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.
The following question struck me by accident.
Would appreciate to let me know the flaw in my logic, no need to deviate from this approach.
We know that on small-scale, experiment such as dropping ball from some height to the earth tells us that space is non-isotropic, because of preferred...
I've been told that the infinitesimal change in coordinates x and y as you rotate along a hyperbola that fits the equation b(dy)^2-a(dx)^2=r takes the form δx=bwy and δy=awx, where w is a function of the angle of rotation (I'm pretty sure it's something like sinh(theta) but it wasn't clarified...
I'm studying Susskind's GR TTM book, in which he gives a nice explanation of why differential geometry is needed for GR. But there is one gap that I want to fill.
The argument is: through a thought experiment, it seems that a uniform gravitation field can be seen as an artifact of going from an...
I got no problem with the arithmetic and solving it, finding the intersection of 2 lines is nothing, but I decided to play around with the problem and explore it more and ran into a strange problem, why can't I shift the X axis up? Let me explain.
The problem can be set up from 2 different...
I need to prove that under an infinitesimal coordinate transformation ##x^{'\mu}=x^\mu-\xi^\mu(x)##, the variation of a vector ##U^\mu(x)## is $$\delta U^\mu(x)=U^{'\mu}(x)-U^\mu(x)=\mathcal{L}_\xi U^\mu$$ where ##\mathcal{L}_\xi U^\mu## is the Lie derivative of ##U^\mu## wrt the vector...
I started by expanding ##dx## and ##dt## using chain rule:
$$dt = \frac{dt}{dX}dX+\frac{dt}{dT}dT$$
$$dx = \frac{dx}{dX}dX+\frac{dx}{dT}dT$$
and then expressing ##ds^2## as such:
$$ds^2 =...
My Progress:
I tried to perform the coordinate transformation by considering a general function ##f(\mathbf{k},\omega,\mathbf{R},T)## and see how its derivatives with respect all variable ##(\mathbf{k},\omega,\mathbf{R},T)## change:
$$
\frac{\partial}{\partial\omega} f =...
Could one derive a set of coordinate transformations that transforms events between different reference frames in the de Sitter metric using the invariant line element, similar to how the Lorentz Transformations leave the line element of the Minkowski metric invariant? Would these coordinate...
Using the transformation for ##t##, I obtained
$$\mathrm{d}t'=\left(1+\frac{\partial f}{\partial t}\right)\mathrm{d}t+\frac{\partial f}{\partial r}\mathrm{d}r$$.
Using this equation, I substituted it into the general line element to obtain
\begin{align*}...
I have come across Cartesian, Polar, Cylindrical & Spherical Coordinate Systems so far and was wondering if someone could tell me which are the uncommon systems used in physics which everyone says that they exist but no one explicitly mentions. Is there a "standard reference" or are they just...
I posted a thread yesterday and I think that I did not formulated it properly.
So I have a metric ##{ds}^{2}=-{dt}^{2}+{dx}^{2}+2{a}^2(t)dxdy+{dz}^{2}##
I was asked to find the the coordinate transformation so that I can get a diagonalized metric.
so what I've done is I assumed a coordinate...
hey there :)
So I had a homework, and I was asked to diagonalize the metric ##{ds}^2=-{dt}^2+{dx}^2+2a^2(t)dxdy+{dz}^2## and to find the coordinate transformation for the coordinates of the new metric.
so I found the coordinate transformation but the lecturer said that what I found is a...
Let us suppose we have a covariant derivative of a contravariant vector such as
$$\nabla_{\mu}V^{\nu}=\partial_{\mu}V^{\nu} + \Gamma^{\nu}_{\mu \lambda}V^{\lambda}$$
If ##\Delta_{\mu}V^{\nu}## is a (1,1) Tensor, it must be transformed as
$$\nabla_{\bar{\mu}}V^{\bar{\nu}} = \frac{ \partial...
In a certain anisotropic conductive material, the relationship between the current density ##\vec j## and
the electric field ##\vec E## is given by: ##\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)## where ##\vec n## is a constant unit vector.
i) Calculate the angle between the...
I kinda know how to do this problem, it is just that I hit a sign problem. If I take the partial derivative of the coordinate transformation with respect to ##x'^\mu##, I get
writing it first in the inverse form, ##x^\alpha = x'^\alpha - \epsilon^\alpha##
##\frac{\partial x^\alpha}{\partial...
Let me begin by stating that I'm aware of the fact that this is a metric of de Sitter spacetime, aka I know the solution, my problem is getting there. My idea/approach so far: in the coordinates ##(u,v)## the metric is given by
$$g_{\mu\nu}= \begin{pmatrix}1 & 0\\ 0 & -u^2\end{pmatrix}.$$
The...
