Curvilinear coordinates
Curvilinear coordinates are a class of coordinate systems in Euclidean space where the coordinate curves are not necessarily straight lines, unlike Cartesian coordinates, and instead consist of intersecting families of curved surfaces that define the position of points through the values of constants on those surfaces.[1] These systems are particularly useful for exploiting symmetries in physical or mathematical problems, such as rotational invariance, by aligning the coordinates with the geometry of the domain.[2] Common examples include cylindrical coordinates, which extend polar coordinates into three dimensions using radial distance r, azimuthal angle \phi, and height z, and spherical coordinates, which use radial distance r, polar angle \theta, and azimuthal angle \phi.[1] A key subclass is orthogonal curvilinear coordinates, where the coordinate directions are mutually perpendicular at every point, forming an orthonormal basis that simplifies vector operations and differential equations.[2] In these systems, scale factors h_i account for the variation in metric along each coordinate direction, defined as the ratio of infinitesimal arc length to coordinate differential (h_i = ds/du_i), and are essential for expressing gradients, divergences, and curls in a form analogous to Cartesian coordinates.[2] Rectangular coordinates serve as a special case where scale factors are unity.[2] Curvilinear coordinates find widespread application in fields like electromagnetism, fluid dynamics, and quantum mechanics, where they facilitate the solution of partial differential equations by matching boundary conditions to coordinate surfaces, such as spheres or cylinders.[1] For instance, in cylindrical coordinates, the position is given by x = r \cos \phi, y = r \sin \phi, z = z, while in spherical coordinates, x = r \sin \theta \cos \phi, y = r \sin \theta \sin \phi, z = r \cos \theta, with conventions varying slightly between physics and mathematics contexts.[1] These transformations enable more intuitive formulations of problems with inherent symmetries, reducing computational complexity.[2]Introduction and Fundamentals
Definition and Motivation
Curvilinear coordinates provide a generalization of the standard Cartesian coordinate system, where the loci of constant coordinate values form families of curved surfaces rather than flat planes. In three dimensions, these coordinates are typically denoted by variables u, v, w, which parameterize points in space through a smooth position vector \mathbf{r}(u, v, w) that maps to the corresponding Cartesian coordinates x, y, z. This mapping assumes an injective and differentiable transformation from the coordinate domain to Euclidean space, allowing for flexible descriptions of geometric configurations that deviate from rectilinear grids. Readers are presumed to be acquainted with Cartesian coordinates and basic vector notation, as curvilinear systems build directly upon these foundations.[3][4] The primary motivation for employing curvilinear coordinates arises in physical and mathematical problems exhibiting inherent symmetries, such as cylindrical or spherical geometries, where Cartesian coordinates lead to cumbersome expressions and inefficient computations. By aligning the coordinate surfaces with the problem's symmetry, curvilinear systems simplify the formulation of governing equations, particularly partial differential equations (PDEs) like those in electrostatics, fluid dynamics, and heat conduction. For instance, in domains with rotational invariance, these coordinates facilitate the separation of variables technique, transforming multidimensional PDEs into a set of ordinary differential equations (ODEs) that are solvable along individual coordinate directions, thereby reducing computational complexity and enhancing analytical tractability. Common examples include polar coordinates in two dimensions and spherical coordinates in three dimensions, which exploit such symmetries effectively.[5][6] Historically, the development of curvilinear coordinates emerged in the 19th century amid advances in potential theory, where solving Laplace's equation \nabla^2 \phi = 0 for gravitational and electrostatic potentials required tools beyond Cartesian frameworks. Pierre-Simon Laplace introduced the equation in his 1822 treatise on celestial mechanics to model fluid equilibrium, but it was Gabriel Lamé who, in the 1830s, coined the term "curvilinear coordinates" and applied them to transform Laplace's equation into separable forms, such as ellipsoidal coordinates, for heat transfer and elasticity problems. Building on earlier work by Carl Friedrich Gauss on surface geometry, Lamé's contributions provided a powerful method to handle curved geometries in potential theory, influencing subsequent developments in vector calculus and differential geometry. These 19th-century innovations remain foundational for modern applications in physics and engineering.[7][8]Basic Coordinate Transformations
