Three-dimensional space
Three-dimensional space, also known as 3D space, is a geometric model in which the position of any point is uniquely determined by three mutually perpendicular coordinates, typically denoted as (x, y, z) in a Cartesian system, representing length, width, and height.[1] This framework, known as Euclidean three-dimensional space or \mathbb{R}^3, consists of all ordered triples of real numbers and assumes a flat, isotropic structure where distances and angles follow the principles of Euclidean geometry, such as the Pythagorean theorem extended to three dimensions.[2] It serves as the foundational setting for describing the arrangement and motion of physical objects in everyday experience, distinct from the one- or two-dimensional spaces used for lines or planes.[3] In mathematics, three-dimensional space enables the study of vectors, which are quantities with magnitude and direction representable as arrows from the origin, and facilitates calculations of distances between points using the formula \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}.[4] Beyond the standard Cartesian coordinates, alternative systems like cylindrical coordinates (r, θ, z) and spherical coordinates (ρ, θ, φ) are employed to simplify descriptions of rotationally symmetric objects or surfaces, such as cylinders or spheres.[5] These tools underpin multivariable calculus, where functions of three variables model volumes, surfaces, and gradients, and linear algebra, where subspaces and transformations preserve the space's dimensionality.[6] From a physical perspective, three-dimensional space forms the arena for classical mechanics, electromagnetism, and fluid dynamics, where phenomena like gravity and light propagation are analyzed assuming spatial isotropy and homogeneity on macroscopic scales.[7] Cosmological models suggest that the universe's large-scale structure stabilized into three spatial dimensions during its early evolution, favoring this dimensionality over others for stable atomic and planetary formations.[8] In modern physics, while general relativity embeds three-dimensional space within four-dimensional spacetime, the spatial component remains fundamentally three-dimensional for describing observable matter and fields.[9]History
Ancient and medieval perspectives
In ancient Greek philosophy, space was conceptualized not as an abstract void but as an integral aspect of the physical world, serving as a container for material bodies. Aristotle, in his Physics (Book IV), defined place (topos) as the innermost boundary of the containing body, emphasizing that space is relational and dependent on the presence of bodies rather than an independent entity.[10] This view portrayed three-dimensional space as a plenum filled with substances, where extension arises from the arrangement of matter, influencing later understandings of spatial containment without invoking empty voids.[11] Euclid's Elements (c. 300 BCE) further shaped early intuitions of three-dimensional geometry through synthetic methods, describing solids such as polyhedra and spheres via axioms and postulates without coordinate systems or algebraic formalism. Books XI–XIII of the Elements establish properties of planes and volumes intuitively, treating space as a continuous medium for geometric constructions observable in everyday objects like buildings and celestial bodies.[12] These works provided a foundational framework for visualizing spatial relations, prioritizing empirical deduction over measurement.[13] Contributions from Indian and Islamic scholars expanded observational approaches to three-dimensional space, particularly through astronomy. Aryabhata, in his Aryabhatiya (499 CE), developed spherical trigonometry for modeling celestial motions, treating the Earth and heavens as embedded in a three-dimensional spherical framework to compute planetary positions and eclipses.[14] In the 11th century, Al-Biruni advanced geodetic measurements by determining the Earth's radius using trigonometric observations from mountain elevations, confirming its sphericity and curvature with an accuracy close to modern values, thus refining conceptions of global spatial extent.