What is Curvilinear coordinates: Definition and 34 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. 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...
  2. 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...
  3. Sian R

    Curvilinear Dynamics and forces

    Homework Statement A wedge with face inclined at an angle θ to the horizontal is fixed to a rotating turntable. A block of mass m rests on the inclined plane and the coefficient of static friction between the block and the wedge is µ. The block is to remain at position R from the centre of...
  4. 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...
  5. sams

    I A Question about Unit Vectors of Cylindrical Coordinates

    I wrote the equations of the Nabla, the divergence, the curl, and the Laplacian operators in cylindrical coordinates ##(ρ,φ,z)##. I was wondering how to define the direction of the unit vector ##\hat{φ}##. Can we obtain ##\hat{φ}## by evaluating the cross-product of ##\hat{ρ}## and ##\hat{z}##...
  6. F

    I Examples of Non-Orthogonal Curvilinear Coordinates

    I am beginning to study the mathematics of curvilinear coordinates and all textbooks and web sites do not have realistic examples of non-othogonal systems. What are some examples of non-orthoganal curvilinear coordinates so that I can practice on actual systems rather than generalized examples...
  7. Robin04

    Differential operators in 2D curvilinear coordinates

    Homework Statement I’m studying orthogonal curvilinear coordinates and practice calculating differential operators. However, I’ve run across an exercise where the coordinate system is only in 2D and I’m confused about how to proceed with the calculations. Homework Equations A point in the...
  8. 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...
  9. E

    Curvilinear coordinate adaptive method in COMSOL

    Hello everyone, In the COMSOL v5.1 doc i haven't found the mathematical description on how does the automatic geometry analysis or the curvilinear coordinates adaptive method (it is mentioned that they are similar) works. It would be convenient to have an idea about how does the Ecoil vector...
  10. davidge

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

    In Weinberg's book, it is said that a given metric ##g_{\mu \nu}## could be describing a true gravitational field or can be just the metric ##\eta_{\alpha \beta}## of special relativity written in curvilinear coordinates. Then it is said that in the latter case, there will be a set of...
  11. 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...
  12. 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...
  13. 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...
  14. 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...
  15. 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...
  16. 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...
  17. 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...
  18. 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...
  19. 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!
  20. 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...
  21. 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...
  22. 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...
  23. 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...
  24. mnb96

    Curvilinear coordinates from orbits

    Hello, I have the following problem where I have two groups of transformations R_\alpha (rotation) and S_\lambda (scaling) acting on the plane, so that the orbits of any arbitrary point x=(x0,y0) under the actions of S_\lambda and R_\alpha are known (in the former case they are straight lines...
  25. L

    Curvilinear Coordinates and Vector Calculus

    Homework Statement With \vec{L} = -i\vec{r} x \nabla, verify the operator identities \nabla = \hat{r}\frac{\partial }{\partial \vec{r}}-i\frac{\vec{r}\times\vec{L}}{r^{2}} and \vec{r} \bigtriangledown ^2 - \bigtriangledown (1+\vec{r}\frac{\partial }{\partial \vec{r}})=i\bigtriangledown \times...
  26. L

    Question about curvilinear coordinates

    Just a quick little question. I was reading a wikipedia article about curvilinear coordinates, as well as some others, and a question popped into my head. Although we take this for granted (at least I do), now I have to ask this. From what I've seen as an engineer, we always define...
  27. A

    Gradient operator in Natural Curvilinear Coordinates

    Hi All, I have been trying to understand some fluid mechanics in a research paper and have been wrestling with the mathematics for quite some time now without success. I want to derive gradient operator with following coordinate system in R^3 space Let and arbitrary curve C be locus of...
  28. P

    Dirac equation in curvilinear coordinates

    I wonder how Dirac equation transform under change of coordinates (in flat spacetime). Should I simply express partial derivaties of one coordinates in another or it is necessary to transform Dirac matrices as well?
  29. mnb96

    Complex curvilinear coordinates(?)

    Hello, a system of curvilinear coordinates is usually expressed by an admissible transformation represented by a set of real scalar functions x_i=x_i(u_1,\ldots,u_n). Does it make sense to form a system of curvilinear coordinates where the [itex]x_i[/tex] and [itex]u_i[/tex] functions are...
  30. mnb96

    Curvilinear coordinates question

    Hello, given a system of curvilinear coordinates x_i=x_i(u_1,\ldots,u_n); u_i=u_i(x_1,\ldots,x_n) and considering the position vector \mathbf{r}=x_1\mathbf{e}_1+\ldots+x_n\mathbf{e}_n there is the well-known identity that defines the reciprocal frame: \frac{\partial \mathbf{r}}{\partial u_i...
  31. mnb96

    Curvilinear coordinates question

    Hi, if we consider a transformation of coordinates Cartesian\rightarrowPolar, it is straightforward to derive r = (x^2 + y^2)^{1/2} and \theta = atan2(y/x), because we actually know what our new coordinate system should be like. Now let's pretend we have never seen polar coordinates, and we...
  32. W

    Gauge covariant derivative in curvilinear coordinates

    If we work in cartesian coordinates, we say for instance, that D_x \phi = \left( \frac{\partial}{\partial x} + i g \sum_a T_a A^a_x \right) \phi where g is the gauge coupling, and \{T^a\} are the generators of the gauge group, and \{A^a_\mu\} is the gauge vector field. But what happens when...
  33. B

    General curvilinear coordinates

    Homework Statement When I work in general curvilinear coordinates and in particular for the computation of line and surface integrals, do I need to do anything apart from working through the 'usual steps?' Homework Equations If I am correct, computation of line and surface integrals is...
  34. H

    Tensor analysis in curvilinear coordinates

    Im taking a course in contiuum mechanics and had some questions that I am sure are pretty basic but I'm not getting. We just started curvilinear coordinates and I was curious if someone could explain in a little simplier language of what the superscript and subscripts mean. Or if you...