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Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s'') are required to determine the position of an element (i.e., point). This is the informal meaning of the term
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
. In mathematics, a tuple of
numbers A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
can be understood as the Cartesian coordinates of a location in a -dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
. The set of these -tuples is commonly denoted \R^n, and can be identified to the -dimensional Euclidean space. When , this space is called three-dimensional Euclidean space (or simply Euclidean space when the context is clear). It serves as a model of the physical
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. ...
(when relativity theory is not considered), in which all known
matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic part ...
exists. While this space remains the most compelling and useful way to model the world as it is experienced, it is only one example of a large variety of spaces in three dimensions called
3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
s. In this classical example, when the three values refer to measurements in different directions ( coordinates), any three directions can be chosen, provided that vectors in these directions do not all lie in the same 2-space ( plane). Furthermore, in this case, these three values can be labeled by any combination of three chosen from the terms '' width/breadth'', ''
height Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For example, "The height of that building is 50 m" or "The height of an airplane in-flight is ab ...
/depth'', and '' length''.


History

Books XI to XIII of
Euclid's Elements The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postulat ...
dealt with three-dimensional geometry. Book XI develops notions of orthogonality and parallelism of lines and planes, and defines solids including parallelpipeds, pyramids, prisms, spheres, octahedra, icosahedra and dodecahedra. Book XII develops notions of similarity of solids. Book XIII describes the construction of the five regular Platonic solids in a sphere. In the 17th century, three-dimensional space was described with Cartesian coordinates, with the advent of analytic geometry developed by
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...
in his work '' La Géométrie'' and
Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
in the manuscript ''Ad locos planos et solidos isagoge'' (Introduction to Plane and Solid Loci), which was unpublished during Fermat's lifetime. However, only Fermat's work dealt with three-dimensional space. In the 19th century, developments of the geometry of three-dimensional space came with William Rowan Hamilton's development of the quaternions. In fact, it was Hamilton who coined the terms scalar and
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
, and they were first defined within his geometric framework for quaternions. Three dimensional space could then be described by quaternions q = a + ui + vj + wk which had vanishing scalar component, that is, a = 0. While not explicitly studied by Hamilton, this indirectly introduced notions of basis, here given by the quaternion elements i,j,k, as well as the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
and cross product, which correspond to (the negative of) the scalar part and the vector part of the product of two vector quaternions. It wasn't until Josiah Willard Gibbs that these two products were identified in their own right, and the modern notation for the dot and cross product were introduced in his classroom teaching notes, found also in the 1901 textbook '' Vector Analysis'' written by
Edwin Bidwell Wilson Edwin Bidwell Wilson (April 25, 1879 – December 28, 1964) was an American mathematician, statistician, physicist and general polymath. He was the sole protégé of Yale University physicist Josiah Willard Gibbs and was mentor to MIT economist ...
based on Gibbs' lectures. Also during the 19th century came developments in the abstract formalism of vector spaces, with the work of
Hermann Grassmann Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mat ...
and
Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The sta ...
, the latter of whom first gave the modern definition of vector spaces as an algebraic structure.


In Euclidean geometry


Coordinate systems

In mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, each perpendicular to the other two at the origin, the point at which they cross. They are usually labeled , and . Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes. Other popular methods of describing the location of a point in three-dimensional space include
cylindrical coordinates A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference d ...
and
spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
, though there are an infinite number of possible methods. For more, see
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
. Below are images of the above-mentioned systems. Image:Coord XYZ.svg, Cartesian coordinate system Image:Cylindrical Coordinates.svg,
Cylindrical coordinate system A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
Image:Spherical Coordinates (Colatitude, Longitude).svg, Spherical coordinate system