I can see that by the tensor transformation law of the Kronecker delta that
##\frac{\partial x^a}{\partial x^b}=\delta^a_b##
And thus coordinates must be independent of each other.
But is there a more straightforward and fundamental reason why we don’t consider dependent coordinates? Is it...
I will start with an example.
Consider components of metric tensor g' in a coordinate system
$$ g'=
\begin{pmatrix}
xy & 1 \\
1 & xy \\
\end{pmatrix}
$$
We can find a transformation rule which brings g' to euclidean metric g=\begin{pmatrix}
1 & 0 \\
0 & 1\\
\end{pmatrix}, namely...
Given the coordinates ##P = (3,4)## , find the coordinates of ##P"(x',y')## when the origin is shifted to (1, –2), and the axes are rotated by 90° in the clockwise direction.
I attempted to solve this problem using the following formulas :
##x = X + h## and ##y = Y + k## for translation of the...
Hi all,
I am currently working on a creating a mathematical model of a longboard and am in need of advice. The pictures describe the sitaution.
Side view
Top view
Back view
The pictures depict a simplified longboard - the brown line is the deck and the black lines represent the...
In Mathematical Methods for Physicists, 6th Edition, by Arfken and Weber, Chapter 1 Vector Analysis, pages 8-9, the authors make the following statement:
"If Ax and Ay transform in the same way as x and y, the components of the general two-dimensional coordinate vector r, they are the...
I am really confused about coordinate transformations right now, specifically, from cartesian to polar coordinates.
A vector in cartesian coordinates is given by ##x=x^i \partial_i## with ##\partial_x, \partial_y \in T_p \mathcal{M}## of some manifold ##\mathcal{M}## and and ##x^i## being some...
This is intuitively very simple problem but I am unable to complete it with Mathematical rigor. Here is the deal:
A coordinate system $(u,v,w,p)$ in which the metric tensor has the following non-zero components, $g_{uv}= g_{ww}=g_{pp}=1$. Find the coordinate transformation between $(u,v,w,p)$...
I've been playing around a bit with the Kerr orbit program I wrote, and have been thinking about ways to set the initial conditions. One thing I'd like to be able to do is specify a launch from some event in terms that would be convenient for an observer at that event with some given...
How author derives these old basis unit vectors in terms of new basis vectors ? Please don't explain in two words.
\hat{e}_x = cos(\varphi)\hat{e}'_x - sin(\varphi)\hat{e}'_y
\hat{e}_y = sin(\varphi)\hat{e}'_x + cos(\varphi)\hat{e}'_y
Hi everyone!I'm trying to obtain the natural and dual basis of a circular paraboloid parametrized by:
$$x = \sqrt U cos(V)$$
$$y = \sqrt U sen(V)$$
$$z = U$$
with the inverse relationship:
$$V = \arctan \frac{y}{x}$$
$$U = z$$
The natural basis is:
$$e_U = \frac{\partial \overrightarrow{r}}...
In the second volume, Field Theory, of popular series of Theoretical Physics by Landau-Lifschitz are given following equations as in attached file from the book. Here is considered metric change under coordinate transformation. How is the new, prime metric expressed in original coordinates is...
I am trying to analyse response of a dynamic system. The source disturbance is about x,y,theta (rotation about x ) & Phi of one coordinate system (red coloured coordinate system in the attached figure).
I need to get the response in another coordinate system ( green coloured coordinate system...
##x= r Cosh\theta##
##y= r Sinh\theta##
In 2D, the radius of hyperbolic circle is given by:
##\sqrt{x^2-y^2}##, which is r.
What about in 3D, 4D and higher dimensions.
In 3D, is the radius
##\sqrt{x^2-y^2-z^2}##?
Does one call them hyperbolic n-Sphere? How is the radius defined in these...
I have the following equations:
\left\{ \begin{array}{l}
x = \sin \theta \cos \varphi \\
y = \sin \theta \cos \varphi \\
z = \cos \theta
\end{array} \right.
Assume \vec r = (x,y,z), which is a 1*3 vector. Obviously, x, y, and z are related to each other. Now I want to calculate \frac{{\partial...
Is there some geometry in which a coordinate transformation of a vector of magnitude zero transforms to a vector that does not have a zero magnitude?
Since the formula for the magnitude of a vector is √(x12+x22+...xn2), I can see no way for it to have magnitude zero unless every component is...
hi, I always see that jacobian matrix is derived for just 2 dimension ( ıt means 2x2 jacobian matrix) in books while ensuring the coordinate transformation. After that kind of derivation, books say that you can use same principle for higher dimensions. But, I really wonder if there is a proof...
