What is Coordinate system: Definition and 200 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 Navier-Stokes equation is solved in a vector grid in a Cartesian coordinate system. That is, rectangular. But does a rectangular mesh relate to what happens in a gas or liquid, and is it better to use a triangular mesh?
Undoubtedly, it is incredibly difficult to take into account all the...
Hi, as test of GR I'm aware of there is the "anomalous" precession of the perihelion of Mercury.
My question is: in which coordinate system are the previsions of GR verified concerning the above ? Thanks.
Hi, I'm trying to solve the problem from here: https://www.physics.uoguelph.ca/chapter-1-newtonian-mechanics
Exercise 1.1: Determine how the coordinates $$x$$ and $$y$$, as well as the basis vectors $$\hat{x}$$ and $$\hat{y}$$, change under a rotation around the $$z$$ axis by an angle $$α$$...
I am looking for more books like this one: https://archive.org/details/MethodOfCoordinateslittleMathematicsLibrary
Method of Coordinaes (Little Mathematics Library) by A. S. Smogorzhevsky
I am also interested in papers if you can suggest any. I am interested in texts, that explore the idea of...
Hi everybody, I encountered a problem simulation of permanent magnets (PM) in Ansys Maxwell. There are many PM in my simulation and I need to define for each of them a proper coordinate system (CS). But I could only defined 255 CS. After that I can create new CS but It won't be possible to...
Hi, so I'm trying to find the volume of a shape using integral, I found the equation of one plane in 3D space but the second one is something like that, which I cannot write in integral as a function: ##\frac{2(2x-a)}{a}=-\frac{2(6y-a\sqrt3)}{a\sqrt3}=\frac{2z-a\sqrt3}{a\sqrt3}##
In the 3D...
Hi I'm wondering if someone can illustrate with an example what I bracketed in blue? I'm having a hard time visualizing how it is that the accelerations of the components are NOT necessarily equal to the components of the acceleration...Much appreciated!
In《Introducing Einstein's Relativity Ed 2》on page 106"lowering the first index with the metric,then it is easy to establish,for example by using geodesic coordinates..."
In 《A First Course in General Relativity - 2nd Edition》on page 159 "If we lower the index a,we get(in the locally flat...
Dear all,
the following problem is not a home-work problem. I have come up with this question for myself. Nevertheless, I am stuck and need your help.
The question is: Can I calculate the distance between points A and B from this information? And if yes, how?
I think it should be possible...
* We've a vector ##\mathbf{A}## lying in space, changing according to some rule.
* We introduce an inertial frame and find ##\left(\frac{d}{d t}
\mathbf{A} \right)_{i n}## in it.
* We also introduce a co located frame rotating with ##\mathbf{\omega}##. In this rotating frame I find...
Goldstein 3 ed, pg 171, under" rate of change of a vector " :
The author derives the relationship between the change of a vector in a stationary and rotating coordinate system.
In the process he uses this assumption :>It is no loss of generality to take the space and body axes as...
If we change the orientation of a coordinate system as shown above, (the standard eluer angles , ##x_1y_1z_1## the initial configuration and ##x_by _b z_b## the final one), then the formula for the coordinates of a vector in the new system is given by
##x'=Ax##
where...
Just want to clarify some concepts.
There seems to be difference between reference frame and coordinate system. See https://en.wikipedia.org/wiki/Frame_of_reference#Definition . A reference frame is something has physical meaning and is related to physical laws, whereas coordinate system...
Hello,
reading the wiki entry for Langevin observers on rotating disk - Born_coordinates I'm struggling with the following quoted sentence: But as we see from Fig. 1, ideal clocks carried by these ring-riding observers cannot be synchronized.
I do not grasp why, starting from the figure...
I have been using a coordinate system that is anchored on an event (rather than a speed reference) in Minkowskian spacetime. This makes it sort of a special case (no gravity or dark energy, just like special relativity) of the cosmological (or CMB-isotropic) coordinate system used to foliate the...
The result equation doesn't fit with the familiar divergence form that are usually used in electrodynamics.
I want to know the reason why I was wrong.
My professor says about transformation of components.
But I cannot close to answer by using this hint, because I don't have any idea about "x"...
