N-sphere

Last updated
2-sphere wireframe as an orthogonal projection Sphere wireframe.svg
2-sphere wireframe as an orthogonal projection
Just as a stereographic projection can project a sphere's surface to a plane, it can also project a 3-sphere into 3-space. This image shows three coordinate directions projected to 3-space: parallels (red), meridians (blue), and hypermeridians (green). Due to the conformal property of the stereographic projection, the curves intersect each other orthogonally (in the yellow points) as in 4D. All of the curves are circles: the curves that intersect <0,0,0,1>  have an infinite radius (= straight line). Hypersphere coord.PNG
Just as a stereographic projection can project a sphere's surface to a plane, it can also project a 3-sphere into 3-space. This image shows three coordinate directions projected to 3-space: parallels (red), meridians (blue), and hypermeridians (green). Due to the conformal property of the stereographic projection, the curves intersect each other orthogonally (in the yellow points) as in 4D. All of the curves are circles: the curves that intersect 0,0,0,1 have an infinite radius (= straight line).

In mathematics, an n-sphere or hypersphere is an n-dimensional generalization of the 1-dimensional circle and 2-dimensional sphere to any non-negative integer n. The n-sphere is the setting for n-dimensional spherical geometry.

Contents

Considered extrinsically, as a hypersurface embedded in (n+1)-dimensional Euclidean space, an n-sphere is the locus of points at equal distance (the radius ) from a given center point. Its interior, consisting of all points closer to the center than the radius, is an (n+1)-dimensional ball. In particular:

Given a Cartesian coordinate system, the unit n-sphere of radius 1 can be defined as:

Considered intrinsically, when n ≥ 1, the n-sphere is a Riemannian manifold of positive constant curvature, and is orientable. The geodesics of the n-sphere are called great circles.

The stereographic projection maps the n-sphere onto n-space with a single adjoined point at infinity; under the metric thereby defined, is a model for the n-sphere.

In the more general setting of topology, any topological space that is homeomorphic to the unit n-sphere is called an n-sphere. Under inverse stereographic projection, the n-sphere is the one-point compactification of n-space. The n-spheres admit several other topological descriptions: for example, they can be constructed by gluing two n-dimensional spaces together, by identifying the boundary of an n-cube with a point, or (inductively) by forming the suspension of an (n1)-sphere. When n ≥ 2 it is simply connected; the 1-sphere (circle) is not simply connected; the 0-sphere is not even connected, consisting of two discrete points.

Description

For any natural number n, an n-sphere of radius r is defined as the set of points in (n + 1)-dimensional Euclidean space that are at distance r from some fixed point c, where r may be any positive real number and where c may be any point in (n + 1)-dimensional space. In particular:

Cartesian coordinates

The set of points in (n + 1)-space, (x1, x2, ..., xn+1), that define an n-sphere, Sn(r), is represented by the equation:

where c = (c1, c2, ..., cn+1) is a center point, and r is the radius.

The above n-sphere exists in (n + 1)-dimensional Euclidean space and is an example of an n-manifold. The volume form ω of an n-sphere of radius r is given by

where is the Hodge star operator; see Flanders (1989 , §6.1) for a discussion and proof of this formula in the case r = 1. As a result,

n-ball

The space enclosed by an n-sphere is called an (n + 1)-ball. An (n + 1)-ball is closed if it includes the n-sphere, and it is open if it does not include the n-sphere.

Specifically:

Topological description

Topologically, an n-sphere can be constructed as a one-point compactification of n-dimensional Euclidean space. Briefly, the n-sphere can be described as Sn = ℝn ∪ {∞}, which is n-dimensional Euclidean space plus a single point representing infinity in all directions. In particular, if a single point is removed from an n-sphere, it becomes homeomorphic to n. This forms the basis for stereographic projection. [1]

Volume and area

Let Sn−1 be the surface area of the unit (n1)-sphere of radius 1 embedded in n-dimensional Euclidean space, and let Vn be the volume of its interior, the unit n-ball. The surface area of an arbitrary (n1)-sphere is proportional to the (n1)st power of the radius, and the volume of an arbitrary n-ball is proportional to the nth power of the radius.

Graphs of volumes (Vn) and surface areas (Sn-1) of n-balls of radius 1. Hypersphere volume and surface area graphs.svg
Graphs of volumes (Vn) and surface areas (Sn−1) of n-balls of radius 1.

The 0-ball is sometimes defined as a single point. The 0-dimensional Hausdorff measure is the number of points in a set. So

A unit 1-ball is a line segment whose points have a single coordinate in the interval [−1, 1] of length 2, and the 0-sphere consists of its two end-points, with coordinate {−1, 1}.