[15] During the medieval European period, scholastic thinkers synthesized these ideas with Christian theology. Thomas Aquinas, drawing on Aristotle in works like Summa Theologica, integrated the notion of space as a bounded container into a cosmological hierarchy where the finite, three-dimensional universe reflects divine order, with heavenly spheres encompassing earthly bodies in a geocentric model.[16] This reconciliation portrayed space as a created medium, harmonious with faith, bridging philosophical inquiry and religious worldview.[17] A pivotal development bridging medieval and Renaissance views occurred in the 15th century through artistic innovations, exemplified by Filippo Brunelleschi's experiments in Florence around 1420. Using mirrors and peepholes to project the Baptistery's facade onto a painted panel, Brunelleschi demonstrated linear perspective, enabling two-dimensional representations that mimicked three-dimensional depth and spatial recession, thus enhancing perceptual understanding of volume and distance.[18] These techniques, while artistic, laid groundwork for later mathematical formalizations of space.Modern mathematical development
The modern mathematical development of three-dimensional space began in the 17th century with René Descartes' introduction of Cartesian coordinates in his 1637 work La Géométrie, which allowed for the algebraic representation of points, lines, and surfaces in 3D space through ordered triples of numbers, transforming geometry into an analytic discipline.[19] This innovation enabled the precise description of spatial relationships using equations, bridging algebra and geometry and laying the foundation for subsequent advancements in vector analysis and coordinate-based modeling.[19] In the 18th century, Leonhard Euler advanced the study of polyhedra and space-filling structures, culminating in his 1752 formulation of the relation V - E + F = 2 for convex polyhedra, where V denotes vertices, E edges, and F faces, providing a topological invariant that characterizes the connectivity of 3D polyhedral forms.[20] Euler's explorations also included analyses of regular polyhedra and tessellations, contributing to understandings of how shapes fill 3D space without gaps or overlaps.[21] The 19th century saw significant progress in projective and differential geometry relevant to 3D space. Carl Friedrich Gauss's 1827 Theorema Egregium demonstrated that the Gaussian curvature of a surface embedded in 3D space is an intrinsic property, independent of its embedding, which was later generalized to surfaces in higher dimensions. August Ferdinand Möbius, in his 1827 Der barycentrische Calcül, introduced barycentric coordinates, facilitating projective treatments of points and figures in 3D projective space by expressing positions as weighted combinations relative to reference points.[22] Julius Plücker extended projective geometry to lines in 3D space through his line coordinates, introduced in works from the 1840s and elaborated in 1868's Theorie der Flächen, representing lines via six homogeneous coordinates and enabling algebraic studies of line complexes. Bernhard Riemann's 1854 habilitation lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen developed the framework of Riemannian geometry, describing curved 3D spaces via metrics on manifolds and providing the mathematical basis for non-Euclidean geometries.[23] Entering the 20th century, Henri Poincaré's foundational work in topology, particularly his 1895 Analysis Situs and subsequent papers, analyzed 3D manifolds as abstract spaces, introducing concepts like fundamental groups to classify their connectivity and homology, which distinguished simply connected spaces and influenced the study of 3D topological structures.Euclidean geometry
Coordinate systems
