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**Definition of Calculus**

The analysis studies the degree of change within functions, such as gradients and curvatures, as well as problems that have to do with surface, such as finding the center of gravity of a geometric figure.

The analysis as a separate branch of mathematics starts with the differential and integral calculus, invented in parallel by Leibniz and Newton, who challenged each other the invention. Leibniz developed these techniques to study geometric objects, while Newton needed them to provide a solid foundation for his celestial mechanics. Also, Barrow, Descartes, The Fermat, and Huygens have made important early contributions.

The differential account studies the concept of derivative, which indicates the degree of change of a function. One of the main reasons for developing the analysis was to solve the tangent problem. The integral account does the opposite of the differential account: based on the initial situation and the degree of change, the original function is found. The area of a geometric figure can be expressed as an integral, even if that figure is curvilinear (the area of linear polygons does not require an integral calculation). A rigorous definition of derivatives and integrals uses limits and continuity.

For centuries the analysis remained without a consistent logical foundation and made use of seemingly contradictory concepts such as “infinitely small quantities different from 0” ( infinitesimals ). It was not until the nineteenth century that mathematicians such as Karl Weierstrass placed the analysis in a strict framework independent of intuitive concepts such as “striving for” and “in the first approach”. It is that evolution that gave rise to a correct study of the set of real numbers and the formal set theory.

**Etymology**

The words analysis and synthesis come from the Greek and mean as much as ‘dissecting’ and ‘merging’. For Ancient Greek mathematicians, they were the two steps in the general method of solving mathematical (typically geometrical) problems. For René Descartes, the term analytic geometry refers to the use of numbers (coordinates) and algebraic functions to solve geometric problems.

**Basic concepts**

See limit, derivative and integral account for the main articles on these topics.

She {\ displaystyle f: I \ subset \ mathbb {R} \ to \ mathbb {R}} a real function that is at least defined in an open interval {\ displaystyle I,} and they {\ displaystyle a} an element of {\ displaystyle I.} We say that {\ displaystyle f}has a limit in{\ displaystyle a} if there is a real number {\ displaystyle b} exists with the property that for randomly small intervals around {\ displaystyle b,} there are always intervals around {\ displaystyle a} exist that through {\ displaystyle f} within the given intervals {\ displaystyle b} are shown:

{\ displaystyle \ forall \ varepsilon> 0, \ exists \ delta> 0: f \ left ((a- \ delta, a + \ delta) \ right) \ subset (b- \ varepsilon, b + \ varepsilon).}

In that case they write {\ displaystyle \ lim _ {x \ to a} f (x) = b} and they say that too {\ displaystyle f} is continuous in{\ displaystyle a.}

A function {\ displaystyle f}called derivable in{\ displaystyle a}as the next limit, called the derivative of{\ displaystyle f} in {\ displaystyle a,} exists:

{\ displaystyle f ‘(a) = \ lim _ {x \ to a} {\ frac {f (x) -f (a)} {xa}}.}[1]

The derivative gives the slope of the tangent to the graph of {\ displaystyle f} in the point {\ displaystyle (a, f (a)).}

The integral of a function{\ displaystyle f} at an interval {\ displaystyle I = (a, b)} is the limit, if any, of the total area of randomly thin vertical strips below the graph:

{\ displaystyle \ int _ {x = a} ^ {b} f (x) \ dx = \ lim _ {\ delta \ to 0} \ delta \ sum _ {n = 0} ^ {[(ba) / \ delta]} f (a + n \ delta).}

The main statement of the integral account is that derivatives and integrals are in a certain sense each other’s reverse processing; slightly more precise: if in the above expression the integral is seen as a function of{\ displaystyle b:}

{\ displaystyle F (c) = \ int _ {x = a} ^ {c} f (x) \ dx \ (c \ in (a, b))}

then that function {\ displaystyle F} derivable and its derivative in one point {\ displaystyle c_ {0}} is again {\ displaystyle f (c_ {0}).}

**Branches**

Starting from the original differential and integral calculation, many specializations have arisen, each with its own problems but with a great deal of mutual interaction.

**Complex analysis**

See the Function theory for the main article on this topic.

If in the definition of the derivative the fraction is interpreted as a division of complex numbers, we obtain the concept of “complex deductibility” for functions of a subset of {\ display style \ mathbb {C}} to {\ displaystyle \ mathbb {C}.}This condition appears to be much stricter in a certain sense than ordinary, real derivability, and the complex functions that meet it are called analytic functions.

