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Spherical coordinate system

The spherical coordinate system is a curvilinear used to specify the position of a point in three-dimensional relative to a fixed , employing a radial ρ (where ρ ≥ 0) from the origin, an azimuthal θ (typically 0 ≤ θ < 2π) measured from the positive x-axis in the xy-plane, and a polar angle φ (0 ≤ φ ≤ π) measured from the positive z-axis. This system extends the two-dimensional polar coordinates into three dimensions, providing a natural framework for describing points on or within spheres centered at the origin. The relationship between spherical coordinates (ρ, θ, φ) and Cartesian coordinates (x, y, z) is given by the transformation equations:
x = ρ sin φ cos θ,
y = ρ sin φ sin θ,
z = ρ cos φ,
with the inverse transformations involving ρ = √(x² + y² + z²), θ = atan2(y, x), and φ = acos(z / ρ). These conversions enable seamless integration with rectangular systems, though notation conventions differ across disciplines: in mathematics, θ is often the azimuthal angle and φ the polar angle, while in physics, the roles are frequently reversed. Points at ρ = 0 or along the z-axis (φ = 0 or π) exhibit non-uniqueness in angular coordinates, requiring careful handling in computations. Spherical coordinates are particularly advantageous for problems exhibiting spherical symmetry, simplifying differential equations and integrals in fields like vector calculus and physics. In mathematics, they facilitate the evaluation of triple integrals over regions bounded by spheres or cones, such as ∭ f(ρ, θ, φ) ρ² sin φ dρ dθ dφ, and describe surfaces like spheres (constant ρ), cones (constant φ), and vertical half-planes (constant θ). In physics, they are essential for analyzing radially symmetric phenomena, including the gravitational potential around point masses, electrostatic fields of charged spheres, the Maxwell-Boltzmann speed distribution in gases, and the Schrödinger equation for the hydrogen atom. This system's utility extends to meteorology for atmospheric modeling and engineering for antenna design, where spherical symmetry aligns with the geometry of the problems.

Fundamentals

Terminology

The spherical coordinate system utilizes three fundamental parameters to locate a point in three-dimensional Euclidean space: the radial distance, the polar angle, and the azimuthal angle. The radial distance, commonly denoted by r or \rho, measures the straight-line distance from the origin to the point. The polar angle, denoted by \theta, quantifies the inclination from the positive z-axis to the radial line connecting the origin to the point, ranging from 0 to \pi radians. The azimuthal angle, denoted by \phi, specifies the orientation in the xy-plane from the positive x-axis to the projection of the radial line, spanning 0 to $2\pi radians. The nomenclature "spherical coordinates," also known as spherical polar coordinates, derives from the geometric property that all points sharing the same radial distance lie on a sphere centered at the origin. In certain disciplines such as physics and astronomy, the polar angle \theta is interchangeably termed colatitude, reflecting its measurement from the pole akin to 90 degrees minus the equatorial latitude in geographic contexts. This system represents a three-dimensional generalization of the two-dimensional , which employs only a radial distance and an azimuthal angle to describe positions within a plane, thereby adding the polar angle to capture out-of-plane variations.

Definition

The spherical coordinate system specifies the position of a point in three-dimensional using three parameters: the radial distance \rho from the origin, the polar angle \theta measured from the positive z-axis, and the azimuthal angle \phi measured from the positive x-axis in the xy-plane. The relationship between spherical coordinates (\rho, \theta, \phi) and Cartesian coordinates (x, y, z) is given by the parametric equations \begin{align} x &= \rho \sin \theta \cos \phi, \\ y &= \rho \sin \theta \sin \phi, \\ z &= \rho \cos \theta. \end{align} These equations describe the position vector from the origin to the point. The parameters satisfy \rho \geq 0, $0 \leq \theta \leq \pi, and $0 \leq \phi < 2\pi, providing a unique representation for every point in space except the origin and points on the z-axis (where \theta = 0 or \theta = \pi, and \phi is arbitrary). Geometrically, \rho is the straight-line distance from the origin to the point, \theta (the polar angle) is the angle between the position vector and the positive z-axis, and \phi (the azimuthal angle) describes the rotation of the position vector around the z-axis.

