What is Curvilinear: Definition and 87 Discussions

In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible (a one-to-one map) at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The name curvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved.
Well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space (R3) are cylindrical and spherical coordinates. A Cartesian coordinate surface in this space is a coordinate plane; for example z = 0 defines the x-y plane. In the same space, the coordinate surface r = 1 in spherical coordinates is the surface of a unit sphere, which is curved. The formalism of curvilinear coordinates provides a unified and general description of the standard coordinate systems.
Curvilinear coordinates are often used to define the location or distribution of physical quantities which may be, for example, scalars, vectors, or tensors. Mathematical expressions involving these quantities in vector calculus and tensor analysis (such as the gradient, divergence, curl, and Laplacian) can be transformed from one coordinate system to another, according to transformation rules for scalars, vectors, and tensors. Such expressions then become valid for any curvilinear coordinate system.
A curvilinear coordinate system may be simpler to use than the Cartesian coordinate system for some applications. The motion of particles under the influence of central forces is usually easier to solve in spherical coordinates than in Cartesian coordinates; this is true of many physical problems with spherical symmetry defined in R3. Equations with boundary conditions that follow coordinate surfaces for a particular curvilinear coordinate system may be easier to solve in that system. While one might describe the motion of a particle in a rectangular box using Cartesian coordinates, the motion in a sphere is easier with spherical coordinates. Spherical coordinates are the most common curvilinear coordinate systems and are used in Earth sciences, cartography, quantum mechanics, relativity, and engineering.

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  1. chwala

    I Cylindrical coordinates -Curvilinear

    Why are the coordinates seemingly used when the symmetry is around ##z## axis? Any particular reason why not ##x## or ##y##. In transforming from Cartesian to cylindrical form; I can see that ##z## is not considered when determining ##r##. Can we also use ##x## and ##z## assuming that the...
  2. 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...
  3. K

    Curvilinear coordinate system: Determine the standardized base vectors

    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!
  4. Like Tony Stark

    Difference between curvilinear and rotational motion

    The solution states that there's no rotational motion when ##C## is cut (the motion is curvilinear), so we can take torques with respect to the centre of mass of the plate. But, isn't it rotating? I think of it as a pendulum, which describes a circular motion. What's the difference? Wouldn't the...
  5. CPW

    Curvilinear.... isn't that a strange word?

    Hi PF. I'm over ten years out from graduate school in physics, and I still enjoy re-reading my textbooks. I came across the word curvilinear, and thought how strange a word it is; a juxtaposition of curved and linear, which are at times understood to be opposites. The textbook was describing...
  6. Z

    A Curvilinear Coordinate System

    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...
  7. 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...
  8. Sian R

    Curvilinear Dynamics and forces

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  9. S

    What is the Velocity of a Rotating Weight?

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  10. C

    I Force fields in curvilinear coordinate systems

    I am trying to solve problems where I calculate work do to force along paths in cylindrical and spherical coordinates. I can do almost by rote the problems in Cartesian: given a force ##\vec{F} = f(x,y,z)\hat{x} + g(x,y,z)\hat{y}+ h(x,y,z)\hat{z}## I can parametricize my some curve ##\gamma...
  11. F

    I Examples of Non-Orthogonal Curvilinear Coordinates

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  12. Robin04

    Differential operators in 2D curvilinear coordinates

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  13. F

    I Demo of cosine direction with curvilinear coordinates

    1) Firstly, in the context of a dot product with Einstein notation : $$\text{d}(\vec{V}\cdot\vec{n} )=\text{d}(v_{i}\dfrac{\text{d}y^{i}}{\text{d}s})$$ with ##\vec{n}## representing the cosine directions vectors, ##v_{i}## the covariant components of ##\vec{V}## vector, ##y^{i}## the...
  14. E

    Curvilinear coordinate adaptive method in COMSOL

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  15. Z

    Finding the acceleration in a curvilinear motion (n-t)

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  16. S

    Cartesian to curvilinear coordinate transformations

    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...
  17. davidge

    I Gravitation vs Curvilinear Coordinates: Analysis of Weinberg's Book

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  18. EastWindBreaks

    Normal force in a curvilnear motion

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  19. T

    I Question About Curvilinear Cooridnates

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  20. D

    Momentum density in curvilinear coordinates

    Hi, In an article on theoretical fluid dynamics I recently came across the following equation: $$M_i = \sqrt{g} \rho v_i$$ where ##M_i## denotes momentum density, ##v_i## velocity, ##\rho## the mass density and g is the determinant of the metric tensor. It is probably quite obvious, but I do...
  21. N