**Size and opportunity**

See Maattheorie for the main article on this subject.

The integral calculation makes it possible to define and calculate the area under a curvilinear graph, and from there also three-dimensional volumes. In both cases, the definitions are based on the basic surface area of a rectangle or the basic volume of a beam, namely the product of their dimensions.

Size theory studies alternative ways of giving content to subsets of an abstract collection, starting from a small number of hypotheses or axioms (see Maat (mathematics) ). The modern probability theory is based on probabilities, a special case of abstract measures.

**Differential equations**

See Differential comparison for the main article on this topic.

A differential equation is a comparison in which a connection is given between an unknown function and one or more of its derived functions. The main difference between a differential equation and a traditional (for example, algebraic) equation is that the unknown is not a separate number, but a function.

Differential equations are applied in almost all branches of science and technology; even Newton’s original motivation to invent derivatives were the differential equations that describe the movements of the planets.

**Functional analysis**

See Functional Analysis for the main article on this topic.

Modern functional analysis studies abstract topological vector spaces, but the original motivation was formed by the vector spaces of functions in which people seek solutions to differential equations. Important classes of topological vector spaces are Hilbert spaces and more general Banach spaces.

**Topology**

See Topology for the main article on this topic.

The elementary definition of limits and continuity gives precise content to the intuitive concept of proximity-based on randomly small intervals. Alternative and abstract versions of the concept of proximity are defined and studied in topology.

A metric space is a collection{\ display style X,}provided with a remote function

{\ displaystyle d: X \ times X \ to \ mathbb {R} ^ {+}}

which is symmetrical in its two arguments, which only assigns the distance 0 to identical couples, and which satisfies the triangular inequality :

{\ displaystyle \ forall x, y, z \ in X: d (x, z) \ leq d (x, y) + d (y, z).}

The triangle inequality represents on an abstract level that one side of a triangle is never longer than the sum of the other two sides.

A topological space is an abstract family of subsets that inherits some properties of metric spaces, but that does not necessarily stem from a distance function.

**Curved spaces**

See Variety (math) for the main article on this topic.

A variety is a topological space whose points can be provided with coordinates locally, although it is not necessary for one coordinate system to cover the entire space. A local coordinate system is called a map. In places where two different maps overlap, there must be a coordinate transformation that has “good” properties. Depending on the choice of what a “good” transformation is, different subareas of the analysis arise, each with their own class of varieties.

With topological varieties, the transformations are continuous functions. They are studied in algebraic topology, among other things.

With differentiable varieties, the transformations can often be deduced arbitrarily. They form the central object of the differential topology.

In differential geometry, a specific distance concept is added to a differentiable variety so that the concept of curvature makes sense.

**Fourier analysis **

See Fourier analysis for the main article on this topic.

In his research of heat equation, a partial differential equation describing heat transfer through conduction, Joseph Fourier discovered that every continuous real function at the interval{\ displaystyle [0.2 \ pi]}can be written as a series consisting of a constant, a series of multiples of the sine and its harmonics, and a series of multiples of the cosine and its harmonics:

{\ displaystyle f (x) = a_ {0} + a_ {1} \ cos (x) + b_ {1} \ sin (x) + a_ {2} \ cos (2x) + b_ {2} \ sin ( 2x) + \ ldots}

It soon became apparent that many linear partial differential equations from physics can be analyzed by expressing the initial and boundary conditions as an infinite sum or integral of “elementary” functions for which the solution is in a certain sense simpler. The Fourier analysis is the result of this determination.

On closer inspection, the success of Fourier analysis appears to be due to the favorable behavior of the convolution product of two functions while taking derivatives (see Convolution # Derivative ). The harmonic analysis, the abstract context in which this takes place, studies the dissolution of functions on locally compact groups provided with a Haar measure, that is, a volume concept that is left invariant by the group processing. In the case of classical Fourier series of periodic functions, that group is the unit circle (sum of angles), and the measure is the normal length measure on the circumference. In the case of Fourier transformations in {\ displaystyle n} variables we look at the group {\ displaystyle \ mathbb {R} ^ {n}} (addition of vectors) with the measure it {\ displaystyle n}-dimensional hypervolume.