Space
The study of space originates with geometry Geometry is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the third century BC geometry was put into an axiomatic form by Euclid, whose treatment†– in particular, Euclidean geometry Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content. The method consists of assuming a small set of. Trigonometry Trigonometry is a branch of mathematics that deals with triangles, particularly those plane triangles in which one angle has 90 degrees (right triangles). Trigonometry deals with relationships between the sides and the angles of triangles and with the trigonometric functions, which describe those relationships is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions; it combines space and numbers, and encompasses the well-known Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem (in British English) is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle in British English). It states:. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries A non-Euclidean geometry is characterized by a non-vanishing Riemann curvature tensor. Examples of non-Euclidean geometries include the hyperbolic and elliptic geometry, which are contrasted with a Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. Euclid's fifth postulate, (which play a central role in general relativity General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a geometric property of space and time, or spacetime) and topology Topology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation. Quantity and space both play a role in analytic geometry Analytic geometry, usually called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra; the modern development of analytic geometry is thus suggestively called algebraic geometry, differential geometry Differential geometry is a mathematical discipline that uses the methods of differential and integral calculus, as well as linear and multilinear algebra, to study problems in geometry. The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis for its initial development in the eighteenth and, and algebraic geometry Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and. Within differential geometry are the concepts of fiber bundles In mathematics, and particularly topology, a fiber bundle is intuitively a space E which locally "looks" like a product space B × F, but globally may have a different topological structure. Specifically, the similarity between the fiber bundle E and a product space B × F is defined using a continuous surjective map and calculus on manifolds In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold. Thus a line and a circle are one-dimensional manifolds, a plane and the surface of a ball are two-dimensional manifolds,, in particular, vector Vector calculus is a branch of mathematics concerned with differentiation and integration of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an and tensor calculus A tensor is an object which includes and extends the notion of scalar, vector, and matrix. The term is slightly ambiguous, and this often leads to misunderstandings; roughly speaking, there are different default meanings in mathematics and physics. In the mathematical fields of multilinear algebra and differential geometry, a tensor is first of. Within algebraic geometry is the description of geometric objects as solution sets of polynomial In mathematics, a polynomial is a finite length expression constructed from variables and constants, by using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents. For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term involves division by the variable equations, combining the concepts of quantity and space, and also the study of topological groups In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. Topological groups allow one to study the notion of continuous symmetries in the form of continuous group actions, which combine structure and space. Lie groups In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after the nineteenth century Norwegian mathematician Sophus Lie, who laid the foundations of the theory of continuous transformation groups are used to study space, structure, and change. Topology Topology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation in all its many ramifications may have been the greatest growth area in 20th century mathematics, and includes the long-standing Poincaré conjecture In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere among three-dimensional manifolds. It began as a popular, important conjecture, but is now considered a theorem to the satisfaction of the awarders of the Fields medal. The claim concerns a space that locally looks like ordinary three and the controversial four color theorem In mathematics, the four color theorem, or the four color map theorem, states that given any separation of a plane into contiguous regions, called a map, the regions can be colored using at most four colors so that no two adjacent regions have the same color. Two regions are called adjacent only if they share a border segment, not just a point, whose only proof, by computer, has never been verified by a human.
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