In three-dimensional Euclidean space, the Cartesian coordinate system provides a standard framework for locating points using three mutually perpendicular axes intersecting at the origin. A point is represented by an ordered triple (x, y, z), where x, y, and z denote the signed distances from the origin along the respective axes, typically oriented as the x-axis (horizontal), y-axis (depth), and z-axis (vertical). This system, introduced by René Descartes in his 1637 work La Géométrie, extends the two-dimensional plane to allow precise positioning in space.[24][25] The distance between two points (x_1, y_1, z_1) and (x_2, y_2, z_2) in this system is given by the Euclidean metric: \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}, which generalizes the Pythagorean theorem to three dimensions.[26] Alternative coordinate systems, such as cylindrical and spherical, simplify representations when symmetry about an axis or radial structure is present. Cylindrical coordinates (r, \theta, z) describe a point by its radial distance r \geq 0 from the z-axis in the xy-plane, the azimuthal angle \theta (measured from the positive x-axis), and the height z along the z-axis. The conversion to Cartesian coordinates is: x = r \cos \theta, \quad y = r \sin \theta, \quad z = z. For volume integrals, the Jacobian determinant yields the volume element r \, dr \, d\theta \, dz, accounting for the scaling in the radial direction.[27][28] Spherical coordinates (r, \theta, \phi) use the radial distance r \geq 0 from the origin, the polar angle \theta (from the positive z-axis, $0 \leq \theta \leq \pi), and the azimuthal angle \phi (from the positive x-axis in the xy-plane, $0 \leq \phi < 2\pi). The conversion formulas are: x = r \sin \theta \cos \phi, \quad y = r \sin \theta \sin \phi, \quad z = r \cos \theta. This system is particularly useful for problems exhibiting radial symmetry, such as those involving spheres or isotropic fields.[29] Coordinate transformations, such as rotations, preserve distances and angles in Euclidean space and are represented by orthogonal matrices with determinant 1. For a counterclockwise rotation by angle \theta around the z-axis, the transformation matrix applied to a point's Cartesian coordinates is: \begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}. Such matrices facilitate changing the orientation of axes or objects while maintaining the underlying geometry.[30]Lines, planes, and distances
In three-dimensional Euclidean space, a line can be defined using parametric equations that describe its position as a function of a parameter t. The parametric form passing through a point (x_0, y_0, z_0) with direction vector \langle a, b, c \rangle is given by \begin{align*} x &= x_0 + a t, \\ y &= y_0 + b t, \\ z &= z_0 + c t, \end{align*} where t \in \mathbb{R}./01%3A_Vectors_and_Geometry_in_Two_and_Three_Dimensions/1.05%3A_Equations_of_Lines_in_3d) The direction vector \langle a, b, c \rangle indicates the orientation and scaling of the line, and any scalar multiple of it yields an equivalent representation.[31] To determine if two lines intersect, their parametric equations are set equal to solve for parameters t and s. For lines \mathbf{r}_1 = \mathbf{p}_1 + t \mathbf{d}_1 and \mathbf{r}_2 = \mathbf{p}_2 + s \mathbf{d}_2, intersection occurs if there exist scalars t and s such that \mathbf{p}_1 + t \mathbf{d}_1 = \mathbf{p}_2 + s \mathbf{d}_2, which rearranges to ( \mathbf{p}_2 - \mathbf{p}_1 ) = t \mathbf{d}_1 - s \mathbf{d}_2; a unique solution implies intersection at that point, while no solution indicates skew or parallel non-intersecting lines.[31] If the direction vectors are parallel (one is a scalar multiple of the other) and the lines do not coincide, they are parallel and do not intersect unless the vector between points on each lies in the span of the direction.[32] A plane in three-dimensional space is defined by the general equation a x + b y + c z + d = 0, where \langle a, b, c \rangle is the normal vector perpendicular to the plane.[33] This normal vector determines the plane's orientation, and the equation can be derived from a point on the plane and the normal via a (x - x_0) + b (y - y_0) + c (z - z_0) = 0./01%3A_Vectors_and_Geometry_in_Two_and_Three_Dimensions/1.05%3A_Equations_of_Lines_in_3d) The distance from a point (x_0, y_0, z_0) to the plane a x + b y + c z + d = 0 is the length of the perpendicular from the point to the plane, calculated as \frac{|a x_0 + b y_0 + c z_0 + d|}{\sqrt{a^2 + b^2 + c^2}}. This formula arises from projecting the vector from a point on the plane to (x_0, y_0, z_0) onto the unit normal vector.