What is Coordinate transformations: Definition and 41 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.
For transformations, A and B are similar if A = S-1BS where S is the change of basis matrix.
For Lie groups, the adjoint representation Adg(b) = gbg-1, describes a group action on itself.
The expressions have similar form except for the order of the inverses. Is there there any connection...
i have successfully transformed the continuity equation using coordinate transform,but having trouble with the momentum equation .
can someone kindly provide the transformation of the right hand sight of equation of the image i have attached.
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:
$$
\frac{\partial}{\partial\omega} f =...
Could one derive a set of coordinate transformations that transforms events between different reference frames in the de Sitter metric using the invariant line element, similar to how the Lorentz Transformations leave the line element of the Minkowski metric invariant? Would these coordinate...
The example is about the transformation between the cartesian coordinates and polar coordinates using the definition
In lewis Ryder's solution, I got confused in this specific line
I really can't see how is that straightforward to find?
The line element given corresponds to the metric:
$$g = \begin{bmatrix}a^2t^2-c^2 & at & 0 & 0\\at & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{bmatrix}$$
Using the adjugate method: ##g^{-1}=\frac{1}{|g|}\tilde{g}## where ##\tilde{g}## is the adjugate of ##g##. This gives me...
T = (x+\frac{1}{\alpha}) sinh(\alpha t)
X = (x+\frac{1}{\alpha}) cosh(\alpha t) - \frac{1}{\alpha}
Objective is to show that
ds^2 = -(1 +\alpha x)^2 dt^2 + dx^2
via finding dT and dX and inserting them into ds^2 = -dT^2 + dX^2
Incorrect attempt #1:
dT= (dx+\frac{1}{\alpha})...
I was studying linearized GR where we make the following coordinate transformation ## \tilde{x}^{a} = x^{a} + \epsilon y^{a}(x) ##
This coordinate transformation is then meant to imply ## g_{ab}(x) = \tilde{g}_{ab}(x) + \epsilon \mathcal{L}_{Y} g_{ab} ##
Would anyone be kind enough to explain...
Hi there
I'm studying GR and I am confused about coordinate transformations.
In my understanding, if I want to study a rotating reference system this is what I do.
In my inertial system the object trajectory is described by
$$
x = r\cos(\theta - \omega t)\\
y = r\sin(\theta - \omega t)
$$...
I was reading this article (https://www.nature.com/articles/srep40083) about designing a panoramic lens with transformation optics, and wanted to try to play around with modifying the coordinate transformations. I contacted one of the authors of the article, and she mentioned that she plotted...
Homework Statement
Consider the metric ds2=(u2-v2)(du2 -dv2). I have to find a coordinate system (t,x), such that ds2=dt2-dx2. The same for the metric: ds2=dv2-v2du2.
Homework Equations
General coordinate transformation, ds2=gabdxadxb
The Attempt at a Solution
I started with a general...
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
Homework Statement
Is there a more intuitive way of thinking or calculating the transformation between coordinates of a field or any given vector?
The E&M book I'm using right now likes to use the vector field
## \vec F\ = \frac {\vec x} {r^3} ##
where r is the magnitude of ## \vec x...
I was reading this article: https://arxiv.org/ftp/arxiv/papers/1005/1005.5206.pdf , regarding the mathematical description of a diamond-shaped cloaking device, and am struggling to understand how the authors found the coordinate transformations in equations (1) and (5).
What is the process for...
Apologies for perhaps a very trivial question, but I'm slightly doubting my understanding of Jacobians after explaining the concept of coordinate transformations to a colleague.
Basically, as I understand it, the Jacobian (intuitively) describes how surface (or volume) elements change under a...
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...
Suppose that I have a two-dimensional coordinate system (x,y) and I change to a new coordinate system (u,v). What I know is that there is some function \theta(u,v) such that:
\dfrac{\partial x}{\partial u} = cos(\theta)
\dfrac{\partial x}{\partial v} = -sin(\theta)
\dfrac{\partial y}{\partial...
I've come up with a curious two-part question while working on a map program:
What is the minimum number of points necessary in order to transform an NxN adjacency matrix into a coordinate matrix in terms of N given Euclidean space?
