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. Athenian

    I Transforming Cartesian Coordinates in terms of Spherical Harmonics

    As the subject title states, I am wondering how would one go about transforming Cartesian coordinates in terms of spherical harmonics. To my understanding, cartesian coordinates can be transformed into spherical coordinates as shown below. $$x=\rho \sin \phi \cos \theta$$ $$y= \rho \sin \phi...
  2. E

    MHB Finding the Coordinates of Point D

    Points A ,B and C have coordinates A(1,3) B(5,1) and C(2,-8). point D is such that the vector AD = Vector BC + (2x) vector AB + (3y) vectorAC = vector AB + (2x) vectorAC + (3y) vector BC find coordinates of D
  3. JorgeM

    A How do I express an equation in Polar coordinates as a Cartesian one.

    I got a polar function. $$ \psi = P(\theta )R(r) $$ When I calculate the Laplacian: $$ \ \vec \nabla^2 \psi = P(\theta)R^{\prime\prime}(r) + \frac{P(\theta)R^{\prime}(r)}{r} + \frac{R(r)P^{\prime\prime}(\theta)}{r^{2}} $$ Now I need to convert this one into cartesian coordinates and then...
  4. M

    Determining an object's velocity in cylindrical coordinates

    I got the answer for velocity and acceleration. But I don't know how to draw the shape of the particle's motion over time. How to draw it? should we change a,b,c,e into a numbers or not? or we may not to change a,b,c,e? Please help me how to draw the shape of particle's motion over time?
  5. M

    Divergence Theorem in Curvilinear Coordinates: Questions & Explanations

    Hi, I was trying to gain an understanding of a proof of the divergence theorem in curvilinear coordinates. I have found these online notes here and am looking at the proof on pages 4-5. The method intuitively makes sense to me as opposed to other proofs which fiddle around with vector...
  6. patric44

    Orthogonal trajectories in polar coordinates

    there is a problem in a book that asks to find the orthogonal trajectories to the curves described by the equation : $$r^{2} = a^{2}\cos(\theta)$$ the attempt of a solution is as following : 1- i defferntiate with respect to ##\theta## : $$2r \frac{dr}{d\theta} = -a^{2}\;\sin(\theta)$$ 2- i...
  7. Another

    Velocity in spherical coordinates

    Why the velocity in spherical coordinates equal to ## v^2 = v \dot{} v = \dot{r}^2 + \dot{r}^2\dot{\theta}^2## maybe ## v^2 = v \dot{} v = (\hat{ \theta } \dot{ \theta } r +\hat{r} \dot{r} + \hat{ \phi } \dot{\phi } r \sin{ \theta}) \dot{} (\hat{ \theta } \dot{ \theta } r +\hat{r} \dot{r} +...
  8. M

    Stokes' Theorem 'corollary' integral in cylindrical polar coordinates

    Hi, I was just working on a homework problem where the first part is about proving some formula related to Stokes' Theorem. If we have a vector \vec a = U \vec b , where \vec b is a constant vector, then we can get from Stokes' theorem to the following: \iint_S U \vec{dS} = \iiint_V \nabla...
  9. B

    Choosing proper coordinates in a complex 2 pulley system

    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...
  10. W

    I Using polar coordinates to describe rose petals

    I encountered a question which asked me to describe the rose petal sketched below in polar coordinates. The complete answer is R = {(r, θ): 0 ≤ r ≤ 6 cos(3θ), 0 ≤ θ ≤ π}. That makes sense to me for the right petal. What about the other two on the left?
  11. U

    I Calculating Surface Area of Schwarzschild Black Hole w/Weyl Coordinates

    Recently, I was tasked to find the surface area of the Schwarzschild Black Hole. I have managed to do so using spherical and prolate spheroidal coordinates. However, my lecturer insists on only using Weyl canonical coordinates to directly calculate the surface area. The apparent problem arises...
  12. T

    Orbital equations in polar coordinates

    The equations of motion are: \ddot{r}-r{\dot{\theta}} ^{2} = -\frac{1}{r^{2}} for the radial acceleration and r\ddot{\theta} + 2\dot{r}\dot{\theta}= 0 for the transverse acceleration When I integrate these equations I get only circles. The energy of the system is constant and the angular...
  13. P

