What is Coordinate transformation: Definition and 97 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. D

    Diffeomorphisms and active transformations

    I've been reading Sean Carroll's notes on General Relativity, http://arxiv.org/pdf/gr-qc/9712019.pdf . I've got to chapter 5 (page 133) and am reading the section on diffeomorphisms in which Sean relates diffeomorphisms to active transformations. When he says this does he mean that one defines a...
  2. T

    Coord. Transf.: V'μ from (dy/dx)*Vν

    V′μ=((∂yμ)/(∂xν))*Vν This is a contravariant vector transformation. (Guys I am really sorry for making the formula above looks so incomprehensible as I still new to this.) For the y in the partial derivative, is y a function in terms of x? In that sense, is it formula that maps x to y? Is it...
  3. 9

    Coordinate transform from sensor to North East Down frame

    I am using an algorithm that transforms from my sensor frame to North West Up and I want to instead use North East Down. I have attached the current algorithm. I also want to skip the first step in my algorithm. Here is the current algorithm: http://www.filedropper.com/transformationalgorithm...
  4. K

    Coordinate Transformation of the equation of continuity for a vaporizing droplet

    Hey there, I trying to understand the following coordinate transformation of the equation of continuity (spherical coordinates) for a vaporizing liquid droplet\frac{\partial \rho}{\partial t} + \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 \rho v) = 0 into \epsilon \sigma \frac{\partial...
  5. throneoo

    Coordinate transformation parameterization

    Homework Statement Suppose two observers O and O', whose positions coincide , each sets up a set of 2D cartesian coordinates (x,y) and (x',y') respectively to describe the position of a certain object at a fixed point . Derive a set of formulae for one observer to convert the other observer's...
  6. K

    What is the Geometrical Proof for the Coordinate Transformation Formula?

    Homework Statement Prove: \cos\alpha\cdot\cos\alpha'+\cos\beta\cdot\cos\beta'+\cos\gamma\cdot\cos \gamma'=\cos\theta See drawing Snap1 Homework Equations None The Attempt at a Solution See drawing Snap2. i make the length of the lines 1 and 2 to equal one, for simplicity. The...
  7. J

    Geometrical interpretation of this coordinate transformation

    How can I geometrically interpret this coordinate transformation (from x,y space to \check{x},\check{y} space)? x = \check{x}cos(β) - \check{y}sin(β) y = \frac{1}{2}(\check{x}2 -\check{y}2)sin(2β) -\check{x}\check{y}cos (2β)
  8. I

    Simple coordinate transformation question

    http://i.imgur.com/MDigPh5.png if i have my original coordinate (white) and i am transforming this into the red coord. , could someone explain to me why y=y'cos\phi is incorrect and why y'=ycos\phi is correct?
  9. E

    Proving The Hamiltonian Is Invariant Under Coordinate Transformation

    Homework Statement (a) Consider a system with one degree of freedom and Hamiltonian H = H (q,p) and a new pair of coordinates Q and P defined so that q = \sqrt{2P} \sin Q and p = \sqrt{2P} \cos Q. Prove that if \frac{\partial H}{\partial q} = - \dot{p} and \frac{\partial H}{\partial p} =...
  10. H

    Coordinate transformation for line integrals; quadrature rules

    Hi all, The context of this problem is as follows: I'm trying to implement a discontinuous finite element method and the formulation calls for the computation of line integrals over the edges of the mesh. Anyway, more generally, I need to evaluate \int_{e}f(x,y)ds, where e is a line segment...
  11. C

    Line element under coordinate transformation to get polar form

    Homework Statement Hello Guys, I am reading Hobson's General Relativity and I have come across an exercise problem, part of which frustrates me: 3.20 (P. 91) In the 2-space with line element ds^2=\frac{dr^{2}+r^{2}d\theta^{2}}{r^{2}-a^{2}}-\frac{r^{2}dr^{2}}{{(r^{2}-a^{2})}^{2}} and...
  12. O

    !Understanding Partial Derivatives of Coordinate Transformation

    Hi Everyone, I was studying coordinate transformation and I came across this equation, that I couldn't understand how it came up. Let me put it this way: x = rcosθ Then if I want to express the partial derivative (of any thing) with respect to x, what would be the expression? i.e. ∂/∂x=...
  13. topsquark

    MHB Coordinate transformation derivatives

    I've had to hit my books to help someone else. Ugh. Say we have the coordinate transformation \bf{x}' = \bf{x} + \epsilon \bf{q}, where \epsilon is constant. (And small if you like.) Then obviously d \bf{x}' = d \bf{x} + \epsilon d \bf{q}. How do we find \frac{d}{d \bf{x}'}? I'm missing...
  14. S

    How should I approach this (coordinate transformation) problem?