Curvilinear coordinates provide a framework for describing points in space that aligns with the symmetries inherent in many physical problems, such as those involving cylindrical or spherical geometries.[9] The fundamental setup for transforming from Cartesian coordinates (x, y, z) to general curvilinear coordinates (u, v, w) involves expressing the position vector \mathbf{r} as a function of the new coordinates. Specifically, the position vector is given by \mathbf{r}(u, v, w) = x(u, v, w) \mathbf{i} + y(u, v, w) \mathbf{j} + z(u, v, w) \mathbf{k}, where \mathbf{i}, \mathbf{j}, and \mathbf{k} are the standard Cartesian unit vectors.[10] This mapping defines the curvilinear system, with coordinate surfaces corresponding to level sets such as u = \text{constant}, v = \text{constant}, and w = \text{constant}, which intersect to form a curvilinear grid in space.[10] At any point in space, the tangent vectors to the coordinate curves provide a natural basis for the tangent space. These are the partial derivatives of the position vector: \mathbf{e}_u = \frac{\partial \mathbf{r}}{\partial u}, \quad \mathbf{e}_v = \frac{\partial \mathbf{r}}{\partial v}, \quad \mathbf{e}_w = \frac{\partial \mathbf{r}}{\partial w}. These vectors point in the directions of increasing u, v, and w, respectively, while holding the other coordinates fixed, and they span the tangent space at that point.[10] For the transformation to be well-defined and useful, the mapping from (u, v, w) to (x, y, z) must be a diffeomorphism, meaning it is smooth, bijective, and has a smooth inverse, ensuring a one-to-one correspondence between points in the coordinate domains locally.[11] This invertibility requires that the Jacobian matrix of the transformation has a non-zero determinant at each point, guaranteeing the local existence of the inverse function.[11] A simple illustration of this setup occurs in one dimension, where a curve is parameterized by arc length s along its path. The position vector \mathbf{r}(s) satisfies |\frac{d\mathbf{r}}{ds}| = 1, so the tangent vector \frac{d\mathbf{r}}{ds} is a unit vector pointing along the curve, demonstrating how the coordinate s directly measures distance without scaling factors in this basic case.[10]Orthogonal Curvilinear Coordinates
Systems in Two Dimensions
In two dimensions, orthogonal curvilinear coordinate systems provide a framework for describing points in the plane using curves that intersect at right angles, facilitating the solution of problems with specific geometric symmetries. One of the most fundamental systems is the polar coordinate system, which employs radial distance r \geq 0 and azimuthal angle \theta \in [0, 2\pi). The transformation to Cartesian coordinates is given byx = r \cos \theta, \quad y = r \sin \theta,
with the inverse relations
r = \sqrt{x^2 + y^2}, \quad \theta = \atan2(y, x). The scale factors for this system are h_r = 1 and h_\theta = r. Geometrically, curves of constant r form concentric circles centered at the origin, while curves of constant \theta are straight rays emanating from the origin. This system is particularly useful for problems exhibiting rotational symmetry, such as wave propagation in circular domains or fluid flow around cylindrical objects. Another important two-dimensional orthogonal system is the parabolic coordinate system, using parameters \sigma \geq 0 and \tau \geq 0. The transformation equations are
x = \sigma \tau, \quad y = \frac{1}{2} (\sigma^2 - \tau^2),
with inverse forms
\sigma = \sqrt{\sqrt{x^2 + y^2} + y}, \quad \tau = \sqrt{\sqrt{x^2 + y^2} - y}. [12] The scale factors are identical for both coordinates: h_\sigma = h_\tau = \sqrt{\sigma^2 + \tau^2}.[12] The coordinate curves consist of confocal parabolas: constant \sigma traces parabolas opening to the right with focus at the origin, while constant \tau traces parabolas opening to the left. These coordinates are applied in quantum mechanics to solve the Schrödinger equation for the hydrogen atom, where they separate variables in the presence of electric fields, as in the Stark effect.[13] The elliptic coordinate system, denoted by \mu \geq 0 and \nu \in [0, 2\pi), is defined relative to two foci separated by distance $2a > 0. The transformation is
x = a \cosh \mu \cos \nu, \quad y = a \sinh \mu \sin \nu.
The scale factors are h_\mu = h_\nu = a \sqrt{\sinh^2 \mu + \sin^2 \nu}.[14] Constant \mu curves are confocal ellipses with foci at (\pm a, 0), and constant \nu curves are confocal hyperbolas sharing the same foci. This system aids in analyzing boundary value problems with elliptical geometries, such as electrostatic potentials around elliptic cylinders or Stokes flow in elliptic domains.[15] The following table compares these three common two-dimensional orthogonal curvilinear systems, highlighting their transformation equations and scale factors:
| System | Coordinates | Transformation to Cartesian | Scale Factors |
|---|---|---|---|
| Polar | r \geq 0, \theta \in [0, 2\pi) | x = r \cos \theta, y = r \sin \theta | h_r = 1, h_\theta = r |
| Parabolic | \sigma \geq 0, \tau \geq 0 | x = \sigma \tau, y = \frac{1}{2} (\sigma^2 - \tau^2) | h_\sigma = h_\tau = \sqrt{\sigma^2 + \tau^2} |
| Elliptic | \mu \geq 0, \nu \in [0, 2\pi) | x = a \cosh \mu \cos \nu, y = a \sinh \mu \sin \nu | h_\mu = h_\nu = a \sqrt{\sinh^2 \mu + \sin^2 \nu} |