A unit 1-sphere is the unit circle in the Euclidean plane, and its interior is the unit disk (2-ball).

The interior of a 2-sphere in three-dimensional space is the unit 3-ball.

In general, Sn−1 and Vn are given in closed form by the expressions

where Γ is the gamma function.

As n tends to infinity, the volume of the unit n-ball (ratio between the volume of an n-ball of radius 1 and an n-cube of side length 1) tends to zero. [2]

Recurrences

The surface area, or properly the n-dimensional volume, of the n-sphere at the boundary of the (n + 1)-ball of radius R is related to the volume of the ball by the differential equation

Equivalently, representing the unit n-ball as a union of concentric (n − 1)-sphere shells ,

We can also represent the unit (n + 2)-sphere as a union of products of a circle (1-sphere) with an n-sphere. Then Since S1 = 2π V0, the equation

holds for all n. Along with the base cases from above, these recurrences can be used to compute the surface area of any sphere or volume of any ball.

Spherical coordinates

We may define a coordinate system in an n-dimensional Euclidean space which is analogous to the spherical coordinate system defined for 3-dimensional Euclidean space, in which the coordinates consist of a radial coordinate r, and n − 1 angular coordinates φ1, φ2, ..., φn−1, where the angles φ1, φ2, ..., φn−2 range over [0, π] radians (or over [0, 180] degrees) and φn−1 ranges over [0, 2π) radians (or over [0, 360) degrees). If xi are the Cartesian coordinates, then we may compute x1, ..., xn from r, φ1, ..., φn−1 with: [3]

Except in the special cases described below, the inverse transformation is unique:

where atan2 is the two-argument arctangent function.

There are some special cases where the inverse transform is not unique; φk for any k will be ambiguous whenever all of xk, xk+1, ... xn are zero; in this case φk may be chosen to be zero. (For example, for the 2-sphere, when the polar angle is 0 or π then the point is one of the poles, zenith or nadir, and the choice of azimuthal angle is arbitrary.)

Spherical volume and area elements

To express the volume element of n-dimensional Euclidean space in terms of spherical coordinates, let and for concision, then observe that the Jacobian matrix of the transformation is:

The determinant of this matrix can be calculated by induction. When n = 2, a straightforward computation shows that the determinant is r. For larger n, observe that Jn can be constructed from Jn1 as follows. Except in column n, rows n 1 and n of Jn are the same as row n 1 of Jn1, but multiplied by an extra factor of cos φn1 in row n 1 and an extra factor of sin φn1 in row n. In column n, rows n 1 and n of Jn are the same as column n 1 of row n 1 of Jn1, but multiplied by extra factors of sin φn1 in row n 1 and cos φn1 in row n, respectively. The determinant of Jn can be calculated by Laplace expansion in the final column. By the recursive description of Jn, the submatrix formed by deleting the entry at (n 1, n) and its row and column almost equals Jn1, except that its last row is multiplied by sin φn1. Similarly, the submatrix formed by deleting the entry at (n, n) and its row and column almost equals Jn1, except that its last row is multiplied by cos φn1. Therefore the determinant of Jn is

Induction then gives a closed-form expression for the volume element in spherical coordinates

The formula for the volume of the n-ball can be derived from this by integration.

Similarly the surface area element of the (n − 1)-sphere of radius R, which generalizes the area element of the 2-sphere, is given by

The natural choice of an orthogonal basis over the angular coordinates is a product of ultraspherical polynomials,

for j = 1, 2, ..., n − 2, and the eisφj for the angle j = n − 1 in concordance with the spherical harmonics.

Polyspherical coordinates

The standard spherical coordinate system arises from writing n as the product ℝ × ℝn1. These two factors may be related using polar coordinates. For each point x of n, the standard Cartesian coordinates

can be transformed into a mixed polar–Cartesian coordinate system:

This says that points in n may be expressed by taking the ray starting at the origin and passing through , rotating it towards by , and traveling a distance along the ray. Repeating this decomposition eventually leads to the standard spherical coordinate system.