    I am starting to deal with optomechanical systems as part of my work, and am faced with what seems to be an uncomplicated problem, however I'm ashamed to admit that I am having great difficulty getting to grips with it. I'd like some pointers and/or advice as to how to go about solving these...
  15. M

    Analyzing a coordinate transformation

    In McCauley's book Classical Mchanics: Transformations, Flows, Integrable and Chaotic Dynamics we are analyzing a coordinate transformation in order to arrive at symmetry laws. A coordinate transformation is given by q_i(\alpha) = F_i(q_1,...,q_f, \alpha). Then, to the first order Mccauley...
  16. Z

    Coordinate Transformation in Special Relativity with Linear Algebra Part A

    Homework Statement In the figure, let S be an inertial frame and let S' be another frame that is boosted with speed v along its x'-axis w.r.t. S, as shown. The frames are pictured at time t = t0 = 0: A) Find the Non-relativistic transformation (Galilean Transformation) between the two...
  17. X

    Help with coordinate transformation problem

    Homework Statement For elliptical cylindrical coordinates: x = a * cosh (u) * cos (v) y = a * sinh (u) * sin (v) z = z Derive the relations analogous to those of Equations (168b-e) for circular cylindrical coordinates. In particular, verify that h_u = h_v = a * sqrt(cosh^2 (u) -...
  18. T

    About coordinate transformation of tensor.

    note:The tensors in the new coordinates system are represented by X'. I have a question about the coordinate transformation of tensor. σmn=amianjσ'ij; (1) ωpq=apkaqlω'kl; (2) In the original coordinates system, we have σmn = Dmpnqωpq, (3)...
  19. P

    Coordinate transformation of a tensor in 2 dimensions

    Homework Statement Given a symmetric tensor T_{\mu\nu} on the flat Euclidean plane (g_{\mu\nu}=\delta_{\mu\nu}), we want to change to complex coordinates z=x+iy, \,\overline{z}=x-iy. Show, that the components of the tensor in this basis are given by...
  20. A

    Use of tensor densities in coordinate transformation

    Why use a tensor density transformation when doing a coordinate transformations? What is the advantage? I've always learn that transforming a tensor involves pre and post multiplying by the transformation tensor and it's inverse respectively, but I've come across ones in my research that use...
  21. M

    Understanding Vector Invariance Under Coordinate Transformation

    What does it mean for a vector to remain "invariant" under coordinate transformation? I think I already know the answer to this question in a foggy, intuitive way, but I'd like a really clear explanation, if someone has it. I know all of multivariable calculus and quite a bit of linear algebra...
  22. 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...
  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. 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)...
  25. 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...
  26. I

    Coordinate transformation and conformal transformation?

    I'm confused by the relation of coordinate transformation and conformal transformation. I found a nice note about conformal field theory written by David Tong. It does contain the demonstration related to my question, but I still don't understand. Here it goes, The definition of conformal...
  27. Rasalhague

    Coordinate Transformation & Jacobian Matrix

    Is the following correct, as far as it goes? Suppose I have a vector space V and I'm making a transformation from one coordinate system, "the old system", with coordinates xi, to another, "the new system", with coordinates yi. Where i is an index that runs from 1 to n. Let ei denote the...
  28. P

    Galilean Coordinate Transformation (Classical Relativity)

    Homework Statement An observer in an inertial reference frame S sees two cameras flash simultaneously. The cameras are 800 m apart. He measures that the first flash occurs at four coordinates given by X1=0, Y1=0, Z1=0 and T1=0, and that the second flash occurs at four coordinates given by...
  29. M

    Coordinate transformation under rotation

    If a system is rotated around Z axis then the new coordinates are X'= xcos() - Y sin(), Y'= Xsin() + Ycos() Z'= Z How is this obtained ?? () --->theta , angle of rotation around Z axis .
  30. M

    Coordinate transformation of nabla operator

    Hi all! I am studying the Galilean group of transformations and I'm not sure how to transform the Nabla operator. Consider the 2 transformations: (x,t)->(x+s,t) (x,t)->(Dx,t) and the expression "nabla (x)" where D is a matrix and x, s are vectors I am pretty sure that I have...
  31. T

    Coordinate transformation of contravariant vectors.

    Note: The derivatives are partial. I've seen the coordinate transformation equation for contravariant vectors given as follows, V'a=(dX'a/dXb)Vb What I don't get is the need for two indices a and b. Wouldn't it be adequate to just write the equation as follows? V'a=(dX'a/dXa)Va...
  32. F

    Coordinate Transformation - velocity?

    If I have a velocity vector V in some coordinate system. Is there anything special that needs to be done to convert V -> V' ? Basically, I have a matrix (N x 3) in north-east-down coordinates. I am trying to convert (a row at a time_ to another matrix (N x 3) to earth-centered-earth-fixed...
  33. D

    Coordinate Transformation (and using line elements)

    This is from Hartle's GR book, in one of the first chapters it talks about diff geom, nothing too advanced, but I am learning on my own. Homework Statement It's part E I have trouble with. Read e. and skip to last para if you want. Consider this coordinate transformation: x=uv ...
  34. J

    How are tensors transformed in non-linear coordinate systems?

