# Mathematics and Geometry

Mathematics is an area of knowledge that includes such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

- MathematicsGeometry is, with arithmetic, one of the oldest branches of mathematics.

- Geometry30 related topics

## Differential geometry

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.

## Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces.

## Calculus

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

## Topological space

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.

## Vector space

[[File: Vector add scale.svg|200px|thumb|right|Vector addition and scalar multiplication: a vector

[[File: Vector add scale.svg|200px|thumb|right|Vector addition and scalar multiplication: a vector

In mathematics, physics, and engineering, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars.

Finite-dimensional vector spaces occur naturally in geometry and related areas.

## Algebra

Quadratic formula.svg expresses the solution of the equation

Quadratic formula.svg expresses the solution of the equation

Algebra is one of the broad areas of mathematics.

Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example).

## 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 spaces of any positive integer dimension, including the three-dimensional space and the Euclidean plane (dimension two).

## 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.

Area can sometimes be found via geometrical compass-and-straightedge constructions of an equivalent square.

## Greek mathematics

Greek mathematics refers to mathematics texts written during and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean.

Greek mathematics constitutes an important period in the history of mathematics: fundamental in respect of geometry and for the idea of formal proof.

## Mathematical analysis

Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.

Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).