Polyspherical coordinate systems arise from a generalization of this construction. [4] The space n is split as the product of two Euclidean spaces of smaller dimension, but neither space is required to be a line. Specifically, suppose that p and q are positive integers such that n = p + q. Then n = ℝp×q. Using this decomposition, a point x ∈ ℝn may be written as

This can be transformed into a mixed polar–Cartesian coordinate system by writing:

Here and are the unit vectors associated to y and z. This expresses x in terms of , , r ≥ 0, and an angle θ. It can be shown that the domain of θ is [0, 2π) if p = q = 1, [0, π] if exactly one of p and q is 1, and [0, π/2] if neither p nor q are 1. The inverse transformation is

These splittings may be repeated as long as one of the factors involved has dimension two or greater. A polyspherical coordinate system is the result of repeating these splittings until there are no Cartesian coordinates left. Splittings after the first do not require a radial coordinate because the domains of and are spheres, so the coordinates of a polyspherical coordinate system are a non-negative radius and n 1 angles. The possible polyspherical coordinate systems correspond to binary trees with n leaves. Each non-leaf node in the tree corresponds to a splitting and determines an angular coordinate. For instance, the root of the tree represents n, and its immediate children represent the first splitting into p and q. Leaf nodes correspond to Cartesian coordinates for Sn1. The formulas for converting from polyspherical coordinates to Cartesian coordinates may be determined by finding the paths from the root to the leaf nodes. These formulas are products with one factor for each branch taken by the path. For a node whose corresponding angular coordinate is θi, taking the left branch introduces a factor of sin θi and taking the right branch introduces a factor of cos θi. The inverse transformation, from polyspherical coordinates to Cartesian coordinates, is determined by grouping nodes. Every pair of nodes having a common parent can be converted from a mixed polar–Cartesian coordinate system to a Cartesian coordinate system using the above formulas for a splitting.

Polyspherical coordinates also have an interpretation in terms of the special orthogonal group. A splitting n = ℝp×q determines a subgroup

This is the subgroup that leaves each of the two factors fixed. Choosing a set of coset representatives for the quotient is the same as choosing representative angles for this step of the polyspherical coordinate decomposition.

In polyspherical coordinates, the volume measure on n and the area measure on Sn1 are products. There is one factor for each angle, and the volume measure on n also has a factor for the radial coordinate. The area measure has the form:

where the factors Fi are determined by the tree. Similarly, the volume measure is

Suppose we have a node of the tree that corresponds to the decomposition n1+n2 = ℝn1 × ℝn2 and that has angular coordinate θ. The corresponding factor F depends on the values of n1 and n2. When the area measure is normalized so that the area of the sphere is 1, these factors are as follows. If n1 = n2 = 1, then

If n1 > 1 and n2 = 1, and if B denotes the beta function, then

If n1 = 1 and n2 > 1, then

Finally, if both n1 and n2 are greater than one, then

Stereographic projection

Just as a two-dimensional sphere embedded in three dimensions can be mapped onto a two-dimensional plane by a stereographic projection, an n-sphere can be mapped onto an n-dimensional hyperplane by the n-dimensional version of the stereographic projection. For example, the point [x,y,z] on a two-dimensional sphere of radius 1 maps to the point [x/1 − z, y/1 − z] on the xy-plane. In other words,

Likewise, the stereographic projection of an n-sphere Sn of radius 1 will map to the (n − 1)-dimensional hyperplane n−1 perpendicular to the xn-axis as

Probability distributions

Uniformly at random on the (n − 1)-sphere

A set of points drawn from a uniform distribution on the surface of a unit 2-sphere, generated using Marsaglia's algorithm. 2sphere-uniform.png
A set of points drawn from a uniform distribution on the surface of a unit 2-sphere, generated using Marsaglia's algorithm.

To generate uniformly distributed random points on the unit (n − 1)-sphere (that is, the surface of the unit n-ball), Marsaglia (1972) gives the following algorithm.

Generate an n-dimensional vector of normal deviates (it suffices to use N(0, 1), although in fact the choice of the variance is arbitrary), x = (x1, x2, ..., xn). Now calculate the "radius" of this point:

The vector 1/rx is uniformly distributed over the surface of the unit n-ball.

An alternative given by Marsaglia is to uniformly randomly select a point x = (x1, x2, ..., xn) in the unit n-cube by sampling each xi independently from the uniform distribution over (–1, 1), computing r as above, and rejecting the point and resampling if r ≥ 1 (i.e., if the point is not in the n-ball), and when a point in the ball is obtained scaling it up to the spherical surface by the factor 1/r; then again 1/rx is uniformly distributed over the surface of the unit n-ball. This method becomes very inefficient for higher dimensions, as a vanishingly small fraction of the unit cube is contained in the sphere. In ten dimensions, less than 2% of the cube is filled by the sphere, so that typically more than 50 attempts will be needed. In seventy dimensions, less than of the cube is filled, meaning typically a trillion quadrillion trials will be needed, far more than a computer could ever carry out.