    MHB Solving Curvilinear Motion: Find Velocity at t=3s

    The acceleration function of an object doing curvilinear motion is a = {(-0.2t)i+2j+1.5k} m/s^2, where t is in s. If its initial velocity v0 = 8i m/s, and initial position is at the origin, determine the magnitude of its velocity when t=3 s. Badly need your help guys. Thanks! All I see is ax...
  22. Auburn2017

    Curvilinear Motion with nonconstant acceleration

    Homework Statement Please refer to both figures. One has a picture and the other is the actual problem. Ignore the pencil writing on the figure as it was for a separate problem. Homework Equations aB+aA/B=aA vB+vA/B=vA an=v2rThe Attempt at a Solution I am actually really at a lost at the...
  23. mertcan

    I Curved space and curvilinear coordinates

    hi, I really wonder what the difference between curvilinear coordinates in a Euclidean space and embedding a curved space into Euclidean space is ? They resemble to each other for me, so Could you explain or spell out the difference? Thanks in advance...
  24. G

    Curvilinear integral along a line segment

    1. Homework Statement Calculate the curvilinear integral ∫C (x2 + y2)ds where C is the line segment [0,0] → [3,4]. Then calculate the maximum M of x2 + y2 along the segment and verify that the inequality ∫C (x2 + y2)ds ≤ M*length(C) holds. Homework Equations ds = (x'(t)2 + y'(t)2)½dt ∇f(x,y)...
  25. A

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    Homework Statement Homework EquationsThe Attempt at a Solution I have stared at this for hours and don't know where to start. I think I need to get r in terms of t but I don't really know how with the information given. I just need a good hint to get started.
  26. I

    Gradient and curvilinear coordinates

    Homework Statement Show that ##\nabla u_i \cdot \frac{\partial \vec r}{\partial u_i} = \delta_{ij}##. (##u_i## is assumed to be a generalized coordinate.) Homework Equations Gradient in curvilinear coordinates ##\nabla \phi = \sum_{i=1}^3 \vec e_i \frac{1}{h_i} \frac{\partial \phi}{\partial...
  27. Jonski

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    Homework Statement At the instant shown, the driver of the truck has just pressed the accelerator pedal down and the truck has suddenly acquired a tangential acceleration of 2.2m/s^2. Coefficient of static friction between crate and tray = 0.4 Coefficient of kinetic friction between crate and...
  28. K

    Curvilinear coordinate, derivative

    https://www.particleincell.com/2012/curvilinear-coordinates/ http://www.jfoadi.me.uk/documents/lecture_mathphys2_05.pdf Hi, I have a question about the curvilineare coordinate system. I wonder why is normal to the isosurfaces?isnt ei a tangent vector to the surface ui since "With these...
  29. manongistong

    Solving Curvilinear Motion Problem - Physics Forums

    Same with problem https://www.physicsforums.com/threads/trigonometric-problem.76696/ . My problems here is I cannot came up with the same answers in the book. I didn't solve using vectors. Homework Equations I would like to know the equation on the 4th hint although I have different solution...
  30. manongistong

    Solve Trigonometric Problem for Dynamics: 4th Hint Equation

    This problem is same as the problem on this link https://www.physicsforums.com/threads/trigonometric-problem.76696/ . I would like to ask the number 4 hint which is "4) We therefore have, for example the equality: " the equations can't be seen on my pc as it will only outputs this...
  31. S

    Orthogonality of a curvilinear coordinate system

    Homework Statement Show that the uvw-system is orthogonal. r, \theta, \varphi are spherical coordinates. $$u=r(1-\cos\theta)$$ $$v=r(1+\cos\theta)$$ $$w=\varphi$$ The Attempt at a Solution So basically I want to show that the scalar products between \frac{\partial \vec{r}}{\partial u}...
  32. 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...
  33. 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...
  34. 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...
  35. F

    Curvilinear motion of force-couple system

    I have a system of forces which I have reduced to a force-couple system. I am trying to predict the motion of the force and couple. This is hard because the force changes direction as the moment turns the particle. I imagine this should be turned into some sort of tangential-normal components...
  36. ShayanJ

    Momentum operator in curvilinear coordinates

    This paper is about momentum operator in curvilinear coordinates. The author says that using \vec p=\frac{\hbar}{i} \vec \nabla is wrong and this form is only limited to Cartesian coordinates. Then he tries to find expressions for momentum operator in curvilinear coordinates. He's starting...
  37. A

    Curvilinear basis in spherical polar coordinates

    Homework Statement As a part of my self study I am trying to find the covariant basis vectors in the spherical polar coordinates. Since I have never done anything like this before I would appreciate if someone could tell me whether I am on the rigth track. Homework Equations...
  38. J