    Polar coordinates of the centroid of a uniform sector

    If I use cartesian co-ordinates, I get: ##\bar{x}=\frac{1}{A}\iint x\, dA=\frac{1}{A} \iint r^2\cos\theta\, dr\, d\theta= \frac{2a\sin\theta}{3\theta}## ##\bar{y}=\frac{1}{A}\iint y\, dA=\frac{1}{A}\iint r^2\sin\theta\, dr\, d\theta= \frac{2a(1-\cos\theta)}{3\theta}## But if I use polar...
  14. Leo Liu

    A particle moving with zero radial acceleration in polar coordinates

    In the example above, the authors claim that when ##r=r_0e^{\beta t}##, the radial acceleration of the particle is 0. I don't quite understand it because they did not assume ##\beta=\pm \omega##. Can anyone please explain it to me? Many thanks.
  15. Leo Liu

    Insight into polar coordinates (Newtonian mechanics)?

    I am learning to use polar coordinates to describe the motions of particles. Now I know how to use polar coordinates to solve problems and the derivations of many equations. However, the big picture of polar coordinates remains unclear to me. Would you mind sharing your insight with me so that I...
  16. tomdodd4598

    I Argument for Existence of Normal Coordinates at a Point

    Hey there, I've been recently been going back over the basics of GR, differential geometry in particular. I was watching one of Susskind's lectures and did not understand the argument made here (26:33 - 35:40). In short, the argument goes as follows (I think): we have some generic metric ##{ g...
  17. S

    B Understanding how coordinates change under the flow of a vector field

    [Ref. 'Core Concepts in Special and General Relativity' by Luscombe] Let ##M,M'## be manifolds and ##\psi:M\to M'## a diffeomorphism. Even if ##\psi## weren't a diffeomorphism, and instead just a smooth map, the coordinates of the pushback of ##\mathbf{t}\in T_p(M)##, would be related to the...
  18. E

    B Helix in intrinsic coordinates?

    If a particle is in a magnetic field ##\vec{B} = B\hat{z}## with velocity ##\vec{v} = v_x \hat{x} + v_y \hat{y} + v_z \hat{z}##, then in Cartesian coordinates we can obtain the pair of differential equations $$\ddot{x} = \frac{qB}{m}\dot{y}$$$$\ddot{y} = -\frac{qB}{m}\dot{x}$$which give the...
  19. C

    MHB How to prove an ideal of a ring R which is defined as a coordinates

    Hi Everyone, I am wondering how to prove an ideal of a ring $R$ which is defined as a coordinates. Let $R$ be the ring of $\mathbb{Z} \times \mathbb{Z}$. Let $I={(a,a)| a\in \mathbb{Z}}$. I determine that the $I$ is a subring of $R$. Next step is to show the multiplication between the elements...
  20. George Keeling

    I What does comoving mean and how is it related to isotropy?

    I come across the adjective 'comoving' quite often. I understand comoving coordinates for the Universe. They are coordinates which expand with the expansion of the Universe(?) but I'm confused about what it means in essence. Here are some examples: In Sean Carrol's book there is a question...
  21. J

    I Changing spherical coordinates in a Lagrangian

    In order to compute de lagrangian in spherical coordinates, one usually writes the following expression for the kinetic energy: $$T = \dfrac{1}{2} m ( \dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \sin^2 \theta \dot{\phi}^2 )\ ,$$ where ##\theta## is the colatitud or polar angle and ##\phi## is the...
  22. T

    Vector Field Transformation to Spherical Coordinates

    I am trying to solve the following problem from my textbook: Formulate the vector field $$ \mathbf{\overrightarrow{a}} = x_{3}\mathbf{\hat{e_{1}}} + 2x_{1}\mathbf{\hat{e_{2}}} + x_{2}\mathbf{\hat{e_{3}}} $$ in spherical coordinates.My solution is the following: For the unit vectors I use the...
  23. Terrycho

    Divergence of a position vector in spherical coordinates

    I know the divergence of any position vectors in spherical coordinates is just simply 3, which represents their dimension. But there's a little thing that confuses me. The vector field of A is written as follows, , and the divergence of a vector field A in spherical coordinates are written as...
  24. E