[34][35] The angle between two lines is found using the cosine of the angle \theta between their direction vectors \mathbf{d}_1 and \mathbf{d}_2, given by \cos \theta = \frac{|\mathbf{d}_1 \cdot \mathbf{d}_2|}{|\mathbf{d}_1| |\mathbf{d}_2|}, where the acute angle is considered.[36] Similarly, the angle between two planes is the angle between their normal vectors \mathbf{n}_1 and \mathbf{n}_2, with \cos \phi = \frac{|\mathbf{n}_1 \cdot \mathbf{n}_2|}{|\mathbf{n}_1| |\mathbf{n}_2|}.[36] For the angle between a line with direction \mathbf{d} and a plane with normal \mathbf{n}, the setup involves the complement of the angle between \mathbf{d} and \mathbf{n}, using \sin \psi = \frac{|\mathbf{d} \cdot \mathbf{n}|}{|\mathbf{d}| |\mathbf{n}|} for the acute angle \psi./01%3A_Vectors_and_Geometry_in_Two_and_Three_Dimensions/1.05%3A_Equations_of_Lines_in_3d) For two skew lines (non-intersecting and non-parallel) with parametric forms \mathbf{r}_1 = \mathbf{p}_1 + t \mathbf{d}_1 and \mathbf{r}_2 = \mathbf{p}_2 + s \mathbf{d}_2, the shortest distance is the length of the common perpendicular, given by \frac{|(\mathbf{p}_2 - \mathbf{p}_1) \cdot (\mathbf{d}_1 \times \mathbf{d}_2)|}{|\mathbf{d}_1 \times \mathbf{d}_2|}. This expression uses the cross product \mathbf{d}_1 \times \mathbf{d}_2 to find the direction perpendicular to both lines, and the scalar triple product to project the vector between points onto that direction.[37]Spheres, balls, and polytopes
In three-dimensional Euclidean space, a sphere is defined as the set of all points equidistant from a fixed center point, with that distance being the radius r. Using Cartesian coordinates, the equation of a sphere centered at (a, b, c) is given by (x - a)^2 + (y - b)^2 + (z - c)^2 = r^2. This locus represents the surface of the sphere.[38] The surface area of the sphere is $4\pi r^2, derived by considering the sphere as the limit of polyhedral approximations or through integration in spherical coordinates.[39] The volume enclosed by the sphere, known as the ball of radius r, is \frac{4}{3}\pi r^3, which can be obtained via triple integration over the region or by the method of Cavalieri's principle.[39] A ball in three dimensions is the solid object comprising the sphere and its interior, defined as the set of points whose distance from the center is at most r.[40] On the sphere's surface, great circles—formed by the intersection of the sphere with any plane passing through its center—represent the geodesics, or shortest paths connecting two points along the surface. These curves are the three-dimensional analogues of straight lines and have constant curvature equal to that of the sphere.[41] Convex polytopes in three dimensions are polyhedra, bounded by flat polygonal faces, straight edges, and vertices. The regular convex polyhedra, called Platonic solids, are classified into five types: the tetrahedron (4 triangular faces), cube (6 square faces), octahedron (8 triangular faces), dodecahedron (12 pentagonal faces), and icosahedron (20 triangular faces); each has congruent regular polygonal faces and the same number meeting at every vertex.[42] For any convex polyhedron that is topologically equivalent to a sphere, the Euler characteristic satisfies \chi = V - E + F = 2, where V, E, and F denote the numbers of vertices, edges, and faces, respectively; this relation holds due to the polyhedron's spherical topology and can be verified inductively by decomposition.[43] Some convex polyhedra admit space-filling tessellations, partitioning three-dimensional space without gaps or overlaps. The cubic honeycomb, consisting of identical cubes arranged in a lattice, is a prominent example, with each cube sharing faces with six neighbors.[44] Volumes of Platonic solids provide concrete measures of their spatial extent; for instance, a cube of side length a has volume a^3, while a regular tetrahedron of side length a has volume \frac{\sqrt{2}}{12} a^3, computed by dividing the tetrahedron into pyramids or using vector cross products for the enclosed space.[45]Quadric surfaces and surfaces of revolution