    Position vector in curvilinear coordinates

    The position vector ##\vec{r}## in cartesian coordinates is: ##\vec{r} = x \hat{x} + y \hat{y}##, in polar coordinates is: ##\vec{r} = r \hat{r}##. But, given a curve s in somewhere of plane, with tangent unit vector ##\hat{t}## and normal unit vector ##\hat{n}## along of s, exist a definition...
  39. J

    R, dr and d²r and curvilinear coordinates

    Hellow everybody! If ##d\vec{r}## can be written in terms of curvilinear coordinates as ##d\vec{r} = h_1 dq_1 \hat{q_1} + h_2 dq_2 \hat{q_2} + h_2 dq_2 \hat{q_2}## so, how is the vectors ##d^2\vec{r}## and ##\vec{r}## in terms of curvilinear coordinates? Thanks!
  40. mnb96

    Curvilinear coordinates on tangent spaces

    Hello, if we consider a diffeomorphism f:M-->N between two manifolds, we can easily obtain a basis for the tangent space of N at p from the differential of f. I was wondering, why should we always express tangent vectors as linear combinations of tangent basis vectors? Could it be useful in...
  41. heycoa

    Mechanics, Tangential force and potential of a curvilinear path

    Homework Statement a) Prove that m (d^2s/dt^2) = Ftang, the tangential component of the net force on the bead. [hint] one way to do this is to take the time derivative of the equation v^2=v(dot)v. The left side should lead you to (d^2s/dt^2), and the right side should lead to Ftang. b)...
  42. mnb96

    Inner product in curvilinear coordinates

    Hello, let's assume we have an admissible change of coordinates \phi:U\rightarrow \mathbb{R}^n. I would like to know how the inner product on ℝn changes under this transformation. In other words, what is \left\langle \phi (u), \phi (v) \right\rangle for some u,v \in U ? I thought that...
  43. ShayanJ

    Vectors in curvilinear coordinate systems

    To specify a vector in cartesian coordinate systems,we assume its tail to be at the origin and give the cartesian coordinates of its head.What about other coordinate systems? For example,in spherical coordinates,is the following correct? a \hat{x}+b \hat{y}+c \hat{z}=\sqrt{a^2+b^2+c^2}...
  44. A

    Curvilinear Motion: Rectangular Components

    Homework Statement What is the magnitude of the velocity at t=4.00s? I would like to see if my approach and answer is correct. Homework Equations Position: r = {-30cos(\frac{\pi}{10}t) i + 30sin(\frac{\pi}{10}t) j - (7t) k} ft The Attempt at a Solution I took the first...
  45. K

    Curvilinear n&t motion question engineering mechanics

    Homework Statement A pin is constrained to move in a circular slot of radius 39mm. At the same time a slotted bar also constrains the pin to move down with constant velocity 8mm/s. (as shown in attached diagram). What is the magnitude of the acceleration of the pin for the position...
  46. mnb96

    Lie groups actions and curvilinear coordinates question

    Hello, let's suppose I have the following system of curvilinear coordinates in ℝ2: x(u,v) = u y(u,v) = v + e^u where one arbitrary coordinate line C_\lambda(u)=u \mathbf{e_1} + (\lambda+e^u) \mathbf{e_2} represents the orbit of some point in ℝ2 under the action of a Lie group. Now I consider...
  47. N

    Electromagnetic Field Tensor in Curvilinear Coordinates

    How to express electromagnetic field tensor in curvilinear coordinates, that is given a curvilinear coordinates (t,\alpha,\beta,\gamma) with metric tensor as follows: n_{\mu \nu }= \left[ \begin{array}{cccc}h_0^2& 0 & 0 & 0 \\ 0 & -h_1^2 & 0 & 0 \\ 0 & 0 & -h_2^2 & 0 \\ 0 & 0 & 0 & -h_3^2...
  48. C

    Angular Velocity in Curvilinear Translation?

    HI there. Some days ago, whyle studying vector mechanics I came across with a rather dazzling doubt. Why isn't there angular velocity and accelaration in a curvilinear translation? Imagine, a small planet in a perfect circular orbit around a star. Let's say, the planet has no form of...
  49. G

    Question about relating basis of curvilinear coordinate systems.

    Wikipedia gives the relationship between a cartesian and curvlinear coordinate system as gi=(partial)x1/(partial)zi +(partial)x2/(partial)zi http://en.wikipedia.org/wiki/Curvilinear_coordinates Where gi is the i'th basis in the curvlinear coordinate system, x1 and x2 are the cartesian...
  50. I

    Finding the x range using curvilinear motion

    Homework Statement The car passes point A with a speed of 25 m/s after which its speed is defined by V = (25-0.15s)m/s. Determine the magnitude of the car's acceleration when it reaches point B, where S = 51.5 m. (the max height of the hill is 16 m, and the function of the hill the car is...
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