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. P

    Why is ##x = r \sin{\phi} \cos{\theta}## in spherical coordinates?

    My question is really about converting between spherical coordinates and cartesian coordinates. Suppose that ##\phi## and ##\theta## are defined as follows: ##\phi## is the angle between the position vector of a point and the ##z##-axis. ##\theta## is the angle between the projection of that...
  2. M

    Integrating Gaussian in polar coordinates problem

    I have a 2D Gaussian: ## f(x,y) = e^{-[(x-x_o)^2 + (y-y_o)^2]/(2*{sigma}^2)}## which I converted into polar coordinates and got: ## g(r,θ) = e^{-[r^2 + r_o^2 - 2*r*r_o(cos(θ)cos(θ_o) + sin(θ)sin(θ_o))]/({2*{sigma}^2})} ## The proof for how this was done is in the attached file, and it would...
  3. D

    General relativity and curvilinear coordinates

    I have just been asked why we use curvilinear coordinate systems in general relativity. I replied that, from a heuristic point of view, space and time are relative, such that the way in which you measure them is dependent on the reference frame that you observe them in. This implies that...
  4. B

    Understanding the Gradient in Different Coordinate Systems

    Sorry again for all these ongoing question as I try to fix my math deficiencies. (Back to working on differential forms.) So... I understand that the equation of steepest ascent/descent in Cartesian coordinates is written: dxi/dt = ∂f/∂xi And I can follow the "physical interpretations" of...
  5. Soumalya

    Accelerating and Non-accelerating Coordinates - Fluid flow

    Referring to the problem in the attachment, the author mentions that if we consider the coordinate system attached to the bicycle and the bicycle accelerates or decelerates, the flow past the bicycle becomes unsteady. For an unsteady flow, we know that nothing changes at a given location on a...
  6. J

    Spacetime Curvature: Which Tensor Gives Coordinates?

    In the Einstein Field Equations: Rμν - 1/2gμνR + Λgμν = 8πG/c^4 × Tμν, which tensor will describe the coordinates for the curvature of spacetime? The equations above describe the curvature of spacetime as it relates to mass and energy, but if I were to want to graph the curvature of spacetime...
  7. ognik

    MHB Discretising Elliptic PDE in cylindrical coordinates

    Given an energy functional $ E=\int_{0}^{\infty} \,dr.r\left[\frac{1}{2}\left(\d{\phi}{r}\right)^2 - S.\phi\right] $ I am told that discretizing on a lattice ri=ih (h=lattice size, i is i axis) leads to : $ 2{r}_{i}{\phi}_{i} - {r}_{i+\frac{1}{2}}{\phi}_{i+1} - {r}_{i-\frac{1}{2}}{\phi}_{i-1}...
  8. flaticus

    What are the non-zero Christoffel symbols for 2D polar coordinates?

    Just started self teaching myself differential geometry and tried to find the christoffel symbols of the second kind for 2d polar coordinates. I am checking to see if I did everything correctly. With a line element of: therefore the metric should be: The christoffel symbols of the second kind...
  9. ash64449

    What are these Gaussian coordinates?

    [Mentor's note: This post was moved from another thread as it raised a new question, off-topic in the originating thread] Albert Einstein in his book Relativity wrote " It is impossible to build up a system(reference body) from rigid bodies and clocks,which shall be of such a nature that...
  10. PWiz

    Infinitesimal displacement in spherical coordinates

    I'm trying to derive what ##ds^2## equals to in spherical coordinates. In Euclidean space, $$ds^2= dx^2+dy^2+dz^2$$ Where ##x=r \ cos\theta \ sin\phi## , ##y=r \ sin\theta \ sin\phi## , ##z=r \ cos\phi## (I'm using ##\phi## for the polar angle) For simplicity, let ##cos...
  11. W

    Differentiation(Finding coordinates of a point on the curve)

    Homework Statement Homework Equations (y-y1)/(x-x1)=mThe Attempt at a Solution I have attempted part i but I don't know how to do part ii. As point B is still part of the curve and the normal, do I still sub with the same normal eqn? :/ I have no idea how to start... Please help thanks[/B]
  12. S

    Double integral on triangle using polar coordinates

    Homework Statement Let R be the triangle defined by -xtanα≤y≤xtanα and x≤1 where α is an acute angle sketch the triangle and calculate ∫∫R (x2+y2)dA using polar coordinates hint: the substitution u=tanθ may help you evaluate the integral Homework EquationsThe Attempt at a Solution so the...
  13. M

    Converting a 2D Gaussian in Cylindrical Coordinates

    Given the equation for a Gaussian as: ##z = f(x,y) = Ae^{[(x-x0)^2 + (y-y0)^2] /2pi*σ^2 }## , how would I go about converting this into cylindrical coordinates? The mean is non-zero, and this seems to be the biggest hurdle. I believe I read earlier that the answer is ~ ##z = f(r,θ) =...
  14. E