Conventions

In standard mathematical convention, the azimuthal angle is denoted by θ, ranging from 0 to 2π, while the polar angle is denoted by φ, ranging from 0 to π. In physics and engineering, a common alternative reverses these roles: θ serves as the polar angle (0 to π) measured from the positive z-axis, and φ as the azimuthal angle (0 to 2π) measured from the positive x-axis in the xy-plane. This notation is prevalent in electromagnetism, as seen in treatments of boundary value problems for Laplace's equation in J. D. Jackson's Classical Electrodynamics, where it facilitates expansions in spherical harmonics. Variations also exist in the radial coordinate symbol, with ρ often used in mathematical contexts to avoid confusion with cylindrical coordinates, whereas r is standard in physics. Additionally, in meteorology, the complement π - θ is employed as colatitude to align with latitude-based geographical descriptions. The International Organization for Standardization (ISO) standard ISO 80000-2 recommends r for the radial distance, θ for the polar angle (0 to π), and φ for the azimuthal angle (0 to 2π), consistent with the physics convention to promote uniformity in mathematical notation.

Visualization

Plotting

To visualize points in spherical coordinates, a standard technique involves converting the coordinates to Cartesian form, which enables the use of conventional 3D graphing tools for accurate representation in space. This conversion allows points specified by radial distance and angular positions to be plotted as scatter points or markers in three-dimensional plots. Alternatively, spherical grids can be generated directly, where points are distributed across latitude-like (φ) and longitude-like (θ) lines, facilitating visualization without intermediate transformations in specialized software. Surfaces in spherical coordinates are represented by fixing one coordinate while varying the others, producing distinct geometric shapes. A constant ρ value describes a sphere centered at the origin, with the radius determined by that fixed distance. A constant θ value forms a half-plane that includes the z-axis and extends radially outward at the specified azimuthal angle. Similarly, a constant φ value yields a half-cone with its apex at the origin and axis along the z-axis, opening at the fixed polar angle. Various software tools support spherical plotting by leveraging Cartesian conversions for rendering. In MATLAB, functions like sph2cart transform coordinates for use with plot3 or surf to create 3D visualizations of points or surfaces. Python's Matplotlib library, through its mplot3d toolkit, enables similar plotting by generating coordinate arrays and applying them to Axes3D methods such as scatter for points or plot_surface for meshes. For basic rendering in such environments, pseudocode might involve creating nested loops over θ and φ ranges to produce a grid of points, computing their spatial positions, and rendering them as a wireframe or filled surface to depict the desired spherical structure. Note that visualizations must account for singularities at the poles, where multiple θ values map to the same point, potentially requiring angular averaging or grid adjustments for smooth rendering.

Unique coordinates

In spherical coordinates, the representation of points is not always unique due to inherent ambiguities in the angular variables. Specifically, at the poles where the polar angle φ is 0 (north pole) or π (south pole), the azimuthal angle θ becomes undefined because all values of θ correspond to the same physical point along the z-axis. This non-uniqueness arises because the mapping from (ρ, θ, φ) to Cartesian coordinates collapses the θ direction to a single point at these locations, as the radial distance in the xy-plane, ρ sin φ, equals zero. Additionally, along the entire z-axis (including the origin where ρ = 0), θ can take any value without altering the position, further contributing to multiplicity. The periodicity of θ, which repeats every 2π, introduces another layer of non-uniqueness, allowing multiple (θ, φ) pairs—such as (θ, φ) and (θ + 2πk, φ) for integer k—to describe the same point off the z-axis. To address these issues, standard handling strategies involve selecting principal values for the angles: typically restricting φ to [0, π] and θ to [0, 2π), which ensures a unique representation for all points except those on the z-axis. For differentiability in applications like vector calculus, limits are often used near the poles; for instance, expressions involving θ are evaluated by approaching φ = 0 or π while fixing the Cartesian coordinates. These singularities have significant implications for numerical computations in algorithms using spherical coordinates. Expressions such as the conversion to cylindrical radius ρ sin φ or certain differential operators often involve divisions by sin φ, which approach zero near the poles and can lead to numerical instability or overflow. To mitigate this, algorithms may employ coordinate transformations, such as the Double Fourier Sphere method, which reformulates functions to avoid explicit pole singularities by extending the domain periodically, or use specialized grids that cluster points away from the poles to preserve accuracy without direct division. Such approaches ensure stable evaluations even for smooth underlying functions that appear singular in the coordinate system.