Quadric surfaces in three-dimensional Euclidean space are defined by second-degree polynomial equations in the variables x, y, and z. The general equation takes the form Ax^2 + By^2 + Cz^2 + axy + bxz + cyz + d_1 x + d_2 y + d_3 z + E = 0, where the coefficients determine the specific type of surface through the eigenvalues of the associated quadratic form or by completing the square and translating coordinates.[46] These surfaces are classified into non-degenerate types—ellipsoids, hyperboloids of one or two sheets, elliptic and hyperbolic paraboloids—and degenerate cases such as cones, cylinders, and pairs of planes, based on the signs and ranks of the quadratic terms after canonical reduction.[47] Among these, the ellipsoid represents a bounded, closed surface analogous to an ellipse stretched in three dimensions, with the standard equation \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1, where a, b, and c are positive semi-axes lengths; when a = b = c, it reduces to a sphere.[47] The hyperbolic paraboloid, a ruled surface known for its saddle-like shape, features hyperbolic cross-sections and is given by z = \frac{x^2}{a^2} - \frac{y^2}{b^2} in canonical form, exhibiting both positive and negative curvatures along principal directions.[48] Surfaces of revolution arise by rotating a curve in a plane around an axis lying in that plane but not intersecting the curve, producing rotationally symmetric surfaces in three dimensions.[49] For instance, revolving a semicircle about its diameter yields a sphere, while rotating a circle offset from the axis generates a torus, whose implicit equation is \left( \sqrt{x^2 + y^2} - R \right)^2 + z^2 = r^2, with R > r > 0 denoting the major and minor radii, respectively.[50] Pappus's centroid theorem provides a method to compute areas and volumes of such surfaces without integration: the lateral surface area equals the arc length of the generating curve times the circumference described by its centroid (i.e., $2\pi times the centroid's distance to the axis), and the enclosed volume equals the area under the curve times the same circumferential distance.[49] This theorem, attributed to Pappus of Alexandria in the 4th century CE, relies on the centroid's definition as the average position weighted by arc length or area.[51]Linear algebra
Vectors, dot product, and norms
In three-dimensional Euclidean space, vectors are commonly represented as ordered triples of real numbers, \vec{v} = (v_x, v_y, v_z), where v_x, v_y, and v_z are the components along the respective Cartesian axes.[25] This representation corresponds to the displacement from the origin to a point in \mathbb{R}^3. Vector addition is performed component-wise: \vec{u} + \vec{v} = (u_x + v_x, u_y + v_y, u_z + v_z), which geometrically corresponds to the parallelogram rule.[52] Scalar multiplication scales the vector by a real number k, yielding k\vec{v} = (k v_x, k v_y, k v_z), altering its magnitude while preserving direction (or reversing it if k < 0).[53] The dot product of two vectors \vec{u} = (u_x, u_y, u_z) and \vec{v} = (v_x, v_y, v_z) in three dimensions is defined algebraically as \vec{u} \cdot \vec{v} = u_x v_x + u_y v_y + u_z v_z.[54] Geometrically, it equals \|\vec{u}\| \|\vec{v}\| \cos \theta, where \theta is the angle between the vectors and \|\cdot\| denotes the Euclidean norm; this relation links the algebraic form to the spatial orientation.[55] Two nonzero vectors are orthogonal if their dot product is zero, as \cos \theta = 0 implies \theta = 90^\circ.[56] The Euclidean norm, or length, of a vector \vec{v} is given by \|\vec{v}\| = \sqrt{\vec{v} \cdot \vec{v}} = \sqrt{v_x^2 + v_y^2 + v_z^2}, providing a measure of magnitude invariant under rotations.[57] A unit vector, with norm 1, is obtained by normalizing: \hat{v} = \vec{v} / \|\vec{v}\| for \vec{v} \neq \vec{0}.[58] The vector projection of \vec{v} onto \vec{u} (nonzero) is \operatorname{proj}_{\vec{u}} \vec{v} = \left( \frac{\vec{v} \cdot \vec{u}}{\|\vec{u}\|^2} \right) \vec{u}, representing the component of \vec{v} parallel to \vec{u}.[59] These concepts find applications in determining the angle between vectors via \cos \theta = \frac{\vec{u} \cdot \vec{v}}{\|\vec{u}\| \|\vec{v}\|}, essential for geometric computations.[60] In physics, the dot product computes work as W = \vec{F} \cdot \vec{d}, where \vec{F} is force and \vec{d} is displacement, capturing only the component of force along the path.[61]Cross product and orientations