Lines and planes

Two distinct points always determine a (straight) line. Three distinct points are either
collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...
or determine a unique plane. On the other hand, four distinct points can either be collinear,
coplanar In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. How ...
, or determine the entire space. Two distinct lines can either intersect, be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a unique plane, so skew lines are lines that do not meet and do not lie in a common plane. Two distinct planes can either meet in a common line or are parallel (i.e., do not meet). Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point, or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel. A line can lie in a given plane, intersect that plane in a unique point, or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line. A hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a three-dimensional space are the two-dimensional subspaces, that is, the planes. In terms of Cartesian coordinates, the points of a hyperplane satisfy a single linear equation, so planes in this 3-space are described by linear equations. A line can be described by a pair of independent linear equations—each representing a plane having this line as a common intersection.
Varignon's theorem Varignon's theorem is a statement in Euclidean geometry, that deals with the construction of a particular parallelogram, the Varignon parallelogram, from an arbitrary quadrilateral (quadrangle). It is named after Pierre Varignon, whose proof was ...
states that the midpoints of any quadrilateral in ℝ3 form a parallelogram, and hence are coplanar.


Spheres and balls

A
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
in 3-space (also called a 2-sphere because it is a 2-dimensional object) consists of the set of all points in 3-space at a fixed distance from a central point . The solid enclosed by the sphere is called a ball (or, more precisely a 3-ball). The volume of the ball is given by V = \frac\pi r^, and the surface area of the sphere is A = 4\pi r^2. Another type of sphere arises from a 4-ball, whose three-dimensional surface is the 3-sphere: points equidistant to the origin of the euclidean space . If a point has coordinates, , then characterizes those points on the unit 3-sphere centered at the origin. This 3-sphere is an example of a 3-manifold: a space which is 'looks locally' like 3D space. In precise topological terms, each point of the 3-sphere has a neighborhood which is homeomorphic to an open subset of 3D space.


Polytopes

In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra.


Surfaces of revolution

A
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
generated by revolving a plane
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
about a fixed line in its plane as an axis is called a
surface of revolution A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on ...
. The plane curve is called the '' generatrix'' of the surface. A section of the surface, made by intersecting the surface with a plane that is perpendicular (orthogonal) to the axis, is a circle. Simple examples occur when the generatrix is a line. If the generatrix line intersects the axis line, the surface of revolution is a right circular
cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...
with vertex (apex) the point of intersection. However, if the generatrix and axis are parallel, then the surface of revolution is a circular cylinder.


Quadric surfaces

In analogy with the
conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...
s, the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely, :Ax^2 + By^2 + Cz^2 + Fxy + Gyz + Hxz + Jx + Ky + Lz + M = 0, where and are real numbers and not all of and are zero, is called a quadric surface. There are six types of non-degenerate quadric surfaces: # Ellipsoid #
Hyperboloid of one sheet In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...
# Hyperboloid of two sheets # Elliptic cone # Elliptic paraboloid # Hyperbolic paraboloid The degenerate quadric surfaces are the empty set, a single point, a single line, a single plane, a pair of planes or a quadratic cylinder (a surface consisting of a non-degenerate conic section in a plane and all the lines of through that conic that are normal to ). Elliptic cones are sometimes considered to be degenerate quadric surfaces as well. Both the hyperboloid of one sheet and the hyperbolic paraboloid are
ruled surface In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, t ...
s, meaning that they can be made up from a family of straight lines. In fact, each has two families of generating lines, the members of each family are disjoint and each member one family intersects, with just one exception, every member of the other family. Each family is called a regulus.


In linear algebra

Another way of viewing three-dimensional space is found in
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...
, where the idea of independence is crucial. Space has three dimensions because the length of a box is independent of its width or breadth. In the technical language of linear algebra, space is three-dimensional because every point in space can be described by a linear combination of three independent
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
s.


Dot product, angle, and length

A vector can be pictured as an arrow. The vector's magnitude is its length, and its direction is the direction the arrow points. A vector in can be represented by an ordered triple of real numbers. These numbers are called the components of the vector. The dot product of two vectors and is defined as: :\mathbf\cdot \mathbf = A_1B_1 + A_2B_2 + A_3B_3 = \sum_^3 A_i B_i. The magnitude of a vector is denoted by . The dot product of a vector with itself is :\mathbf A\cdot\mathbf A = \, \mathbf A\, ^2 = A_1^2 + A_2^2 + A_3^2, which gives : \, \mathbf A\, = \sqrt = \sqrt, the formula for the Euclidean length of the vector. Without reference to the components of the vectors, the dot product of two non-zero Euclidean vectors and is given by :\mathbf A\cdot\mathbf B = \, \mathbf A\, \,\, \mathbf B\, \cos\theta, where is the
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...
between and .