Hello,
This question is with regards to the discussion around page 56 (1971 Edition) in Anthony French's Newtonian Mechanics. He is discussing the choice of a coordinate system where the axes are not necessarily perpendicular to each other. Here is the summary of what I read (as applied to...
Maybe my question is naive and due to my not deep enough knowledge of particle physics. I imagine we entangle two particles on Earth and then send one on spaceship going from Earth - two coordinate frames moving in relation to one another. Moments of simultaneity are different for them. When...
In celestial coordinate system (right ascension/declination), how to check if an object with position RA and dec is within a given circular field of view of radius R (in arcminutes) and centred at (0,0)?
R is small in this case so I assumed that I could compute the distance d of the object from...
To my understanding, to get the basis vectors for a given coordinate system (in this case being the elliptic cylindrical coordinate system), I need to do something like shown below, right?
$$\hat{\mu}_x = \hat{\mu} \cdot \hat{x}$$
$$\hat{v}_z = \hat{v} \cdot \hat{z}$$
And do that for...
How I would have guessed you were supposed to solve it:
What you are supposed to do is just take the gradients of all the u:s and divide by the absolute value of the gradient? But what formula is that why is the way I did not the correct way to do it?
Thanks in advance!
FBD Block 1
FBD Block 2
FBD Pulley B
I'm mainly concerned with the coordinate system direction in this problem, but just to show my attempt, here are the equations I got from the system.
##-T_A + m_1g = m_1a_1##
##T_B - m_2g = m_2a_2##
##T_A - 2T_B = 0##
Using the fact that the lengths...
The attached file is the coordinate system I've used
a) $$\vec{E}=\dfrac{\vec{F_e}}{q}=\dfrac{1,10\cdot{10^{-13}}\hat{j}\;N}{1,6\cdot{10^{-19}}\;C}=6,88\cdot{10^5}\hat{j}\;N/C$$
b) $$\sum{\vec{F_{net}}}=\vec{0}=\vec{F_e}+\vec{F_m}$$...
##\vec F= 2x^2y \hat i - y^2 \hat j + 4xz^2 \hat k ##
## \Rightarrow \vec \nabla \cdot \vec F= 4xy-2y+8xz##
Let's shift to a rotated cylindrical system with axis on x axis:
##x \to h, y \to \rho cos \phi, z \to \rho sin \phi ##
Then our flux, as given by the Divergence theorem is the volume...
Hello,
the physical domain in the (y, z) space is mapped to a rectangular computational region in the (ŋ,Ƹ)-space, where (ŋ,Ƹ) are the new coordinates. This technique frees the computational simulation from geometry restriction.
after transforming the governing equations ( PDEs) to the...
Good Morning
I have a very vague memory of having read (about 40 years ago) that there are only 11 coordinate systems in which the field equations of physics can be separated.
I can no longer be sure if my memory has failed me. But this issue has been in my head for all these years. (Gotta do...
These are images from the book Introduction to Electrodynamics by David J. Griffiths .
. .
My problem is that I'm unable to understand how the current has zero ##\phi## component (I have underlined it in the first image)? I do understand cylindrical coordinates, I know...
I am teaching myself math and have a question about cartesian coordinate systems. How is time illustrated in such a graph?
[Moderator's note: Moved from a math forum after post #13.]
When you graph something like ##θ=\frac{π}{4}## on a Polar Coordinate System:
Why does the line go into the opposite quadrant as well?
I can intuitively understand why it is in the first quadrant: ##θ = 45°## there and so all possible values of ##r## would apply there, giving you a straight line...
hi all,
firstly i need to calculate point location in new coordinate system. ı have 2 line segments and a point(x,y,z) in word cartesian coordinates system. For example, my first line segment is (0,0,5) , (50,0,3) locations and second line segment is (0,6,3),(0,-6,7) locations in cartesian...
As far as I understand, when we want to differentiate a vector field along the direction of another vector field, we need to define either further structure affine connection, or Lie derivative through flow. However, I don't understand why they are needed. If we want to differentiate ##Y## in...
What is the advantage of using a polar coordinate system with rotating unit vectors? Kleppner's and Kolenkow's An Introduction to Mechanics states that base vectors ##\mathbf{ \hat{r}}## and ##\mathbf{\hat{\theta}}## have a variable direction, such that for a Cartesian coordinates system's base...