Transformations

Cartesian coordinates

The forward transformation from spherical coordinates (\rho, \theta, \phi) to Cartesian coordinates (x, y, z) arises from the geometric projection of the position vector \vec{r} with magnitude \rho. The z-component is the vertical projection along the positive z-axis, given by z = \rho \cos \phi, where \phi is the polar angle measured from the positive z-axis (with $0 \leq \phi \leq \pi). The remaining radial distance in the xy-plane is \rho \sin \phi, which forms a right triangle with the x- and y-axes, where \theta is the azimuthal angle in the xy-plane (with $0 \leq \theta < 2\pi). Thus, x = (\rho \sin \phi) \cos \theta and y = (\rho \sin \phi) \sin \theta. These relations yield the complete forward transformation: \begin{align*} x &= \rho \sin \phi \cos \theta, \\ y &= \rho \sin \phi \sin \theta, \\ z &= \rho \cos \phi. \end{align*} The inverse transformation from Cartesian to spherical coordinates begins with the radial distance \rho = \sqrt{x^2 + y^2 + z^2}, which follows from the Euclidean distance formula in three dimensions. The polar angle is then \phi = \arccos\left(\frac{z}{\rho}\right), ensuring \phi lies in [0, \pi] as the arccosine range matches the geometric constraint. For the azimuthal angle, \theta = \atantwo(y, x), where the two-argument arctangent function returns values in (-\pi, \pi] and correctly handles all quadrants by considering the signs of x and y, avoiding ambiguities in the principal value of the standard arctangent. The Jacobian matrix for the forward transformation is the 3×3 matrix of partial derivatives \mathbf{J} = \frac{\partial(x,y,z)}{\partial(\rho,\theta,\phi)}, with elements: \mathbf{J} = \begin{pmatrix} \sin \phi \cos \theta & -\rho \sin \phi \sin \theta & \rho \cos \phi \cos \theta \\ \sin \phi \sin \theta & \rho \sin \phi \cos \theta & \rho \cos \phi \sin \theta \\ \cos \phi & 0 & -\rho \sin \phi \end{pmatrix}. Its determinant is \det(\mathbf{J}) = -\rho^2 \sin \phi. Since \sin \phi \geq 0 for $0 \leq \phi \leq \pi and \rho > 0, the absolute value is |\det(\mathbf{J})| = \rho^2 \sin \phi, reflecting the orientation-reversing nature of this variable order but used positively in integration. In multivariable calculus, this absolute value facilitates change of variables in triple integrals, yielding the volume element dV = \rho^2 \sin \phi \, d\rho \, d\theta \, d\phi.

Cylindrical coordinates

Cylindrical coordinates, denoted as (\rho_\text{cyl}, \theta, z), describe a point in three-dimensional space using the radial distance \rho_\text{cyl} from the z-axis in the xy-plane, the azimuthal angle \theta around the z-axis, and the height z along the z-axis, where \rho_\text{cyl} = \sqrt{x^2 + y^2}. The transformation from spherical coordinates (\rho, \theta, \phi) to cylindrical coordinates simplifies by preserving the azimuthal angle \theta and projecting the spherical radial distance \rho onto the cylindrical components, yielding \rho_\text{cyl} = \rho \sin \phi and z = \rho \cos \phi. The inverse transformation recovers the spherical parameters as \rho = \sqrt{\rho_\text{cyl}^2 + z^2} and \phi = \arccos\left(\frac{z}{\sqrt{\rho_\text{cyl}^2 + z^2}}\right), with \theta unchanged. Geometrically, spherical coordinates extend cylindrical coordinates by tilting the radial segment from the z-axis toward the , forming a where the cylindrical radius \rho_\text{cyl} and z serve as the legs, and the spherical \rho as the . This relation highlights cylindrical coordinates' utility in axisymmetric problems, where the shared azimuthal angle \theta aligns with the planar from Cartesian systems.