    Vector identity proof in general curvilinear coordinates

    Homework Statement Need to prove that: ,b means partial differentation with respect to b, G is the metric tensor and Γ is Christoffel symbol. I think I could proceed with this quite well if I could understand the hint given, that I should lower the index j. Homework Equations am=Gmjaj...
  15. H

    Triple integral in spherical coordinates

    Homework Statement Evaluate \int \int \int _R (x^2+y^2+z^2)dV where R is the cylinder 0\leq x^2+y^2\leq a^2, 0\leq z\leq h Homework Equations [/B] x = Rsin\phi cos\theta y = Rsin\phi sin\theta z = Rcos\phiThe Attempt at a Solution [/B] 2*\int_{0}^{\pi/2}d\phi \int_{0}^{2\pi}d\theta...
  16. gfd43tg

    Particle in a box in cartesian coordinates

    Homework Statement Homework EquationsThe Attempt at a Solution a) The schrödinger equation $$i \hbar \frac {\partial \Psi}{\partial t} = - \frac {\hbar^{2}}{2m} \nabla^{2} \psi + V \psi $$ For the case ##0 \le x,y,z \le a##, ##V = 0## $$i \hbar \frac {\partial \Psi}{\partial t} = - \frac...
  17. M

    MHB Integrating (triple) over spherical coordinates

    Hi, Set up the triple integral in spherical coordinates to find the volume bounded by z = \sqrt{4-x^2-y^2}, z=\sqrt{1-x^2-y^2}, where x \ge 0 and y \ge 0. \int_0^{2\pi} \int_0^2 \int_{-\sqrt{4-x^2-y^2}}^{\sqrt{4-x^2-y^2}} r\ dz\ dr\ d\theta
  18. YogiBear

    Show that w is solenoidal having spherical polar coordinates

    Homework Statement The gradient, divergence and curl in spherical polar coordinates r, ∅, Ψ are nablaΨ = ∂Φ/∂r * er + ∂Φ/∂∅ * e∅ 1/r + ∂Φ/∂Ψ * eΨ * 1/(r*sin(∅)) nabla * a = 1/r * ∂/∂r(r2*ar) + 1/(r*sin(∅)*∂/∂∅[sin(∅)a∅] + 1/r*sin(∅) * ∂aΨ/∂Ψ nabla x a = |er r*e∅ r*sin(∅)*eΨ | |∂/∂r ∂/∂∅...
  19. K

    Curl of a field in spherical polar coordinates

    Homework Statement I have a field w=wφ(r,θ)eφ^ (e^ is supposed to be 'e hat', a unit vector) Find wφ(r,θ) given the curl is zero and find a potential for w. Homework Equations I can't type the matrix for curl in curvilinear, don't even know where to start! I've been given it in the form...
  20. H

    Solving Laplace's equation in spherical coordinates

    The angular equation: ##\frac{d}{d\theta}(\sin\theta\,\frac{d\Theta}{d\theta})=-l(l+1)\sin\theta\,\Theta## Right now, ##l## can be any number. The solutions are Legendre polynomials in the variable ##\cos\theta##: ##\Theta(\theta)=P_l(\cos\theta)##, where ##l## is a non-negative integer...
  21. R

    New coordinates from the rotation of an axis

    Homework Statement There is a point P(x,y) and now I rotate the x-y axis, say by θ degree. What will be the coordinates of P from this new axis. I have google but found formula for new coordinates when the points is rotated by θ degree. So I tried my own. So is there other simplified formula...
  22. H

    The x,y,z coordinates of CM of a solid cylinder

    I have found via integration that the y coordinate is $$y =h/2 = 120 mm$$. The x coordinate is $$x = \frac{-4r}{3\pi} = -51.9mm$$ and the z coordinate is $$z = r - \frac{4r}{3\pi} = 69.1 mm$$. I have no answers in my textbook so can't confirm whether i am correct or not.
  23. U

    Finding limits of integral in spherical coordinates

    Homework Statement The question asks me to convert the following integral to spherical coordinates and to solve it Homework EquationsThe Attempt at a Solution just the notations θ = theta and ∅= phi dx dy dz = r2 sinθ dr dθ d∅ r2 sinθ being the jacobian and eventually solving gets me ∫ ∫ ∫...
  24. binbagsss

    Algebra Question: Where Does the 2 Come From?

    Probably a really stupid question.. ##u=t+r+2M ln(\frac{r}{2M}-1) ## From this I get ##\frac{du}{dr}=(1-\frac{2M}{r})^{-1}## But, 1997 Sean M. Carroll lectures notes get ##\frac{du}{dr}=2(1-\frac{2M}{r})^{-1}## . (equation 7.71). No idea where this factor of 2 comes from. Thanks
  25. jalessandrom

    Instantaneous acceleration from coordinates?