Cross product

The cross product or vector product is a binary operation on two
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
s in three-dimensional space and is denoted by the symbol ×. The cross product A × B of the vectors A and B is a vector that is
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It ca ...
to both and therefore
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
to the plane containing them. It has many applications in mathematics,
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, and
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
. In function language, the cross product is a function \times: \mathbb^3 \times \mathbb^3 \rightarrow \mathbb^3. The components of the cross product are \mathbf\times\mathbf = _2B_3 - B_2A_3, A_3B_1 - B_3A_1, A_1B_2 - B_1A_2/math>, and can also be written in components, using Einstein summation convention as (\mathbf\times\mathbf)_i = \epsilon_A_jB_k where \epsilon_ is the
Levi-Civita symbol In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the parity of a permutation, sign of a permutation of the n ...
. It has the property that \mathbf\times \mathbf = -\mathbf\times \mathbf. Its magnitude is related to the angle \theta between \mathbf and \mathbf by the identity , , \mathbf\times \mathbf, , = , , \mathbf, , \cdot, , \mathbf, , \cdot , \sin\theta, . The space and product form an algebra over a field, which is not
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
nor associative, but is a Lie algebra with the cross product being the Lie bracket. Specifically, the space together with the product, (\mathbb^3,\times) is isomorphic to the Lie algebra of three-dimensional rotations, denoted \mathfrak(3). In order to satisfy the axioms of a Lie algebra, instead of associativity the cross product satisfies the
Jacobi identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
. For any three vectors \mathbf, \mathbf and \mathbf \mathbf\times(\mathbf\times\mathbf) + \mathbf\times(\mathbf\times\mathbf) + \mathbf\times(\mathbf\times\mathbf) = 0 One can in ''n'' dimensions take the product of vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions.


Abstract description

It can be useful to describe three-dimensional space as a three-dimensional vector space V over the real numbers. This differs from \mathbb^3 in a subtle way. By definition, there exists a basis \mathcal = \ for V. This corresponds to an isomorphism between V and \mathbb^3: the construction for the isomorphism is found here. However, there is no 'preferred' or 'canonical basis' for V. On the other hand, there is a preferred basis for \mathbb^3, which is due to its description as a Cartesian product of copies of \mathbb, that is, \mathbb^3 = \mathbb\times \mathbb\times \mathbb. This allows the definition of canonical projections, \pi_i:\mathbb^3 \rightarrow \mathbb, where 1 \leq i \leq 3. For example, \pi_1(x_1,x_2,x_3) = x. This then allows the definition of the standard basis \mathcal_ = \ defined by \pi_i(E_j) = \delta_ where \delta_ is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
. Written out in full, the standard basis is E_1 = \begin1 \\ 0\\ 0\end, E_2 = \begin0 \\ 1\\ 0\end, E_3 = \begin0 \\ 0\\ 1\end. Therefore \mathbb^3 can be viewed as the abstract vector space, together with the additional structure of a choice of basis. Conversely, V can be obtained by starting with \mathbb^3 and 'forgetting' the Cartesian product structure, or equivalently the standard choice of basis. As opposed to a general vector space V, the space \mathbb^3 is sometimes referred to as a coordinate space. Physically, it is conceptually desirable to use the abstract formalism in order to assume as little structure as possible if it is not given by the parameters of a particular problem. For example, in a problem with rotational symmetry, working with the more concrete description of three-dimensional space \mathbb^3 assumes a choice of basis, corresponding to a set of axes. But in rotational symmetry, there is no reason why one set of axes is preferred to say, the same set of axes which has been rotated arbitrarily. Stated another way, a preferred choice of axes breaks the rotational symmetry of physical space. Computationally, it is necessary to work with the more concrete description \mathbb^3 in order to do concrete computations.


Affine description

A more abstract description still is to model physical space as a three-dimensional affine space E(3) over the real numbers. This is unique up to affine isomorphism. It is sometimes referred to as three-dimensional Euclidean space. Just as the vector space description came from 'forgetting the preferred basis' of \mathbb^3, the affine space description comes from 'forgetting the origin' of the vector space. Euclidean spaces are sometimes called ''Euclidean affine spaces'' for distinguishing them from Euclidean vector spaces. This is physically appealing as it makes the translation invariance of physical space manifest. A preferred origin breaks the translational invariance.


