculate Displacement Between Two Oases Without a Coordinate System

    Homework Statement Starting from one oasis, a camel walks 82.021 ft in a direction 30 degrees south of west and then walks 30 km toward the north to a second oasis. Without using a coordinate system, calculate the magnitude and the direction of the displacement from the first oasis to the...
  31. F

    About coordinate transformations in general

    I want to make sure of my understanding of coordinate transformations. First of all, is it true that if \[{x_i}\] is a coordinate system on a manifold, then \[{q_j} = {q_j}({x_i})\] is a coordinate transform from "x" space to "q" space? If so can "x" be a flat space and "q" a curved...
  32. R

    Charged particles and cylindrical coordinate system

    A charged particle in a magnetic field is spiralling along a path defined in cylindrical coordinates by r = 1 m and θ = 2z rad (where z is in meters). The speed along the path is constant at 3.87 km/s. What is the z-component of the velocity, vz, in cylindrical coordinates? My attempt...
  33. H

    Unknown coordinate given the distance

    Homework Statement The distance of point X (k,2) from the line y=x+4 is four. Find the values of k. Homework Equations The Attempt at a Solution (y-2)/(x-k)=-1 y=-x+k+2 I think this gives me the equation of the line from the point to the line y=x+4, but not sure what to do next...
  34. W

    Transforming angular velocity between different coordinate frames

    Homework Statement There are two coordinate frames i.e. frame A and frame B. The relationship between them is that frame A is rotated w.r.t to frame B . This relationship remains fixed i.e. rigid body. This rigid body relationship is given by rotation matrix R_BA which transforms the vector in...
  35. L

    What are coordinate covalent bonds and how do they form?

    How does one determine when a coordinate covalent bond will occur?
  36. F

    Euclidean space, euclidean topology and coordinate transformation

    Hi, I have some doubts about the precise meaning of Euclidean space. I understand Euclidean space as the metric space (\mathbb{R}^n,d) where d is the usual distance d(x,y)=\sqrt{\sum_i(x_i-y_i)^2}. Now let's supose that we have our euclidean space (in 3D for simplicity) (\mathbb{R}^3,d)...
  37. P

    Coordinate conjugate to momentum.

    Let's take a system, for simplicity with only one degree of freedom, described by a certain lagrangian L[x,\dot x] I define the momentum p=\frac{\partial L}{\partial\dot x} Now, if I make a change of coordinates x\longmapsto y\qquad\qquad\qquad(1) I obtain a second lagrangian M[y,\dot...
  38. T

    Coordinate Systems: 3D Angles Explained

    Hi! I see there are three 3D coordinate systems based on either 3 number (cartesian), 2 numbers and 1 angle (cylindrical) and 1 number and 2 angles (spherical). So can't there be a system based on 3 angles? Thank you.
  39. D

    Angles of a Vector on the Coordinate Plane

    Homework Statement This question is on my physics homework, I guess it could be posted elsewhere, but I'm not sure where. Here is the problem: Which of the following are the same as 18 degrees below the -x-axis? Choose all that apply: !The choices I selected are in red! A) 72 degrees left...
  40. M

    What are the general requirements for defining a coordinate system in R^3?

    Say we have a vector field defined in R^3. That is, at every point p in R^3, we have the corresponding set (p, v(p)). In representing this field, as far as I can tell, we have a certain list of very general requirements. That seems to be a.) an origin, b.) three everywhere non-coplanar curves...
  41. T

    Solve Isosceles Triangle in Coordinate System

    Homework Statement I will to explain this without a diagram. Consider a coordinate system(1st quadrant) where the x and y-axis both stop at 7 units (i mean the boundary), there are two points P(3,3) and Q(4,4). How many points can R be positioned such that PQR is an isosceles triangle...
  42. agro

    Double integral in polar coordinate

    Homework Statement With a > 0, b > 0, and D the area defined by D: \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 Change the integral expression below: \iint\limits_D (x^2+y^2) dx\,dy by using x = a r cos θ, y = b r sin θ. After that evaluate the integral. The Attempt at a Solution...
  43. S

    Playing with Coordinate Systems (Spherical Geometry)

    I'm working on a problem that involves two Earth stations that scan the skies. I'm writing a simulation program (no physics involved) that simply finds the az/alt of an event observed simultaneously by each station. At this point, I'm warming up to the mathematics, spherical geo, etc. to pull...
  44. S

    Can someone explain a polar coordinate conversion?

    I am having trouble understanding how (2x - x2)1/2 becomes 2 cos θ. Thanks
  45. H

    How to define a parabola in 3d coordinate system.

    Currently I am using a graphying application called "Autograph" and modeling a building with a dome shaped roof on top. I need to define parabolic shapes in 3d system. But i can't do it ( my math knowledge is pretty elementary) What would be the basic parabolic function in 3d that i can base...
  46. E

    Finding Volume Using Polar Coordinates: Inside Sphere and Outside Cylinder

    Homework Statement Use polar coordinates to find the volume of the given solid. Inside the sphere x²+y²+z²=16 and outside the cylinder x²+y²=4. Homework Equations x=rcosΘ,y=rsinΘ, x²+y²=r² The Attempt at a Solution 2∫∫ (√(16-r²)r)drdΘ R{(r,Θ)l 0<Θ<2∏, 2<r<4} I was...
  47. P

    Relating 2nd order partial derivatives in a coordinate transformation.

    Homework Statement Could some mathematically minded person please check my calculation as I am a bit suspicious of it (I'm a physicist myself). This isn't homework so feel free to reveal anything you have in mind. Suppose I have two functions \phi(t) and \chi(t) and the potential V which...
  48. A

    Cylindrical coordinate convertion

    Homework Statement cylindrical coordinates: r=2cscƟ, give both rectangular and spherical cordinates Homework Equations I know this: From rectangular to cylinder z=z r2=x2+y2 tanƟ=y/x From Cyl to rectangle x=cosƟ y=sinƟ z=z From cyl to spherical Ɵ=Ɵ...
  49. D

    Coordinate transformations question

    Hi all, I've been struggling with this for a couple of days now and am positing a question in the hope that someone can help me out. I have a global cartesian coordinate system X, Y, Z and a cube with it's centre at (0,0,0) and dimension 1. Hence it's corners are: (0.5, -0.5, 0.5), (-0.5...
  50. 0

    Bending of manifold and coordinate change

    I am new to this subject of topology. I want to know if bending and stretching of a manifold is same as a general transformation of a coordinate system drawn on the manifold. Or the mathematical definition of bending and stretching shall equally help.
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