In three-dimensional Euclidean space, the cross product of two vectors \vec{u} = (u_x, u_y, u_z) and \vec{v} = (v_x, v_y, v_z) is a vector defined by the determinant-like formula \vec{u} \times \vec{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_x & u_y & u_z \\ v_x & v_y & v_z \end{vmatrix} = (u_y v_z - u_z v_y, \, u_z v_x - u_x v_z, \, u_x v_y - u_y v_x). /01:_Vectors_in_Euclidean_Space/1.04:_Cross_Product) This operation yields a vector perpendicular to both \vec{u} and \vec{v}, with magnitude \|\vec{u} \times \vec{v}\| = \|\vec{u}\| \|\vec{v}\| \sin \theta, where \theta is the angle between them; geometrically, this magnitude equals the area of the parallelogram formed by \vec{u} and \vec{v} as adjacent sides./01:_Vectors_in_Euclidean_Space/1.04:_Cross_Product) Key properties of the cross product include anticommutativity, \vec{u} \times \vec{v} = -\vec{v} \times \vec{u}, and orthogonality to its input vectors, \vec{u} \cdot (\vec{u} \times \vec{v}) = 0 and \vec{v} \cdot (\vec{u} \times \vec{v}) = 0./01:_Vectors_in_Euclidean_Space/1.04:_Cross_Product) The direction follows the right-hand rule: aligning the fingers of the right hand with \vec{u} and curling them toward \vec{v} points the thumb in the direction of \vec{u} \times \vec{v}.[62] These attributes make the cross product useful for determining a normal vector to the plane spanned by \vec{u} and \vec{v}, essential in applications like surface parameterization./01:_Vectors_in_Euclidean_Space/1.04:_Cross_Product) In physics, the cross product computes torque as \vec{\tau} = \vec{r} \times \vec{F}, where \vec{r} is the position vector from the pivot to the force application point and \vec{F} is the force, yielding a vector whose magnitude is r F \sin \theta and direction indicates the rotation axis.[63] For volumes, the scalar triple product \vec{a} \cdot (\vec{b} \times \vec{c}) gives the signed volume of the parallelepiped spanned by \vec{a}, \vec{b}, and \vec{c}, with the absolute value representing the actual volume.[64] The cross product inherently encodes orientations through its handedness, distinguishing chiral (handed) structures in 3D space via the right-hand rule, which selects one of two possible perpendicular directions.[62] This vector-valued binary operation is unique to three dimensions; in higher dimensions, analogous constructions yield higher-rank tensors or subspaces rather than vectors.[65]Abstract vector spaces
In the context of three-dimensional space, the algebraic structure can be abstracted to a finite-dimensional vector space over the real numbers \mathbb{R}, providing a foundation for linear operations independent of specific geometric embeddings. A vector space V over \mathbb{R} is a set equipped with two operations: vector addition +\colon V \times V \to V and scalar multiplication \cdot\colon \mathbb{R} \times V \to V, satisfying the following axioms: for all \mathbf{u}, \mathbf{v}, \mathbf{w} \in V and \alpha, \beta \in \mathbb{R},- Associativity of addition: (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}),
- Commutativity of addition: \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u},
- Existence of zero vector: there exists \mathbf{0} \in V such that \mathbf{u} + \mathbf{0} = \mathbf{u},
- Additive inverses: for each \mathbf{u}, there exists -\mathbf{u} such that \mathbf{u} + (-\mathbf{u}) = \mathbf{0},
- Distributivity over vector addition: \alpha (\mathbf{u} + \mathbf{v}) = \alpha \mathbf{u} + \alpha \mathbf{v},
- Distributivity over scalar addition: (\alpha + \beta) \mathbf{u} = \alpha \mathbf{u} + \beta \mathbf{u},
- Compatibility: \alpha (\beta \mathbf{u}) = (\alpha \beta) \mathbf{u},
- Identity for scalar multiplication: $1 \cdot \mathbf{u} = \mathbf{u}.[66]
Calculus
Vector calculus operators