Related Systems

Ellipsoidal coordinates

Ellipsoidal coordinates (\mu, \nu, \phi) constitute a three-dimensional orthogonal curvilinear coordinate system that extends the spherical coordinate system to accommodate non-spherical geometries, particularly those exhibiting ellipsoidal symmetry. Unlike the spherical system's reliance on confocal spheres for constant radial distance, ellipsoidal coordinates are founded on three mutually confocal families of quadric surfaces: ellipsoids (constant \mu), hyperboloids of one sheet (constant \nu), and hyperboloids of two sheets (constant \phi). These surfaces share the same focal conics, determined by the distinct semi-axes a > b > c > 0 of a reference triaxial ellipsoid, enabling separation of variables in partial differential equations for problems with such symmetry. The ranges of the coordinates are typically -a^2 < \phi < -b^2, -b^2 < \nu < -c^2, and -c^2 < \mu < \infty, ensuring coverage of the entire space while maintaining . This setup contrasts sharply with spherical coordinates, where all constant-radius surfaces are spheres without focal structure beyond the origin. The ellipsoidal system's confocal property facilitates analytical solutions in , gravitation, and wave propagation within ellipsoidal domains. In the limiting case where the eccentricity approaches zero—achieved when the semi-axes a, b, and c become equal—the ellipsoidal coordinates degenerate to spherical coordinates, with \mu approaching the radial distance r, and \nu, \phi mapping to the polar and azimuthal angles \theta, \varphi. The scale factors h_\mu, h_\nu, h_\phi then simplify to $1, r, and r \sin \theta, respectively, yielding the familiar spherical metric ds^2 = dr^2 + r^2 d\theta^2 + r^2 \sin^2 \theta \, d\varphi^2. For the general case, the scale factors are h_\mu = \sqrt{\frac{(\mu - \nu)(\mu - \phi)}{4(\mu + a^2)(\mu + b^2)(\mu + c^2)}}, \quad h_\nu = \sqrt{\frac{(\nu - \phi)(\nu - \mu)}{4(\nu + a^2)(\nu + b^2)(\nu + c^2)}}, \quad h_\phi = \sqrt{\frac{(\phi - \mu)(\phi - \nu)}{4(\phi + a^2)(\phi + b^2)(\phi + c^2)}}, which define the line element ds^2 = h_\mu^2 d\mu^2 + h_\nu^2 d\nu^2 + h_\phi^2 d\phi^2 and underpin the orthogonality of the system. The transformation from ellipsoidal to Cartesian coordinates (x, y, z) employs parametric expressions derived from the inversion of elliptic integrals, utilizing to parameterize the confocal s. Alternatively, direct inversion solves the system of equations \frac{x^2}{a^2 + \mu} + \frac{y^2}{b^2 + \mu} + \frac{z^2}{c^2 + \mu} = 1, with analogous equations for \nu and \phi, yielding x^2 = \frac{(a^2 + \mu)(a^2 + \nu)(a^2 + \phi)}{(b^2 - a^2)(c^2 - a^2)}, and similar for y^2 and z^2, from which the coordinates are obtained by choosing appropriate signs. This approach, rooted in elliptic integrals, highlights the system's complexity but its power for exact solutions in ellipsoidal geometries.