Uniformly at random within the n-ball

With a point selected uniformly at random from the surface of the unit (n − 1)-sphere (e.g., by using Marsaglia's algorithm), one needs only a radius to obtain a point uniformly at random from within the unit n-ball. If u is a number generated uniformly at random from the interval [0, 1] and x is a point selected uniformly at random from the unit (n − 1)-sphere, then u1/nx is uniformly distributed within the unit n-ball.

Alternatively, points may be sampled uniformly from within the unit n-ball by a reduction from the unit (n + 1)-sphere. In particular, if (x1, x2, ..., xn+2) is a point selected uniformly from the unit (n + 1)-sphere, then (x1, x2, ..., xn) is uniformly distributed within the unit n-ball (i.e., by simply discarding two coordinates). [5]

If n is sufficiently large, most of the volume of the n-ball will be contained in the region very close to its surface, so a point selected from that volume will also probably be close to the surface. This is one of the phenomena leading to the so-called curse of dimensionality that arises in some numerical and other applications.

Distribution of the first coordinate

Let be the square of the first coordinate of a point sampled uniformly at random from the (n-1)-sphere, then its probability density function, for , is

Let be the appropriately scaled version, then at the limit, the probability density function of converges to . This is sometimes called the Porter–Thomas distribution. [6]

Specific spheres

0-sphere
The pair of points R} with the discrete topology for some R > 0. The only sphere that is not path-connected. Parallelizable.
1-sphere
Commonly called a circle. Has a nontrivial fundamental group. Abelian Lie group structure U(1) ; the circle group. Homeomorphic to the real projective line.
2-sphere
Commonly simply called a sphere. For its complex structure, see Riemann sphere . Homeomorphic to the complex projective line
3-sphere
Parallelizable, principal U(1)-bundle over the 2-sphere, Lie group structure Sp(1) .
4-sphere
Homeomorphic to the quaternionic projective line, HP1. SO(5) / SO(4).
5-sphere
Principal U(1)-bundle over CP2. SO(6) / SO(5) = SU(3) / SU(2). It is undecidable whether a given n-dimensional manifold is homeomorphic to Sn for n ≥ 5. [7]
6-sphere
Possesses an almost complex structure coming from the set of pure unit octonions. SO(7) / SO(6) = G2 / SU(3). The question of whether it has a complex structure is known as the Hopf problem, after Heinz Hopf. [8]
7-sphere
Topological quasigroup structure as the set of unit octonions. Principal Sp(1)-bundle over S4. Parallelizable. SO(8) / SO(7) = SU(4) / SU(3) = Sp(2) / Sp(1) = Spin(7) / G2 = Spin(6) / SU(3). The 7-sphere is of particular interest since it was in this dimension that the first exotic spheres were discovered.
8-sphere
Homeomorphic to the octonionic projective line OP1.
23-sphere
A highly dense sphere-packing is possible in 24-dimensional space, which is related to the unique qualities of the Leech lattice.

Octahedral sphere

The octahedral n-sphere is defined similarly to the n-sphere but using the 1-norm

In general, it takes the shape of a cross-polytope.

The octahedral 1-sphere is a square (without its interior). The octahedral 2-sphere is a regular octahedron; hence the name. The octahedral n-sphere is the topological join of n + 1 pairs of isolated points. [9] Intuitively, the topological join of two pairs is generated by drawing a segment between each point in one pair and each point in the other pair; this yields a square. To join this with a third pair, draw a segment between each point on the square and each point in the third pair; this gives a octahedron.