In light of the modern definition of what is a coordinate system, namely it's a pair (U, f) with U a region of a m-dimensional manifold, and f a bijection from U to ##\mathbb R^m##, can we say that the polar coordinates on ##\mathbb R^2## are a coordinate system?
I was thinking about this and...
I was asking myself what is the definition of a Cartesian Coordinate System. Can we say that it's a coordinate system such that
- the basis vectors are the same ##\forall x \in R^n##
- the basis vectors are orthonormal at each ##x \in R^n##
So for instance, normalized polar coordinates do not...
There is remarkably little information on the internet including Wikipedia on this topic. Can someone point me in the right direction as I want to build a visualization software that illustrates the supergalactic plane and the coordinate system with any kind of celestial sphere involved.
All I...
Hey! :o
We have the basis $B=\left \{\begin{pmatrix}1 \\ 1 \\ 1\end{pmatrix},\begin{pmatrix}2 \\ 1 \\ 0\end{pmatrix}, \begin{pmatrix}1 \\ 2 \\ 1\end{pmatrix} \right \}$ of $\mathbb{R}^3$ and the vector $v$ can we written as a linear combination of the elements of the basis as follows...
Hello,
For example, an electric field vector, such as the gradient of the potential, passes in the following way to any coordinate system:$$E = -\triangledown{}V = - \frac{{\partial V}}{{\partial x^i}}g^{ij}e_j$$
But what about a vector of a magnetic field? How would it be expressed in any...
Knowing that a scalar quantity doesn't change under rotation of a coordinate system. Do we consider a point in a Cartesian coordinate system (i.e. A (4,5)) a scalar quantity? If yes, why do the components of point A change under rotation of the coordinate system?
According to my understanding...
Hello! I am reading "A First Course in General Relativity" by Schutz and in chapter 8 (second edition) he introduces Nearly Lorentz coordinate system. He says that we can always find some coordinates such that the metric is: $$g_{\alpha\beta}=\eta_{\alpha\beta}+h_{\alpha\beta}$$ with...
1. The problem statement, all variables and given/known dana
I was revisiting University physics textbook and came across this problem. We learned new coordinate systems in classical mechanics classes so I wanted to see if I can apply this to the problem of force on semicircular part of the...
Hey guys, I would appreciate some help with the math behind creating a working coordinate system for a robotic arm. I am currently trying to determine what servo angles are necessary to align a robotic arm's claw to the given coordinates. Geometrically simplified, the robotic arm is a...
Homework Statement
Two points in a plane have polar coordinates P1(2.500m, pie/6) and P2(3.800m, 2pie/3) .
Determine their Cartesian coordinates and the distance between them in the Cartesian coordinate system. Round the distance to a nearest centimeter.Homework Equations
Ax=Acosθ
Ay=Asinθ...
Ignoring special relativity theory，maxwell equation are deduced in which coordinate system?In most electrodynamics textbook,maxwell equation are deduced without specifying which coordinate we are using.For example,when we are solving poisson equation in static case，it seems we can freely choose...
So, I've been studying some tensor calculus for general theory of relativity, and I was reading d'Inverno's book, so out of all exercises in this area(which I all solved), this 6.30. exercise is causing quite some problems, so far. Moreover, I couldn't find anything relevant on the internet that...
Homework Statement
I have to calculate the partial derivative of an arctan function. I have started to calculate it but I wonder if there is any simpler form, because if the simplest solution is this complex then it would make my further calculation pretty painful...
Homework Equations
$$\beta...
Homework Statement
Homework EquationsThe Attempt at a Solution
Jacobian of the coordinate- system (## u_1, u_2##) with respect to another coordinate- system (x,y ) is given by
J = ## \begin{vmatrix}
\frac { \partial {u_1 } } {\partial {x } } & \frac { \partial {u_1 } } {\partial {y} } \\...
Homework Statement
With respect to a given Cartesian coordinate system S , a vector A has components Ax= 5 , Ay= −3 , Az = 0 . Consider a second coordinate system S′ such that the (x′, y′) x y z coordinate axes in S′ are rotated by an angle θ = 60 degrees with respect to the (x, y) coordinate...