Geodetic coordinates

Geodetic coordinates represent a practical extension of spherical coordinates adapted to the oblate spheroid model of , incorporating , , and relative to a reference rather than a . This system is fundamental in for precise positioning on Earth's irregular surface. In geodetic coordinates, the position is defined by geodetic latitude \phi, longitude \lambda, and ellipsoidal height h. Geodetic latitude \phi is the angle between the equatorial plane and the normal to the ellipsoid surface at the point, ranging from -90^\circ to $90^\circ. Longitude \lambda is the angle in the equatorial plane from a reference meridian (typically Greenwich) to the meridian through the point, ranging from -180^\circ to $180^\circ or $0^\circ to $360^\circ. Ellipsoidal height h measures the distance along this normal from the ellipsoid surface to the point, positive outward. Geodetic latitude \phi relates to the parametric latitude \beta (also called reduced latitude), which parameterizes positions on an auxiliary sphere of radius \sqrt{ab} where a and b are the . The relation is given by \tan \beta = (1 - e^2) \tan \phi, where e^2 = (a^2 - b^2)/a^2 is the squared ; this facilitates certain projections and computations. Unlike , which assume a perfect and measure as the angle from the to the point along a , geodetic coordinates account for Earth's oblateness by using an , where is defined by the surface normal rather than the equatorial plane directly. This leads to a maximum difference of about $11.5' between geodetic and geocentric latitudes at $45^\circ. Conversion from geodetic coordinates (\phi, \lambda, h) to Earth-Centered Earth-Fixed (ECEF) Cartesian coordinates (X, Y, Z) uses the ellipsoid parameters a (semi-major axis) and e (), with the prime vertical N = a / \sqrt{1 - e^2 \sin^2 \phi}: \begin{align*} X &= (N + h) \cos \phi \cos \lambda, \\ Y &= (N + h) \cos \phi \sin \lambda, \\ Z &= [N (1 - e^2) + h] \sin \phi. \end{align*} The modern standard for is the (WGS 84), developed by the U.S. Department of Defense in to support the (GPS). WGS 84 defines an Earth-centered, Earth-fixed reference frame with specific parameters (a = 6{,}378{,}137 m, f = 1/298.257223563) and has been maintained through GPS observations for alignment with global networks, enabling accurate , , and determination worldwide.

Applications

Geography and astronomy

In geography, the spherical coordinate system provides a framework for locating points on Earth's surface by approximating the planet as a perfect . , which measures the north or south of the , corresponds to 90° minus the polar angle θ in standard spherical coordinates, ranging from 0° at the to ±90° at the poles. , the east or west of the , aligns with the azimuthal angle φ, spanning from 0° to 360° or -180° to 180°. This system enables precise positioning for mapping and geospatial analysis, treating Earth's radius as approximately constant at about 6,371 km. A key application in involves calculating great-circle distances, the shortest paths between two points on , which form the basis for efficient routing in , shipping, and . These distances represent arcs along circles passing through Earth's center, contrasting with straight-line approximations on flat maps, and are essential for determining travel times and fuel requirements over long distances. For instance, the great-circle route from to curves northward across the Pacific, minimizing the actual surface distance compared to a (constant bearing) path. In astronomy, spherical coordinates describe positions on the , an imaginary sphere of infinite radius centered on the observer, where stars and other objects appear fixed relative to distant backgrounds. , analogous to , approximates the azimuthal φ and measures eastward from the vernal along the , typically in hours (0 to 24, equivalent to 0° to 360°). , similar to , equals 90° minus the polar θ, ranging from +90° at the north to -90° at the south, with 0° on the . This equatorial system allows consistent cataloging of celestial bodies, independent of the observer's location or time. Celestial sphere projections using these coordinates facilitate star charts and telescope alignments, projecting the three-dimensional sky onto two-dimensional surfaces for observation planning. In practice, astronomers convert an object's right ascension and declination—along with the observer's latitude and local sidereal time—into alt-azimuth coordinates to direct s, where altitude is the angle above the horizon and is the horizontal direction from north. For example, locating ( ≈ +89°) from a mid-latitude site like 40° N requires adjusting for the local horizon to achieve precise pointing, enabling accurate tracking during observations.