    Let's say I have a non-linear transformation for (ct,x,y,z) in one coordinate system to (cT,X,Y,Z) of another coordinate system. Despite being nonlinear, I assume I can transform all four-vectors using the same non-linear transformation (correct?), but how in the world do I transform the...
  35. A

    Coordinate transformation and choice of a suitable coordinate system.

    Consider the line element: ds^2=-f(x)dt^2+g(x)dx^2 in a coordinate system (t,x) where f(x) and g(x) are two functions to be determined by solving Einstein equation. But I can always make a transformation g(x)dx^2=dy^2 and then calculate everything in the (t,y) coordinate system. My...
  36. P

    Unitary coordinate transformation = rotation?

    Homework Statement Suppose I define a linear coordinate transformation that I can describe with a matrix U. If U is unitary. i.e. U^{-1}U = UU^{-1}=1 does that necessarily imply that the transformation corresponds to a pure rotation (plus maybe a translation), so that I may assume that...
  37. G

    Coordinate transformation and multiplying with size of J

    Hi, I am using the book "Advanced Engineering Mathematics" by Erwin Kreyszig where I am reading on the transformation of coordinates - when changing from \int f(x,y) to \int f(v(x,y),v(x,y) it is necessary to multiply with the size of the jacobian, |J| - I cannot find the proof in the book...
  38. P

    Lorentz coordinate transformation

    Homework Statement Frame S' has an x component of velocity u relative to the frame S and at t=t'=0 the two frames coincide. A light pulse with a spherical wave front at the origin of S' at t'=0. Its distance x' from the origin after a time t' is given by x'^2=(c^2)(t'^2). Transform this...
  39. R

    How Do You Reverse a Vector Transformation?

    If we want to transform vector A from cooedinate ei to ei', then this formula occur: Aj' = aij Ai But I have a question, if I have found all components of Aj', then I want to transform it back to Ai, what should I do? I have tried Ai = aij Aj' but it won't give me the same number. Thanks...
  40. S

    Coordinate transformation to flat space

    Homework Statement find the transformation that turns this metric: ds^2=-X^2dT^2+dX^2 into the minkowski metric: diag(-1,1). The Attempt at a Solution attempt 1: I transformed the above metric into the coordinates that use the lines that define the light cone as the axes. Namely, u=T+ln(X)...
  41. N

    Coordinate transformation of lagrangian

    Hey all, According to my physics textbook, if the potential energy of a particle is a homogeneous function of the spatial coordinate r, one can transform r by some factor a and t by some factor b=a^(1-.5k) such that the Lagrangian of the particle is multiplied by a^k. I understand all of this...
  42. L

    What Coordinate Transformation Converts a Complex Metric to Minkowski Space?

    How can I identify the coordinate transformation that turns ds^2 = \left(1+\frac{\epsilon}{1+c^2t^2}\right)^2c^2dt^2 - \left(\frac{\epsilon}{1+x^2}\right)^2x^2 - \left(\frac{\epsilon}{1+y^2}\right)^2y^2 - \left(\frac{\epsilon}{1+z^2}\right)^2z^2 into the Minkowski metric ds^2 = c^2dt^2...
  43. D

    Mathematica Mathematical Proof For Lorenz Coordinate Transformation

    Hi, I have a pretty in depth understanding of special relativity. Recently I have been searching for mathematical proof of the Lorenz transformation. I found some information about it, but to tell the truth I didn't understand much of it. Maybe one of you guys can shed some light on the proof...
  44. Q

    Coordinate Transformation Help

    Hello, I was wondering how one would go about finding the coordinate transformation that diagonalizes: 2(x1)^2+2(x2)^2+(x3)^2+2(x1)(x3)+2(x2)(x3) Thanks a bunch.
  45. P

    Finding Eigenvectors for Coordinate transformation

    I'm wondering if anyone here might have a solution to a problem I've having. This is a Quantum Mechanics problem I'm doing. I calculate a 4 by 4 complex Hermitian matrix (H = Hamiltonian) in a basis where it is not diagonal. I diagonalize it numerically (using eispack) and get eigenvalues...
  46. W

    Coordinate transformation matrix?

    Can anyone tell me: 1) How to understand the defination to orthogonal transformation matrix? Defination: A(i,j)A(k,j)=q(i,k) where q is Kronecker delta. 2) Why the inverse of this orthogonal matrix is equal to its transpose? Will.
  47. K

    What is the Differential of a Vector?

    Greeting A TA has got me very and utterly confused. He won't be avaible for a few days, so I'm asking you guys. Consider the transformation to cilindrical coord. x-->r.con[the] y-->r.sin[the] z-->z I have the Jabobian (no problems here). He then asks the differential da , where a...
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