Inner product space

The above discussion does not involve the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
. The dot product is an example of an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
. Physical space can be modelled as a vector space which additionally has the structure of an inner product. The inner product defines notions of length and angle (and therefore in particular the notion of orthogonality). For any inner product, there exist bases under which the inner product agrees with the dot product, but again, there are many different possible bases, none of which are preferred. They differ from one another by a rotation, an element of the group of rotations
SO(3) 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 \R^3 under the operation of composition. By definition, a rotation about the origin is a tr ...
.


In calculus


Gradient, divergence and curl

In a rectangular coordinate system, the gradient of a (differentiable) function f: \mathbb^3 \rightarrow \mathbb is given by :\nabla f = \frac \mathbf + \frac \mathbf + \frac \mathbf and in index notation is written (\nabla f)_i = \partial_i f. The divergence of a (differentiable) vector field F = ''U'' i + ''V'' j + ''W'' k, that is, a function \mathbf:\mathbb^3 \rightarrow \mathbb^3, is equal to the scalar-valued function: :\operatorname\,\mathbf = \nabla\cdot\mathbf =\frac +\frac +\frac. In index notation, with Einstein summation convention this is \nabla \cdot \mathbf = \partial_i F_i. Expanded in Cartesian coordinates (see
Del in cylindrical and spherical coordinates This is a list of some vector calculus formulae for working with common curvilinear coordinate systems. Notes * This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reve ...
for
spherical A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
and cylindrical coordinate representations), the curl ∇ × F is, for F composed of 'F''x, ''F''y, ''F''z :\begin \mathbf & \mathbf & \mathbf \\ \\ & & \\ \\ F_x & F_y & F_z \end where i, j, and k are the unit vectors for the ''x''-, ''y''-, and ''z''-axes, respectively. This expands as follows: :\left(\frac - \frac\right) \mathbf + \left(\frac - \frac\right) \mathbf + \left(\frac - \frac\right) \mathbf. In index notation, with Einstein summation convention this is (\nabla \times \mathbf)_i = \epsilon_\partial_j F_k, where \epsilon_ is the totally antisymmetric symbol, the
Levi-Civita symbol In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the parity of a permutation, sign of a permutation of the n ...
.


Line integrals, surface integrals, and volume integrals

For some scalar field ''f'' : ''U'' ⊆ R''n'' → R, the line integral along a
piecewise smooth In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
''C'' ⊂ ''U'' is defined as :\int\limits_C f\, ds = \int_a^b f(\mathbf(t)) , \mathbf'(t), \, dt. where r:
, b The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
→ ''C'' is an arbitrary
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
parametrization of the curve ''C'' such that r(''a'') and r(''b'') give the endpoints of ''C'' and a < b. For a vector field F : ''U'' ⊆ R''n'' → R''n'', the line integral along a
piecewise smooth In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
''C'' ⊂ ''U'', in the direction of r, is defined as :\int\limits_C \mathbf(\mathbf)\cdot\,d\mathbf = \int_a^b \mathbf(\mathbf(t))\cdot\mathbf'(t)\,dt. where · is the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
and r:
, b The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
→ ''C'' is a
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
parametrization of the curve ''C'' such that r(''a'') and r(''b'') give the endpoints of ''C''. A surface integral is a generalization of
multiple integral In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number ...
s to integration over
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
s. It can be thought of as the
double integral In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number ...
analog of the line integral. To find an explicit formula for the surface integral, we need to parameterize the surface of interest, ''S'', by considering a system of
curvilinear coordinates In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inve ...
on ''S'', like the
latitude and longitude The geographic coordinate system (GCS) is a spherical or ellipsoidal coordinate system for measuring and communicating positions directly on the Earth as latitude and longitude. It is the simplest, oldest and most widely used of the various ...
on a
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
. Let such a parameterization be x(''s'', ''t''), where (''s'', ''t'') varies in some region ''T'' in the plane. Then, the surface integral is given by : \iint_ f \,\mathrm dS = \iint_ f(\mathbf(s, t)) \left\, \times \right\, \mathrm ds\, \mathrm dt where the expression between bars on the right-hand side is the
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
of the cross product of the partial derivatives of x(''s'', ''t''), and is known as the surface element. Given a vector field v on ''S'', that is a function that assigns to each x in ''S'' a vector v(x), the surface integral can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector. A
volume integral In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many ...
refers to an
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
over a 3-
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
al domain. It can also mean a triple integral within a region ''D'' in R3 of a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
f(x,y,z), and is usually written as: :\iiint\limits_D f(x,y,z)\,dx\,dy\,dz.