Physics and engineering

In quantum mechanics, spherical coordinates are essential for describing systems with rotational symmetry, particularly in the context of angular momentum operators. The eigenfunctions of the angular momentum operators \mathbf{L}^2 and L_z are the spherical harmonics Y_{l m}(\theta, \phi), where l is the orbital quantum number and m is the magnetic quantum number, satisfying \mathbf{L}^2 Y_{l m} = \hbar^2 l(l+1) Y_{l m} and L_z Y_{l m} = \hbar m Y_{l m}. These functions form a complete orthonormal basis for expanding angular-dependent wave functions on the sphere, enabling the separation of variables in the Schrödinger equation for central potentials. A key application is the , where the time-independent in spherical coordinates separates into radial and angular parts, yielding wave functions of the form \psi_{n l m}(r, \theta, \phi) = R_{n l}(r) Y_{l m}(\theta, \phi), with the R_{n l}(r) depending on the principal quantum number n and l. This separation exploits the potential's spherical symmetry, leading to quantized levels E_n = -\frac{13.6 \mathrm{[eV](/page/EV)}}{n^2} independent of l and m. The angular part, governed by the , determines the probability distribution's directional dependence, crucial for understanding atomic orbitals and selection rules in . In , spherical coordinates facilitate solving value problems for electrostatic potentials with spherical symmetry, such as the potential around a charged . \nabla^2 \Phi = 0 in spherical coordinates separates into radial and angular solutions, expressed as \Phi(r, \theta, \phi) = \sum_{l=0}^\infty \sum_{m=-l}^l \left( A_{l m} r^l + \frac{B_{l m}}{r^{l+1}} \right) Y_{l m}(\theta, \phi), where the coefficients are determined by conditions like a surface charge on the . For a in a external , this expansion yields the induced dipole moment, illustrating image charge methods adapted to spherical geometry. Such solutions are foundational for analyzing dielectrics and conductors with spherical shapes. In engineering, spherical coordinates are used to model radiation patterns of antennas, where the far-field electric field is expressed in terms of \theta and \phi, such as E(\theta, \phi) = E_0 f(\theta, \phi) \frac{e^{-j k r}}{r} for directive patterns like those of dipole antennas. This coordinate system aligns with the natural symmetry of radiating sources, enabling computation of gain, directivity, and beamwidth via integration over spherical surfaces. For fluid dynamics in engineering applications, such as low-Reynolds-number flow around a sphere (Stokes flow), the velocity field satisfies the Stokes equations in spherical coordinates, yielding an analytical solution \mathbf{v}(r, \theta) = \left(1 - \frac{3 a}{2 r} + \frac{a^3}{2 r^3}\right) U \cos \theta \, \hat{r} - \left(1 - \frac{3 a}{4 r} - \frac{a^3}{4 r^3}\right) U \sin \theta \, \hat{\theta}, where a is the sphere radius and U is the free-stream velocity, critical for drag force calculations in particle sedimentation and microfluidics. This symmetry simplifies simulations of aerodynamic or hydrodynamic problems involving spherical objects.

Mathematical Operations

Integration and differentiation

In spherical coordinates, integration over a volume requires the appropriate volume element, which accounts for the geometry of the coordinate system. The volume element dV is derived from the Jacobian determinant of the transformation from spherical coordinates (\rho, \theta, \phi) to Cartesian coordinates (x, y, z), where x = \rho \sin \phi \cos \theta, y = \rho \sin \phi \sin \theta, and z = \rho \cos \phi. The Jacobian matrix consists of the partial derivatives of these transformations, and its determinant is \rho^2 \sin \phi. Thus, the infinitesimal volume element is dV = \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta. This form ensures that multiple integrals, such as \iiint f(\rho, \theta, \phi) \, dV, correctly compute volumes or other scalar quantities in spherical symmetry. Differentiation in spherical coordinates follows the general expressions for orthogonal curvilinear systems, adapted to the scale factors h_\rho = 1, h_\phi = \rho, and h_\theta = \rho \sin \phi. The gradient of a scalar f is \nabla f = \frac{\partial f}{\partial \rho} \hat{e}_\rho + \frac{1}{\rho} \frac{\partial f}{\partial \phi} \hat{e}_\phi + \frac{1}{\rho \sin \phi} \frac{\partial f}{\partial \theta} \hat{e}_\theta. These components reflect the varying along each direction, with the \phi and \theta terms scaled by the respective arc lengths. For vector fields, the divergence measures the net flux out of a small volume and is given by \nabla \cdot \mathbf{A} = \frac{1}{\rho^2} \frac{\partial}{\partial \rho} (\rho^2 A_\rho) + \frac{1}{\rho \sin \phi} \frac{\partial}{\partial \phi} (\sin \phi \, A_\phi) + \frac{1}{\rho \sin \phi} \frac{\partial A_\theta}{\partial \theta}, where \mathbf{A} = A_\rho \hat{e}_\rho + A_\phi \hat{e}_\phi + A_\theta \hat{e}_\theta. The curl, which quantifies local rotation, has components (\nabla \times \mathbf{A})_\rho = \frac{1}{\rho \sin \phi} \left( \frac{\partial}{\partial \phi} (A_\theta \sin \phi) - \frac{\partial A_\phi}{\partial \theta} \right), (\nabla \times \mathbf{A})_\phi = \frac{1}{\rho \sin \phi} \frac{\partial A_\rho}{\partial \theta} - \frac{1}{\rho} \frac{\partial}{\partial \rho} (\rho A_\theta), (\nabla \times \mathbf{A})_\theta = \frac{1}{\rho} \frac{\partial}{\partial \rho} (\rho A_\phi) - \frac{1}{\rho} \frac{\partial A_\rho}{\partial \phi}. These operators are derived by applying the general formulas for orthogonal curvilinear coordinates to the spherical scale factors. The , essential for solving partial differential equations such as those in and conduction, is the of the : \Delta f = \nabla^2 f = \frac{1}{\rho^2} \frac{\partial}{\partial \rho} \left( \rho^2 \frac{\partial f}{\partial \rho} \right) + \frac{1}{\rho^2 \sin \phi} \frac{\partial}{\partial \phi} \left( \sin \phi \frac{\partial f}{\partial \phi} \right) + \frac{1}{\rho^2 \sin^2 \phi} \frac{\partial^2 f}{\partial \theta^2}. This expression facilitates in problems with spherical .

