# What is Coordinates: Definition and 1000 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. ### Rotation of Spherical Mirrors

I think the given solution is wrong. The lens forms image at ##(+75,0)## which is ##25 cm## from pole of the convex mirror which acts as virtual object for mirror. It is true that the reflected ray is rotated by ##2\theta## as in case of plane mirror. Rotation of Spherical Mirrors But that...
2. ### Maple Physics Package Question: Trouble contracting indices

I have been using Maple for a decade, but only recently started using the Physics Package. I unfortunately ran into trouble to contract indices ( Maple calls it SumOverRepeatedIndices ). Below I give an example that will execute if you paste it into a Maple worksheet. A metric is defined. A...

16. ### Einstein relativity between 2 coordinates systems

I calculated the speed using the information provided through the above equation and finding V' = 1.2 m/s. However, the first solution must be -1,2 m/s. I don't know how to reach it, any suggestion?
17. ### Optimizing Polar Axis for Dipole in Polar Coordinates

I don't know how to get the result referring to the previous task. Is my decision correct?
18. ### Find the coordinates of D which lies on the vector BC

Not sure on howto proceed here?
19. ### I Dot product of two vector operators in unusual coordinates

Hi. I hope everyone is well. I'm just an old person struggling to make sense of something I've read and I would be very grateful for some assistance. This is one of my first posts and I'm not sure all the LaTeX encoding is working, sorry. Your help pages suggested I add as much detail as...

23. ### Coordinates of a point on a rotating wheel

My issue is in deriving the coordinates of a point on a wheel that rotates without slipping. In Morin's solution he says that: My attempt at rederiving his equation: I do not understand how the triangle on the bottom with sides indicated in green is the same as the triangle on top that is...

25. ### Angle between normal force and radial line for cylindrical coordinates

so I was wondering. there is this normal force on the can from the path. And there's this formula to find the angle between the radial line and the tangent or also between the normal force and either the radial or theta axis. the formula is ##\psi = r/dr/d\theta##. The thing is that here they...
26. ### A Two Questions about Novikov Coordinates

The general intention of Novikov coordinates on Schwarzschild spacetime is to construct a "comoving" coordinate chart for purely radial timelike geodesics, i.e., every such geodesic should have a constant radial coordinate, and the time coordinate should be the same as proper time for observers...
27. ### Solving Curl A in Spherical Coordinates: Tips & Hints

I've tried writing the curl A (in spherical coord.) and equating the components, but I end up with something that is beyond me: {\displaystyle {\begin{aligned}{B_r = \dfrac{1}{4 \pi} \dfrac{-3}{r^4} ( 3\cos^2{\theta} - 1) =\frac {1}{r\sin \theta }}\left({\frac {\partial...
28. ### How to calculate a sink using spherical coordinates

The issue is that the singularity is not in the center of the sphere. So how would I calculate it? I have a few questions: 1. Can I calculate the terms separately like so: $$A = grad(a+b) = grad(a) + grad(b)$$ 2. If I use a spherical coordinate system with the center being at the singularity I...
29. ### Using Right-Handed Coordinates: Exploring Solutions

Hi! For this problem, Why did the solutions choose to use a different coordinate system? I choose to use the right-handed coordinate system. Many thanks!
30. ### B Calc. Christoffel Symbols of Hiscock Coordinates

The Hiscock coordinates read: $$d\tau=(1+\frac{v^2(1-f)}{1-v^2(1-f)^2})dt-\frac{v(1-f)}{1-v^2(1-f)^2}dx$$ ##dr=dx-vdt## Where ##f## is a function of ##r##. Now, in terms of calculating the christoffel symbol ##\Gamma^\tau_{\tau\tau}## of the new metric, where ##g_{\tau\tau}=v^2(1-f)^2-1## and...
31. ### Can the positive direction be different for each particle in a system?

I'm not really struggling with the question but the coordinate systems involved more so. So due to the modelling assumptions we know that the tension will be equal throughout the rope so we can use f = ma on each particle respectively and solve the resulting equation (as acceleration will be...
32. ### Find the coordinates of intersection between tangents and given curve

ooops...this was a bit tricky but anyway my approach; ... ##\dfrac{dy}{dx}=-2x## therefore; ##\dfrac{y-7}{x+1}=-2x## and given that, ##y=4-x^2## then; ##4-x^2-7=-2x^2-2x## ##x^2+2x-3=0## it follows that, ##(x_1,y_1)=(-3,-5)## and ##(x_2,y_2)=(1,3)##. There may be another approach...
33. ### I Equation of motion: choice of generalized coordinates

I am looking at a textbook solution to the following problem of finding the equation of motion of a half disk. In the solution, the author considers the half disk has a COM at the black dot, and to find the instantaneous translational velocity of the center of mass (he considers rotational...
34. ### Question about vector coordinates

here i found AB to be (-3, 2) and then i thought to do 2/5 multiplied by AB to find AC, however this is incorrect and instead i would have to involve the origin. Why and how can i involve the origin?
35. ### B Method of images and spherical coordinates