    I Velocity Vector Transformation from Cartesian to Spherical Coordinates

    Hi all, I can't find a single thing online that translates a cartesian velocity vector directly to spherical vector coordinate system. If I am given a cartesian point in space with a cartesian vector velocity and I want to convert it straight to spherical coordinates without the extra steps of...
  25. archaic

    Calculating Area in Polar Coordinates

    $$-2\sin\theta=1\Leftrightarrow\theta=-\frac{\pi}{6},\,-\frac{5\pi}{6}\\ \begin{align*} \int_{-\frac{\pi}{6}}^{-\frac{5\pi}{6}}\frac 12\left(4\sin^2\theta-1\right)d\theta &=\int_{-\frac{\pi}{6}}^{-\frac{5\pi}{6}}\frac 12\left(1-2\cos2\theta\right)d\theta\\...
  26. G

    Line integral where a vector field is given in cylindrical coordinates

    What I've done so far: From the problem we know that the curve c is a half-circle with radius 1 with its center at (x,y) = (0, 1). We can rewrite x = r cos t and y = 1 + r sin t, where r = 1 and 0<t<pi. z stays the same, so z=z. We can then write l(t) = [x(t), y(t), z ] and solve for dl/dt...
  27. Replusz

    I Changing to Polar coordinates in order to calculate this integral

    m a
  28. S

    Kinematics problem - changing coordinates

    Hi folks, See below for a solved question finding the down slope distance of an arrow. How easy would it be to solve this question by making the x-axis the slope direction?
  29. ttpp1124

    Determining Coordinates - lines and planes

  30. Tony Hau

    How to find the length of a vector expressed in polar coordinates?

    The velocity of a particle below is expressed in polar coordinates, with bases e r and e theta. I know that the length of a vector expressed in i,j,k is the square of its components. But here er and e theta are not i,j,k. Plus they are changing as well. Can someone help convince me that the...
  31. L

    A Volume element in Spherical Coordinates

    For me is not to easy to understand volume element ##dV## in different coordinates. In Deckart coordinates ##dV=dxdydz##. In spherical coordinates it is ##dV=r^2drd\theta d\varphi##. If we have sphere ##V=\frac{4}{3}r^3 \pi## why then dV=4\pi r^2dr always?
  32. K

    Using polar coordinates in 1-dimensional problems

    If I have a physical problem, say, a particle which is constrained to move in the ##y## direction, which means that its ##x## coordinate remains fixed, does it make sense to write ##y## in terms of polar coordinates? That is, ##y = r \sin\theta##. Since now I have two parameters ##r,\theta##...
  33. A

    I Quadrupole Moment Time Variation: Does Coordinate Choice Matter?

    [Moderator's note: Thread spun off from previous discussion due to topic change.] Does the observed quadrapole moment change over time when considering a relatively moving object, for certain choices of observer coordinates? My suspicion is that it does (Terrell-Penrose rotation implies...
  34. JD_PM

    What are the coordinates of the upper cylinder? (and some algebra)

    I was solving a problem and got stuck in two aspects: 1) Geometric issue. Alright, I understand that the coordinates of the lower cylinder are $$( -R \theta_1, R)$$ The coordinates of the upper cylinder are: $$( x_1 + 2R \sin \theta, 3R - 2(R-\cos \theta))$$ I get that the ##x##...
  35. jk22

    I Solving the Wave Equation via complex coordinates

    I'm looking for material about the following approach : If one suppose a function over complex numbers ##f(x+iy)## then ##\frac{df}{dz}=\frac{\partial f}{\partial x}\frac{1}{\frac{\partial z}{\partial x}}+\frac{\partial f}{\partial y}\frac{1}{\frac{\partial z}{\partial y}}=\frac{\partial...
  36. MathDestructor

    Mechanics Problem using Polar Coordinates

    This is what I have so far, please need urgent help. I don't understand and know what to do. For the first part, I got a really long answer, for the second part I am trying in terms of mv^2/r = mg, or mg = m*(answer to first), but I am getting nowhere. PLease help
  37. K

    MHB Finding 2D Polygon Coordinates from a point

    Suppose that I have the coordinates of x and y on a plane. I am writing a piece of software where the user can select a polygon of 3, 4, 5, 6 or 8 sides. All of the polygon points are equidistant from the x, y point. In other words, if you drew a circle where the center was the x, y point, all...
  38. P

    Is the Chain Rule Applied to Spherical Polar Coordinates Different?