See also

Notes

  1. James W. Vick (1994). Homology theory, p. 60. Springer
  2. Smith, David J.; Vamanamurthy, Mavina K. (1989). "How Small Is a Unit Ball?". Mathematics Magazine. 62 (2): 101–107. doi:10.1080/0025570X.1989.11977419. JSTOR   2690391.
  3. Blumenson, L. E. (1960). "A Derivation of n-Dimensional Spherical Coordinates". The American Mathematical Monthly. 67 (1): 63–66. doi:10.2307/2308932. JSTOR   2308932.
  4. N. Ja. Vilenkin and A. U. Klimyk, Representation of Lie groups and special functions, Vol. 2: Class I representations, special functions, and integral transforms, translated from the Russian by V. A. Groza and A. A. Groza, Math. Appl., vol. 74, Kluwer Acad. Publ., Dordrecht, 1992, ISBN   0-7923-1492-1, pp. 223–226.
  5. Voelker, Aaron R.; Gosmann, Jan; Stewart, Terrence C. (2017). Efficiently sampling vectors and coordinates from the n-sphere and n-ball (Report). Centre for Theoretical Neuroscience. doi:10.13140/RG.2.2.15829.01767/1.
  6. Livan, Giacomo; Novaes, Marcel; Vivo, Pierpaolo (2018), Livan, Giacomo; Novaes, Marcel; Vivo, Pierpaolo (eds.), "One Pager on Eigenvectors", Introduction to Random Matrices: Theory and Practice, SpringerBriefs in Mathematical Physics, Cham: Springer International Publishing, pp. 65–66, doi:10.1007/978-3-319-70885-0_9, ISBN   978-3-319-70885-0 , retrieved 2023-05-19
  7. Stillwell, John (1993), Classical Topology and Combinatorial Group Theory, Graduate Texts in Mathematics, vol. 72, Springer, p. 247, ISBN   9780387979700 .
  8. Agricola, Ilka; Bazzoni, Giovanni; Goertsches, Oliver; Konstantis, Panagiotis; Rollenske, Sönke (2018). "On the history of the Hopf problem". Differential Geometry and Its Applications. 57: 1–9. arXiv: 1708.01068 . doi:10.1016/j.difgeo.2017.10.014. S2CID   119297359.
  9. Meshulam, Roy (2001-01-01). "The Clique Complex and Hypergraph Matching". Combinatorica. 21 (1): 89–94. doi:10.1007/s004930170006. ISSN   1439-6912. S2CID   207006642.

Related Research Articles

<span class="mw-page-title-main">Polar coordinate system</span> Coordinates comprising a distance and an angle

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. Angles in polar notation are generally expressed in either degrees or radians.

<span class="mw-page-title-main">Spherical coordinate system</span> 3-dimensional coordinate system

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a given point in space is specified by three numbers, : the radial distance of the radial liner connecting the point to the fixed point of origin ; the polar angle θ of the radial line r; and the azimuthal angle φ of the radial line r.

<span class="mw-page-title-main">Laplace's equation</span> Second-order partial differential equation

In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as

<span class="mw-page-title-main">3-sphere</span> Mathematical object

In mathematics, a 3-sphere, glome or hypersphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensions is an ordinary sphere, the boundary of a ball in four dimensions is a 3-sphere. A 3-sphere is an example of a 3-manifold and an n-sphere.

<span class="mw-page-title-main">Navier–Stokes equations</span> Equations describing the motion of viscous fluid substances

The Navier–Stokes equations are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).

<span class="mw-page-title-main">Ellipsoid</span> Quadric surface that looks like a deformed sphere

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

<span class="mw-page-title-main">Unit vector</span> Vector of length one

In mathematics, a unit vector in a normed vector space is a vector of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in .

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition.

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols , (where is the nabla operator), or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δf (p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f (p).

Unit quaternions, known as versors, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation about an arbitrary axis. Rotation and orientation quaternions have applications in computer graphics, computer vision, robotics, navigation, molecular dynamics, flight dynamics, orbital mechanics of satellites, and crystallographic texture analysis.

In vector calculus, the Jacobian matrix of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and the determinant are often referred to simply as the Jacobian in literature.

<span class="mw-page-title-main">Spherical harmonics</span> Special mathematical functions defined on the surface of a sphere

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. A list of the spherical harmonics is available in Table of spherical harmonics.

<span class="mw-page-title-main">Bloch sphere</span> Geometrical representation of the pure state space of a two-level quantum mechanical system

In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch.

In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface. In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric. However, when a metric is available, these concepts can be directly tied to the "shape" of the manifold itself; that shape is determined by how the tangent space is attached to the cotangent space by the metric tensor. Abstractly, one would say that the manifold has an associated (orthonormal) frame bundle, with each "frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle is the orthogonal group O(p, q). As a result, such a manifold is necessarily a (pseudo-)Riemannian manifold. The Christoffel symbols provide a concrete representation of the connection of (pseudo-)Riemannian geometry in terms of coordinates on the manifold. Additional concepts, such as parallel transport, geodesics, etc. can then be expressed in terms of Christoffel symbols.

In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body. This theorem has particular application to astronomy.

In calculus, the Leibniz integral rule for differentiation under the integral sign states that for an integral of the form

<span class="mw-page-title-main">Multiple integral</span> Generalization of definite integrals to functions of multiple variables

In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Physical (natural philosophy) interpretation: S any surface, V any volume, etc.. Incl. variable to time, position, etc.

In geometry, various formalisms exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.

In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.

In fluid dynamics, the Oseen equations describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.

References