Fundamental theorem of line integrals

The fundamental theorem of line integrals, says that a line integral through a
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
field can be evaluated by evaluating the original scalar field at the endpoints of the curve. Let \varphi : U \subseteq \mathbb^n \to \mathbb. Then : \varphi\left(\mathbf\right)-\varphi\left(\mathbf\right) = \int_ \nabla\varphi(\mathbf)\cdot d\mathbf.


Stokes' theorem

Stokes' theorem relates the surface integral of the curl of a vector field F over a surface Σ in Euclidean three-space to the line integral of the vector field over its boundary ∂Σ: : \iint_ \nabla \times \mathbf \cdot \mathrm\mathbf = \oint_ \mathbf \cdot \mathrm \mathbf.


Divergence theorem

Suppose is a subset of \mathbb^n (in the case of represents a volume in 3D space) which is compact and has a piecewise smooth boundary (also indicated with ). If is a continuously differentiable vector field defined on a neighborhood of , then the divergence theorem says: : The left side is a
volume integral In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many ...
over the volume , the right side is the surface integral over the boundary of the volume . The closed manifold is quite generally the boundary of oriented by outward-pointing normals, and is the outward pointing unit normal field of the boundary . ( may be used as a shorthand for .)


In topology

Three-dimensional space has a number of topological properties that distinguish it from spaces of other dimension numbers. For example, at least three dimensions are required to tie a
knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...
in a piece of string. In differential geometry the generic three-dimensional spaces are
3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
s, which locally resemble ^3.


In finite geometry

Many ideas of dimension can be tested with
finite geometry Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
. The simplest instance is PG(3,2), which has
Fano plane In finite geometry, the Fano plane (after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and lines ...
s as its 2-dimensional subspaces. It is an instance of Galois geometry, a study of
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...
using finite fields. Thus, for any Galois field GF(''q''), there is a projective space PG(3,''q'') of three dimensions. For example, any three
skew lines In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron. Two lines that both lie in the sa ...
in PG(3,''q'') are contained in exactly one regulus. Albrecht Beutelspacher & Ute Rosenbaum (1998) ''Projective Geometry'', page 72,
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pre ...


See also

*
Dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as mi ...
*
Distance from a point to a plane In Euclidean space, the distance from a point to a plane is the distance between a given point and its orthogonal projection on the plane, the perpendicular distance to the nearest point on the plane. It can be found starting with a change of varia ...
* Four-dimensional space * *
Three-dimensional graph A three-dimensional graph may refer to * A graph (discrete mathematics) In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense ...
*
Solid geometry In mathematics, solid geometry or stereometry is the traditional name for the geometry of three-dimensional, Euclidean spaces (i.e., 3D geometry). Stereometry deals with the measurements of volumes of various solid figures (or 3D figures), inc ...
*
Two-dimensional space In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...


Notes


References

* * Arfken, George B. and Hans J. Weber. ''Mathematical Methods For Physicists'', Academic Press; 6 edition (June 21, 2005). . *


External links

* *
Elementary Linear Algebra - Chapter 8: Three-dimensional Geometry
Keith Matthews from
University of Queensland , mottoeng = By means of knowledge and hard work , established = , endowment = A$224.3 million , budget = A$2.1 billion , type = Public research university , chancellor = Peter Varghese , vice_chancellor = Deborah Terry , city = B ...
, 1991 {{Dimension topics * Analytic geometry Multi-dimensional geometry Three-dimensional coordinate systems 3 (number)