What is the trace of a second rank tensor covariant in both indices?
For a tensor covariant in one index and contravariant in another ##T^i_j##, the trace is ##T^k_k## but what is the trace for ##T_{ij}## because ##T_{kk}## is not even a tensor?
Homework Statement
Given the line element ##ds^2## in some space, find the transformation relating the coordinates ##x,y ## and ##\bar x, \bar y##.
Homework Equations
##ds^2 = (1 - \frac{y^2}{3}) dx^2 + (1 - \frac{x^2}{3}) dy^2 + \frac{2}{3}xy dxdy##
##ds^2 = (1 + (a\bar x + c\bar y)^2) d\bar...
Hi..
I am new to this forum and not sure whether this is the right place to ask for a help.
I have to transform angular velocity vector of a satellite from Earth Centered Inertial (ECI) coordinate system to Local Vertical Local Horizontal(LVLH) system. How can I do that..? Any help appreciated..
I want ask another basic question related to this paper - http://www.tandfonline.com/doi/pdf/10.1080/16742834.2011.11446922
If I have basis vectors for a curvilinear coordinate system(Euclidean space) that are completely orthogonal to each other(basis vectors will change from point to point)...
Iam having trouble understanding how one arrives at the transformation matrix for spherical to rectangular coordinates.
I understand till getting the (x,y,z) from (r,th
ie.,
z = rcos@
y = rsin@sin#
x = rsin@cos#
Note:
@ - theta (vertical angle)
# - phi (horizontal angle)
Please show me how...
Homework Statement
Evening all, I'm trying to solve the 2-D diffusion equation in a region bounded by y = m x + b, and y = -m x -b. The boundary condition makes it complicated to work with numerically, and I recall a trick that involves a coordinate transformation so that y = m x + b, and y =...
Evening all,
I'm trying to solve the 2-D diffusion equation in a region bounded by y = m x + b, and y = -m x -b. The boundary condition makes it complicated to work with numerically, and I recall a trick that involves a coordinate transformation so that y = m x + b, and y = -m x -b are mapped...
Homework Statement
I'm trying to find the direction and magnitude of Earth's gravity on some projectile. The question states that I can ignore z, and that the origins of the x and y axes should be on the surface of the planet. I should then use Newton's law of Gravity to find the direction and...
Homework Statement
I have three angles that relate the sensory body frame to the Earth centered Earth fixed frame:
roll_ecef, pitch_ecef, yaw_ecef
I have the longitude and latitude.
Find the roll, pitch and yaw angles that relate the sensor body frame to the local North East Down frame...
Hi,
I have a reference device that outputs euler angles, which are angles that relate the sensor body frame to the north east down frame. These angles are called pitch roll and yaw. The sensor is an accelerometer. I know how to get the rotation matrix that will put accelerations from the...
Homework Statement
Let B1={([u][/1]),([u][/2]),([u][/3])}={(1,1,1),(0,2,-1),(1,0,2)} and
B2={([v][/1]),([v][/2]),([v][/3])}={(1,0,1),(1,-1,2),(0,2,1)}
a) Show that B1 is a basis for [R][/3]
b) Find the coordinates of w=(2,3,1) relative to B1
c)Given that B2 is a basis for [R[/3], find...
Homework Statement
Transform the coordinates from the red c-system to the blue system. (Picture)
Homework Equations
Using(X Y) for the red cartesian system and (x y) for the blue system
The Attempt at a Solution
The solution to this problem gives
x=Xcos▼ + Ysin▼
y=-Xsin▼+Ycos▼
Im not sure...
Hello,
I'm running a galaxy formation simulation. The output specifies the coordinates in (x, y, z) of all the particles in a galaxy, which usually fall in a disk. The orientation of the disk depends on the initial conditions, but it is generally not aligned with any of the coordinate axes...
So I know that this involves using the chain rule, but is the following attempt at a proof correct.
Let M be an n-dimensional manifold and let (U,\phi) and (V,\psi) be two overlapping coordinate charts (i.e. U\cap V\neq\emptyset), with U,V\subset M, covering a neighbourhood of p\in M, such that...
So I have the metric as ##ds^{2}=-(1-\frac{2m}{r})dt^{2}+(1-\frac{2m}{r})^{-1}dr^{2}+r^{2}d\Omega^{2}##*
I have transformed to coordinate system ##u,r,\phi, \theta ##, where ##u=t-r*##(2),
where ##r*=r+2m In(\frac{r}{2m}-1)##
and to the coordinate system ##v,r,\phi, \theta ##,
where...