Distance and kinematics

In spherical coordinates, the shortest distance between two points on a of \rho is the , given by d = \rho \[arccos](/page/Arccos)(\cos \phi_1 \cos \phi_2 + \sin \phi_1 \sin \phi_2 \cos(\theta_2 - \theta_1)), where (\phi_1, \theta_1) and (\phi_2, \theta_2) are the polar and azimuthal of the points, respectively. This formula arises from the applied to the angular separation between the points, with the arccosine yielding the in radians before scaling by the . The infinitesimal line element ds in spherical coordinates, which describes arc lengths along paths in space, is expressed as ds^2 = d\rho^2 + \rho^2 d\phi^2 + \rho^2 \sin^2 \phi \, d\theta^2. This metric form derives from the orthogonal basis vectors in spherical coordinates, where the coefficients reflect the scaling factors for radial, polar, and azimuthal displacements. For paths confined to a sphere of fixed \rho, the line element simplifies to ds^2 = \rho^2 (d\phi^2 + \sin^2 \phi \, d\theta^2), enabling computation of geodesic lengths such as great-circle arcs. For kinematics, the velocity \mathbf{v} of a particle at (\rho, \theta, \phi) is \mathbf{v} = \dot{\rho} \, \hat{\mathbf{e}}_\rho + \rho \dot{\phi} \, \hat{\mathbf{e}}_\phi + \rho \sin \phi \, \dot{\theta} \, \hat{\mathbf{e}}_\theta, where dots denote time derivatives and \hat{\mathbf{e}}_\rho, \hat{\mathbf{e}}_\phi, \hat{\mathbf{e}}_\theta are the orthonormal unit vectors. This expression accounts for the varying magnitudes of the basis vectors with . The acceleration \mathbf{a} includes both local changes in velocity components and terms from the time variation of the unit vectors, yielding a_\rho = \ddot{\rho} - \rho \dot{\phi}^2 - \rho \sin^2 \phi \, \dot{\theta}^2, a_\phi = \rho \ddot{\phi} + 2 \dot{\rho} \dot{\phi} - \rho \sin \phi \cos \phi \, \dot{\theta}^2, a_\theta = \rho \sin \phi \, \ddot{\theta} + 2 \dot{\rho} \sin \phi \, \dot{\theta} + 2 \rho \cos \phi \, \dot{\phi} \dot{\theta}. The terms -\rho \dot{\phi}^2 and -\rho \sin^2 \phi \, \dot{\theta}^2 represent centripetal accelerations toward the local centers of in the \phi and \theta directions, while \rho \sin \phi \cos \phi \, \dot{\theta}^2 is a centrifugal term coupling the azimuthal motion to the polar direction; the Coriolis-like terms $2 \dot{\rho} \dot{\phi}, $2 \dot{\rho} \sin \phi \, \dot{\theta}, and $2 \rho \cos \phi \, \dot{\phi} \dot{\theta} arise from the non-commuting time derivatives in .