As this question relates to map-making, where I don't...
Hey
So, I was wondering how to convert from one coordinate axes to another... in particular, where the new axes are y = x and y = -x, as seen by the picture below
I want it so that the Red dot in the new coordinate system will be (\sqrt2,0). Is there an easy way to do this? (My lookings on...
I have read over and over in various places about coordinate transformations, and understand the theory (really!), but can't find any worked examples of actual use of the transformation equations. Does anyone know of any web references or tutorials on the subject?
To make things a little more...
Homework Statement
I'm having trouble understanding coordinate transformations for vector fields. There are two 'coordinate pieces', the coordinates pieces of the vector at a point changes, and the function describing the field can also be rewritten in terms of the new coordinates. I'm...
I have seen in various locations different conventions regarding the location of a prime symbol denoting a tensor represented in a new frame. For example, if the position four-vector is
x^{\mu}
then this four-vector in a different frame is often written as either
x'^{\mu}
or...
I've studied classical physics and never heard this before until recently...the allowable coordinate transformations for classical mechanics are rotations and translations. Could someone explain why this is so? What makes these "allowable" (I know they are orthogonal transformations).
Hi,
My question is the following. In special relativity, the Lorentz transformations correspond to a physical situation in which two frames of reference move with uniform rectilinear motion one with respect to the other. In general relativity, given the physical situation in which one frame...
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?
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...
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...
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...
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...
Hi there. This isn't so much a math question as it is a conceptual question. I can't seem to wrap my head around the need for coordinate transformations. *Why* do they need to be done? I think I really need a picture for this, so this might not be the right place to ask, but if you can...
Homework Statement
On a supermarket parking lot, a car is pulling out and bumping into an oncoming car. The car pulls out with 0.8 m/s, while the oncoming car has a speed of 1.2 m/s. The angle between the velocities is 24 degrees, as indicated in the figure. What is the collision speed...
Can anyone explain why it's important to be able to take vectors in an x,y,z coordinate system and be able to transform them into other coordinate systems. Could not all vector considerations be grappled with in the standard x,y,z coordinate systems? How important is this ability to physicists...
Hi,
I would like to transform a vector from Spherical to cartesian coordinate system. But the question is probably not that straight forward. :(
I have a vector say E = E_r~\hat{r}+E_{\theta}~\hat{\theta}+E_{\phi}~\hat{\phi}.
But I know only the cartesian coordinate from where it...
As I try to understand GR, I find coordinate transformations just about everywhere. My question is simply: What is the reason coordinate transformations play such an important role in GR? Thanks.
Sorry to bring up again a question that I asked before but I am still confused about this.
In SR we have Lorentz invariance.
Now we go to GR and one says that the theory is invariant under general coordinate transformations (GCTs). But, as far as I understand, this is simply stating that...
I have a vector (<1, 0, 0>) that needs to be transformed from an initial 3d rectangular coordinate system M1 through M2 and M3 to a final 3d rectangular system M4.
I'm currently doing this by applying sequential rotations omega, phi, and kappa about the x', y', and z' axes, respectively, for...
So often students question the validity of the twin paradox and how acceleration is involved in looking at round trip scenarios that I am asking why not just give them the tools to transform between the coordinates of an inertial frame and those of an accelerating frame. It is not hard to do and...
These problems are actually for my classical mechanics class, but they are linear-algebra based. I can construct a transformation matrix, but I have trouble visualizing the rotations, particularly in 3-space. So if someone could help me get a pictorial idea of what's actually happening, then...
Gents,
Could you please help me:
Speaking about General Coordinate Transformations, one speaks always generally. Are there any explicit expressions for General Coordinate Transformations? Like in SR speaking about Lorentz Transfrmations one recalls Lorentz Matrixes.
Maybe I'm not quite...
Hi.
I’ve just started learning about tensors on my own and am still trying to understand coordinate transformations.
If I begin with a vector whose Cartesian components are (x, y, z) and apply the tensor transformation to cylindrical polars, I end up with (r, 0, z) – is this right? I...