I am finding the potential everywhere in space due to a point charge a distance 'a' on the z-axis above an infinite xy-plane held at zero potential. This problem is fairly straight forward; place an image charge q' = -q at position -a on the z-axis. I have the solution in cartesian coordinates...
36. ### I Instantaneous coordinates of an event in space (special relativity)

In relativity, no signal travels faster than light, and hence if something happened away from me, I will only know about it after some time. This means I cannot measure instantly the position and time of something as it happens; this would violate special relativity. I however imagine that I...
37. ### Integration of acceleration in polar coordinates

I made this exercise up to acquire more skill with polar coordinates. The idea is you're given the acceleration vector and have to find the position vector corresponding to it, working in reverse of the image. My attempts are the following, I proceed using 3 "independent" methods just as you...
38. ### I Understand 4-Vectors & Spacetime: Hartle Gravity Chapter 5

Hartle, gravity. Chapter 5 "A four-vector is defined as a directed line segment in four-dimensional flat spacetime in the same way as a three-dimensional vector (to be called a three-vector in this chapter) can be defined as a direcied line segment in three-dimensional Euclidean Space"For...
39. ### Troubleshooting Coordinates System from Cone

I tried using coordinates system from cone, but not got what actually want to get. Any idea from you will greatly appreciated. Thanks
40. ### A Independence of generalized coordinates and generalized velocities

How can I make sense of this and further how to think of this in the context of phase space diagrams?
41. ### I Formula for integration of natural coordinates over an element

In a textbook I own a formula is given for the integration of natural coordinates over an element. In this case it is a 1 dimensional element (i.e. a line segment) with coordinates ##x_i## and ##x_j##. The coordinate ##x## over the element is written as: $$x = L_1(x) x_i + L_2(x) x_j$$ with...
42. ### I Can I always consider velocities and coordinates to be independent?

It's a topic that's been giving be a headache for some time. I'm not sure if/why/whether I can always consider velocities and (independent) coordinates to be independent, whether in case of cartesian coordinates and velocities or generalized coordinates and velocities.
43. ### Motion in Cylindrical Coordinates

7:03 what is second component of a(theta)? this -> 2 * r' * (theta)' I understand everything except that.
44. ### I Co-Moving Coordinates & Lapse Function N(t) in ADM Decomposition

In the ADM decomposition, like in the construction of the FRW metric, the coordinates are defined to be co-moving, so we know $$d\tau = dt$$ (i.e. the lapse function is normalized away) Starting from a five-dimensional embedded hyperboloid (as in carroll pg. 324) ## -u^2 + x^2 + y^2 + z^2 + w^2...
45. M

### Mathematica Plot a vector valued function in cylindrical coordinates

Hi PF! I have a function ##f(s,\theta) = r(s,\theta)\hat r + t(s,\theta)\hat \theta + z(s,\theta)\hat z##. How can I plot such a thing in Mathematica? Surely there's an easier way than decomposing ##\hat r, \hat \theta## into their ##\hat x,\hat y## components and then using ParametricPlot3D?
46. ### Calculating the partial derivative in polar coordinates

Hello, I am trying to solve the following problem: If ##z=f(x,y)##, where ##x=rcos\theta## and ##y=rsin\theta##, find ##\frac {\partial z} {\partial r}## and ##\frac {\partial z} {\partial \theta}## and show that ##\left( \frac {\partial z} {\partial x}\right){^2}+\left( \frac {\partial z}...
47. ### A Curl in cylindrical coordinates -- seeking a deeper understanding

I calculate that \mbox{curl}(\vec{e}_{\varphi})=\frac{1}{\rho}\vec{e}_z, where ##\vec{e}_{\rho}##, ##\vec{e}_{\varphi}##, ##\vec{e}_z## are unit vectors of cylindrical coordinate system. Is there any method to spot immediately that ##\mbox{curl}(\vec{e}_{\varphi}) \neq 0 ## without employing...
48. ### I Polygon Coordinates given the Area and Center point

I’m wondering if there is a formula for calculating the coordinate points of a polygon given the following - Center point is known - area is known - Point A is known - Points B, C, and D are UNKNOWN I am NOT a math pro - this is for a puzzle I’m trying to solve and I can’t remember if this...

50. ### I Wavefunction in polar coordinates and its bra ket notation

The wavefunction of ##|\psi\rangle## is given by the bra ket ##\psi (x,y,z)= \langle r| \psi\rangle## I can convert the wavefunction from Cartesian to polar and have the wavefunction as ## \psi (r,\theta,\phi)## What bra should act on the ket ##|\psi\rangle## to give me the wavefunction as ##...