Real coordinate space
With component-wise addition and scalar multiplication, it is a real vector space.So, in multivariable calculus, the domain of a function of several real variables and the codomain of a real vector valued function are subsets of Rn for some n. The real n-space has several further properties, notably: These properties and structures of Rn make it fundamental in almost all areas of mathematics and their application domains, such as statistics, probability theory, and many parts of physics.The use of the real n-space, instead of several variables considered separately, can simplify notation and suggest reasonable definitions.In standard matrix notation, each element of Rn is typically written as a column vectorTo see that this is a basis, note that an arbitrary vector in Rn can be written uniquely in the formAny full-rank linear map of Rn to itself either preserves or reverses orientation of the space depending on the sign of the determinant of its matrix.Diffeomorphisms of Rn or domains in it, by their virtue to avoid zero Jacobian, are also classified to orientation-preserving and orientation-reversing.It has important consequences for the theory of differential forms, whose applications include electrodynamics.In the language of universal algebra, a vector space is an algebra over the universal vector space R∞ of finite sequences of coefficients, corresponding to finite sums of vectors, while an affine space is an algebra over the universal affine hyperplane in this space (of finite sequences summing to 1), a cone is an algebra over the universal orthant (of finite sequences of nonnegative numbers), and a convex set is an algebra over the universal simplex (of finite sequences of nonnegative numbers summing to 1).As for vector space structure, the dot product and Euclidean distance usually are assumed to exist in Rn without special explanations.Actually, any positive-definite quadratic form q defines its own "distance" √q(x − y), but it is not very different from the Euclidean one in the sense thatThis also implies that any full-rank linear transformation of Rn, or its affine transformation, does not magnify distances more than by some fixed C2, and does not make distances smaller than 1 / C1 times, a fixed finite number times smaller.[clarification needed] The aforementioned equivalence of metric functions remains valid if √q(x − y) is replaced with M(x − y), where M is any convex positive homogeneous function of degree 1, i.e. a vector norm (see Minkowski distance for useful examples).Although the definition of a manifold does not require that its model space should be Rn, this choice is the most common, and almost exclusive one in differential geometry.All these structures, although can be defined in a coordinate-free manner, admit standard (and reasonably simple) forms in coordinates.Rn is also a real vector subspace of Cn which is invariant to complex conjugation; see also complexification.There are three families of polytopes which have simple representations in Rn spaces, for any n, and can be used to visualize any affine coordinate system in a real n-space.Vertices of a hypercube have coordinates (x1, x2, ..., xn) where each xk takes on one of only two values, typically 0 or 1.As an n-dimensional subset it can be described with a single inequality which uses the absolute value operation:Actually, it does not depend much even on the linear structure: there are many non-linear diffeomorphisms (and other homeomorphisms) of Rn onto itself, or its parts such as a Euclidean open ball or the interior of a hypercube).An immediate consequence of this is that Rm is not homeomorphic to Rn if m ≠ n – an intuitively "obvious" result which is nonetheless difficult to prove.Despite the difference in topological dimension, and contrary to a naïve perception, it is possible to map a lesser-dimensional[clarification needed] real space continuously and surjectively onto Rn.However, it is useful to include these as trivial cases of theories that describe different n. The case of (x,y) where x and y are real numbers has been developed as the Cartesian plane P. Further structure has been attached with Euclidean vectors representing directed line segments in P. The plane has also been developed as the field extension, where the actor has been expressed as j, uses the line y=x for the involution of flipping the plane (x,y) ↦ (y,x), an exchange of coordinates.These numbers, with the coordinate-wise addition and multiplication according to jj=+1, form a ring that is not a field.The action of e on P reduces the plane to a line: It can be decomposed into the projection into the x-coordinate, then quarter-turning the result to the y-axis: e (x + y e) = x e since e2 = 0.The projective line P1(R) is a topological manifold covered by two coordinate charts, [z : 1] → z or [1 : z] → z, which form an atlas.One application of the real projective line is found in Cayley–Klein metric geometry.Euclidean R4 also attracts the attention of mathematicians, for example due to its relation to quaternions, a 4-dimensional real algebra themselves.Some common examples are A really surprising and helpful result is that every norm defined on Rn is equivalent.
Cartesian coordinatesEuclidean planemathematicsdimensionn-tuplesreal numberscoordinate vectorsreal linereal vector spacecoordinatesEuclidean spaceEuclidean lineEuclidean three-dimensional spaceone to one correspondencesgeometriccalculusRené Descartesnatural numbertuplesmultivariable calculusdomainfunction of several real variablesvector valued functionsubsetscomponentwisescalar multiplicationisomorphicdot productinner product spacetopological spacetopological vector spaceaffine spaceanalytic manifoldmanifoldsopen subsetalgebraic varietyreal algebraic varietystatisticsprobability theoryphysicsReal multivariable functionfunction compositioncontinuousvector spacelinearityzero vectoradditive inverseMatrix (mathematics)matrixcolumn vectorrow vectorcolumn vectorsrow vectorsLinear transformationsmatrix multiplicationcontinuous functionopen maprank of the matrixStandard basisfieldsordered fieldorientation structurefull-rankdeterminantpermutesparity of the permutationDiffeomorphismsdomains in itJacobiandifferential formselectrodynamicspoint reflectionevenness of nimproper rotationtranslationsdifferenceline segmentcanonicaloriginConvex analysisconvex setconvex combinationsuniversal algebraorthantsimplexconvex functioninequalitypointsCartesian coordinate systemEuclidean normmetric spacecoordinate systemCartesianpositive-definite quadratic formcomplete metric spaceaffine transformationhomogeneous functionvector normMinkowski distancemanifolddifferential geometryWhitney embedding theoremsdifferentiable m-dimensional manifoldembeddedpseudo-Euclidean spacesymplectic structurecontact structurecomplex conjugationcomplexificationLinear programmingConvex polytopepolytopeshypercubeintervalsunit intervalsystem of 2n inequalitiescross-polytopedual polytopeabsolute valuestandard simplexthe origintopological structurenatural topologyif and only ifopen balllinear topological space