In three-dimensional Euclidean space, vector calculus operators such as the gradient, divergence, curl, and Laplacian provide essential tools for analyzing scalar and vector fields, capturing local properties like rates of change, sources, rotations, and diffusion.[71] These operators, collectively denoted using the del (or nabla) symbol ∇, are defined primarily in Cartesian coordinates but extend to other systems like cylindrical and spherical coordinates, facilitating computations in diverse geometric contexts.[71] The gradient of a scalar field f(x, y, z), denoted \nabla f, is a vector field that points in the direction of the steepest ascent of f and whose magnitude |\nabla f| equals the rate of that ascent.[72] In Cartesian coordinates, it is expressed as \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right). This operator transforms a scalar into a vector, highlighting directional derivatives aligned with the field's increase.[72] The divergence of a vector field \vec{F} = (F_x, F_y, F_z), denoted \nabla \cdot \vec{F}, quantifies the net flux emanating from a point, positive for sources and negative for sinks, representing the rate at which the field's density exits a local volume.[73] In Cartesian coordinates, it takes the form \nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}. This scalar operator measures expansion or contraction within the field.[73] The curl of a vector field \vec{F}, denoted \nabla \times \vec{F}, is a vector field whose magnitude indicates the maximum rotation (circulation per unit area) at a point and whose direction aligns with the axis of that rotation, following the right-hand rule.[74] A field is irrotational if \nabla \times \vec{F} = \vec{0}. In Cartesian coordinates, it is given by \nabla \times \vec{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right). This antisymmetric operator detects local vorticity.[74] The Laplacian of a scalar field f, denoted \nabla^2 f or \Delta f, is the divergence of the gradient, \nabla \cdot (\nabla f), and serves as a measure of the field's variation or diffusivity, appearing in equations for heat, waves, and potentials.[75] Functions satisfying \nabla^2 f = 0 are harmonic, exhibiting mean-value properties over spheres. In Cartesian coordinates, \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}. This second-order scalar operator is fundamental in many physical laws.[75] These operators adapt to curvilinear coordinates for problems with symmetry. The table below summarizes their expressions in Cartesian, cylindrical (r, \theta, z), and spherical (r, \theta, \phi) coordinates, where scale factors account for the geometry.[76][77]| Operator | Cartesian (x, y, z) | Cylindrical (r, \theta, z) | Spherical (r, \theta, \phi) |
|---|---|---|---|
| Gradient \nabla f | \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) | \left( \frac{\partial f}{\partial r}, \frac{1}{r} \frac{\partial f}{\partial \theta}, \frac{\partial f}{\partial z} \right) | \left( \frac{\partial f}{\partial r}, \frac{1}{r} \frac{\partial f}{\partial \theta}, \frac{1}{r \sin \theta} \frac{\partial f}{\partial \phi} \right) |
| Divergence \nabla \cdot \vec{F} | \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} | \frac{1}{r} \frac{\partial (r F_r)}{\partial r} + \frac{1}{r} \frac{\partial F_\theta}{\partial \theta} + \frac{\partial F_z}{\partial z} | \frac{1}{r^2} \frac{\partial (r^2 F_r)}{\partial r} + \frac{1}{r \sin \theta} \frac{\partial (F_\theta \sin \theta)}{\partial \theta} + \frac{1}{r \sin \theta} \frac{\partial F_\phi}{\partial \phi} |
| Curl \nabla \times \vec{F} | \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) | \left( \frac{1}{r} \frac{\partial F_z}{\partial \theta} - \frac{\partial F_\theta}{\partial z}, \frac{\partial F_r}{\partial z} - \frac{\partial F_z}{\partial r}, \frac{1}{r} \left( \frac{\partial (r F_\theta)}{\partial r} - \frac{\partial F_r}{\partial \theta} \right) \right) | \frac{1}{r \sin \theta} \left( \frac{\partial (F_\phi \sin \theta)}{\partial \theta} - \frac{\partial F_\theta}{\partial \phi} \right) \hat{r} + \frac{1}{r} \left( \frac{1}{\sin \theta} \frac{\partial F_r}{\partial \phi} - \frac{\partial (r F_\phi)}{\partial r} \right) \hat{\theta} + \frac{1}{r} \left( \frac{\partial (r F_\theta)}{\partial r} - \frac{\partial F_r}{\partial \theta} \right) \hat{\phi} |
| Laplacian \nabla^2 f | \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} | \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial f}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 f}{\partial \theta^2} + \frac{\partial^2 f}{\partial z^2} | \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial f}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \phi^2} |