    Homework Statement The coordinate of an automobile in meters is x(t) = 5 + 3t + 2t2 and y(t) = 7 + 2t + t3, where t is in seconds. What is the instant acceleration of the car at time t = 2 s? ANSWERS: A. 10.2 m/s2 B. 9.5 m/s2 C. 7.9 m/s2 D. 15.0 m/s2 E. 12.6 m/s2 Homework Equations ains =...
  26. H

    Derive grad T in spherical coordinates

    Homework Statement ##x=r\sin\theta\cos\phi,\,\,\,\,\,y=r\sin\theta\sin\phi,\,\,\,\,\,z=r\cos\theta## ##\hat{x}=\sin\theta\cos\phi\,\hat{r}+\cos\theta\cos\phi\,\hat{\theta}-\sin\phi\,\hat{\phi}## ##\hat{y}=\sin\theta\sin\phi\,\hat{r}+\cos\theta\sin\phi\,\hat{\theta}+\cos\phi\,\hat{\phi}##...
  27. W

    Integrating in Polar Coordinates: Ω Region

    Homework Statement ∫∫dydx Where the region Ω: 1/2≤x≤1 0 ≤ y ≤ sqrt(1-x^2) Homework EquationsThe Attempt at a Solution The question asked to solve the integral using polar coordinates. The problem I have is getting r in terms of θ. I solved the integral in rectangular ordinates using a trig...
  28. F

    Fourier transform in curvilinear coordinates

    Hello, can you suggest a good book reference to find this: I have a 3D coordinate system where the axis are: 1) locally tangential to a spiral in the equatorial plane; 2) perpendicular to 1 in the equatorial plane; 3) colatitude. The direction of axes 1 and 2 changes with position. I need to...
  29. RJLiberator

    Spherical Coordinates - Help me find my bounds

    Homework Statement A vase is filled to the top with water of uniform density f = 1. The side profile of the barrel is given by the surface of revolution obtained by revolving the graph of g(z) = 2 + cos(z) over the z-axis, and bounded by 0 ≤ z ≤ π. Find the mass of the vase. Homework...
  30. K

    Poisson PDE in polar coordinates with FDM

    I want to solve a Laplace PDE in a polar coordinate system with finite difference method. and the boundary conditions: Here that I found in the internet: and the analytical result is: The question is how its works? Can I give an example or itd?Thanks
  31. Calpalned

    Spherical coordinates - phi vs theta

    Homework Statement My textbook states that when ##\theta = c ## where c is the constant angle with respect to the x-axis, the graph is a "half-plane". However, when ##\phi = c ## it is a half-cone. The only difference I see is that ##\phi## is the angle with respect to the z-axis, rather than...
  32. Calpalned

    Triple integral in cylindrical coordinates

    Homework Statement Evaluate ## \int \int \int_E {x}dV ## where E is enclosed by the planes ##z=0## and ##z=x+y+5## and by the cylinders ##x^2+y^2=4## and ##x^2+y^2=9##. Homework Equations ## \int \int \int_E {f(cos(\theta),sin(\theta),z)}dzdrd \theta ## How do I type limits in for...
  33. A

    Acceleration in Rindler coordinates

    [Mentors note: this thread was split off from an older discussion of Rindler coordinates] Can somebody help me understand why acceleration along the hyperbola is constant? To be more precise: assume (X,T) is the inertial coordinates and (x,t) the corresponding Rindler transformations...
  34. Coffee_

    Can I substitute the new coordinates in the old hamiltonian?

    We went over this concept quite fast in class and there is one thing that confused me: When transforming from a set of ##q_i## and ##p_i##to ##Q_i## and ##P_i##, if one checks that the transormations are canonical the new Hamiltonian ##K(Q_i, P_i)## obeys exactly the same equations.This has...
  35. M

    Equation (with polar coordinates) of circle on a sphere

    hi, i'm a newbie... i have this problem: i have a sphere with known and constant R (obvious), i have two point with spherical coordinates P1=(R,p_1,t_1) and P0=(R, p_0, t_0) p_x = phi x = latitude x t_x = theta x =longitude x the distance between point is D=...
  36. N

    Path of a Projectile in Polar Coordinates

    Homework Statement A projectile is launched from a mountain at a given angle and velocity (which is large). Using polar coordinates find the position of the particle at time t. I'm ignoring drag (for now). Homework Equations I tried using the polar kinematic equations...
  37. L

    Lin. Algebra: Find coordinates on a, b, c, d such that AB=BA

    Homework Statement [/B] Matrix A = 1 1 0 1 Matrix B = a b c d Find coordinates on a, b, c, d such that AB = BA. Homework EquationsThe Attempt at a Solution I calculated AB and BA with simple matrix multiplication, but am not sure where to go from here. AB = a + c a + b...
  38. J

    What are momentum,configuration and coordinate spaces?