    Ive found ##\delta x/\delta r## as ##sin\theta cos\phi## ##\delta r/\delta x## as ##csc\theta sec\phi## But unsure how to do the second part? Chain rule seems to give r/x not x/r?
  39. M

    Transformation from de Sitter to flat spacetime coordinates

    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...
  40. S

    Guide to Aircraft Coordinates: Roll, Pitch & Heading

    Hi, I’ve read a set of coordinate elements: „roll, pitch and heading“ for aircrafts. What’s the difference to „roll, pitch and yaw“? Senmeis
  41. A

    Integration in polar coordinates

    In spherical poler coordinates the volume integral over a sphere of radius R of $$\int^R_0\vec \nabla•\frac{\hat r}{r^2}dv=\int_{surface}\frac{\hat r}{r^2}•\vec ds$$ $$=4\pi=4\pi\int_{-\inf}^{inf}\delta(r)dr$$ How can it be extended to get $$\vec \nabla•\frac{\hat r}{r^2}=4\pi\delta^3(r)??$$
  42. T

    MHB What are the coordinates of point D?

    B is the midpoint of AC( LIne segment) and E is the midpoint of BD( LIne segment). If A(-9, -4), C(-1,6) and E(-4,-3), find the cooridinate of D. I got really lost on that
  43. A

    Double integral with polar coordinates

    Hello there, I'm struggling in this problem because i think i can't find the right ##\theta## or ##r## Here's my work: ##\pi/4\leq\theta\leq\pi/2## and ##0\leq r\leq 2\sin\theta## So the integral would be: ##\int_{\pi/4}^{\pi/2}\int_{0}^{2\sin\theta}\sin\theta dr d\theta## Which is equal to...
  44. S

    I Component definition in curvilinear coordinates

    I'm watching this lecture that gives an introduction to tensors. If we have a coordinate system that's an affine transformation of the Cartesian coordinate system, then the projection of a vector ##v## (onto a particular axis) is defined as ##v_m = v.e_m## or the dot product of the vector with...
  45. C

    Element of surface area in spherical coordinates

    r,θ,ϕ For integration over the ##x y plane## the area element in polar coordinates is obviously ##r d \phi dr ## I can also easily see ,geometrically, how an area element on a sphere is ##r^2 sin\theta d\phi ## And I can verify these two cases with the Jacobian matrix. So that's where I'm at...
  46. L

    I The free particle in spherical coordinates

    Hi! I'm studying Shankar's Principle of quantum mechanics I didn't get the last conclusion, can someone help me understand it, please. Where did the l over rho come from?
  47. Athenian

    I Lorentz Boosts: Finding Speed, Coordinates & Rotation w/ Matrix Multiply

    Recently, I've been studying about Lorentz boosts and found out that two perpendicular Lorentz boosts equal to a rotation after a boost. Below is an example matrix multiplication of this happening: $$ \left( \begin{array}{cccc} \frac{2}{\sqrt{3}} & 0 & -\frac{1}{\sqrt{3}} & 0 \\ 0 & 1 & 0 & 0...
  48. Z

    I Area Differential in Cartesian and Polar Coordinates

    The area differential ##dA## in Cartesian coordinates is ##dxdy##. The area differential ##dA## in polar coordinates is ##r dr d\theta##. How do we get from one to the other and prove that ##dxdy## is indeed equal to ##r dr d\theta##? ##dxdy=r dr d\theta## The trigonometric functions are used...
  49. F

    I Deriving AdS Poincare Coordinates from Global Coordinates

    Is there a straight-forward, motivated, derivation of AdS Poincare coordinates, e.g. as given here: https://en.wikipedia.org/wiki/Anti-de_Sitter_space#Poincar.C3.A9_coordinates starting from global coordinates, as given here...
  50. G

    MHB Rewriting vectors in different coordinates

    Lets say you have a vector in spherical coordinates; how do you rewrite this vector into a cartesian one and vice versa? Im fine with rewriting coordinates but vectors have got me confused. I've tried digging through info online but I couldn't find any good examples. In the following task...
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