    What is a momentum space,a coordinate space and a configuration space? Are they in classical or quantum mechanics or both? What are their similarities and differences and when,where and how are they used? thank you in advance!
  39. P

    Components of vectors (polar coordinates)

    I have always been under the impression that a vector is not "fixed" in space. Given any vector, we could just move it around and it would still have the same components (in a cartesian coordinate system). What confuses me, however, is how we define the components of a vector in polar...
  40. geezer73

    Double Integrals in Polar Coordinates

    I'm in the middle of the Great Courses Multivariable Calculus course. A double integral example involves a quarter circle, in the first quadrant, of radius 2. In Cartesian coordinates, the integrand is y dx dy and the outer integral goes from 0 to 2 and the inner from 0 to sqrt(4-y^2). In...
  41. P

    Vector components in polar coordinates

    The magnitude of the parallel component of the time derivative of a vector ##\vec{A}## is given by: $$|\frac{d\vec{A}_{\parallel}}{dt}| = |\frac{dA}{dt}|$$ Where ##A## is the magnitude of the vector. Can we write the actual derivative in vector form as ##\frac{dA}{dt} \hat{A}##? Notice how I...
  42. M

    MHB Spherical coordinates - Orthonormal system

    Hey! :o Using spherical coordinates and the orthonormal system of vectors $\overrightarrow{e}_{\rho}, \overrightarrow{e}_{\theta}, \overrightarrow{e}_{\phi}$ describe each of the $\overrightarrow{e}_{\rho}$, $\overrightarrow{e}_{\theta}$ and $\overrightarrow{e}_{\phi}$ as a function of...
  43. andrewkirk

    Proper Frame of Observer O: Unique Foliation?

    By 'proper frame' of observer O, I mean any reference frame (coordinate system) in which (Condition A:) The worldline of O is always at the spatial origin for every time coordinate. Clearly such a frame is not unique because spatial rotations do not invalidate (A). What I am interested in is...
  44. P

    Polar coordinates, sign ambiguity

    The position of a point in cartesian coordinates is given by: $$\vec{r} = x \hat{\imath} + y \hat{\jmath}$$ In polar coordinates, it is given by: $$\vec{r} = r \hat{r}$$ Now, ##x = r \cos{θ}## and ##y = r \sin{θ}## assuming ##θ## is measured counterclockwise from the ##x##-axis. Equating the two...
  45. R

    How Do You Convert Cartesian Vectors to Cylindrical Coordinates?

    Homework Statement I am trying to convert the following vector at (1, 1, 0) to cylindrical polar coordinates, and show that in both forms it has the same direction and magnitude: ##4xy\hat{x}+2x^2\hat{y}+3z^2\hat{z}## Homework Equations ##\rho^2=x^2+y^2## ##tan \phi = \frac{y}{x}## ##z=z##...
  46. S

    Expressing electric field in cylindrical coordinates.

    Hi everyone, I am new to the physics forums and I need your help :) I understand that depending on the symmetry of the problem, it may be easier to change the coordinate system you are using. My question is, how would I convert the electric field due to a point charge at the origin, from...
  47. M

    MHB Cylindrical coordinates - Orthonormal system

    Hey! :o Using cylindrical coordinates and the orthonormal system of vectors $\overrightarrow{e}_r, \overrightarrow{e}_{\theta}, \overrightarrow{e}_z$ describe each of the $\overrightarrow{e}_r$, $\overrightarrow{e}_{\theta}$ and $\overrightarrow{e}_z$ as a function of $\overrightarrow{i}...
  48. M

    MHB Converting Cylindrical Coordinates to Orthogonal and Spherical Coordinates?

    Hey! :o We are given the following point in cylindrical coordinates. We have to write in orthogonal and spherical coordinates. The point is $\left (2, \frac{\pi}{2}, -4\right )$. First of all, do orthogonal coordinates mean cartesian coordinates?? (Wondering) The cylindrical coordinates...
  49. RJLiberator

    Partial Derivatives and Polar Coordinates

    Homework Statement Write the chain rule for the following composition using a tree diagram: z =g(x,y) where x=x(r,theta) and y=y(r,theta). Write formulas for the partial derivatives dz/dr and dz/dtheta. Use them to answer: Find first partial derivatives of the function z=e^x+yx^2, in polar...
  50. S

    MHB Barycentric coordinates in a triangle - proof

    I want to prove that the barycentric coordinates of a point $P$ inside the triangle with vertices in $(1,0,0), (0,1,0), (0,0,1)$ are distances from $P$ to the sides of the triangle. Let's denote the triangle by $ABC, \ A = (1,0,0), B=(0,1,0), C= (0,0,1)$. We consider triangles $ABP, \ BCP, \...
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