1.
Mathematics
–
Mathematics is the study of topics such as quantity, structure, space, change. There is a range of views among philosophers as to the exact scope and definition of mathematics. Mathematicians use them to formulate new conjectures. Mathematicians resolve the falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of logic, mathematics developed from counting, calculation, measurement, the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Galileo Galilei said, "The universe can not become familiar with the characters in which it is written. Without these, one is wandering about in a dark labyrinth." Carl Friedrich Gauss referred as "the Queen of the Sciences". Benjamin Peirce called mathematics "the science that draws necessary conclusions". David Hilbert said of mathematics: "We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules.

Mathematics
–
Euclid (holding calipers), Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.

Mathematics
–
Greek mathematician Pythagoras (c. 570 – c. 495 BC), commonly credited with discovering the Pythagorean theorem

Mathematics
–
Leonardo Fibonacci, the Italian mathematician who established the Hindu–Arabic numeral system to the Western World

Mathematics
–
Carl Friedrich Gauss, known as the prince of mathematicians

2.
Equation
–
In mathematics, an equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true. Variables are also called the values of the unknowns which satisfy the equality are called solutions of the equation. There are two kinds of equations: conditional equations. An equation is true for all values of the variable. A conditional equation is true for only particular values of the variables. Each side of an equation is called a member of the equation. Each member will contain one or more terms. A x 2 + B x + C = y has two members: A x 2 + B x + C and y. The left member has the right member one term. The parameters are A, B, C. An equation is analogous to a scale into which weights are placed. In geometry, equations are used to describe geometric figures. This is the starting idea of an important area of mathematics. Algebra studies two main families of equations: polynomial equations and, among them the special case of linear equations.

Equation
–
A strange attractor which arises when solving a certain differential equation.

3.
System of linear equations
–
In mathematics, a system of linear equations is a collection of two or more linear equations involving the same set of variables. A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. The word "system" indicates that the equations are to be considered collectively, rather than individually. Linear programming is a collection of methods for finding the "best" integer solution. Gröbner theory provides algorithms when coefficients and unknowns are polynomials. Also tropical geometry is an example of linear algebra in a more exotic structure. The simplest kind of linear system involves two variables: 2 x + 3 y = 6 4 x + 9 y = 15. One method for solving such a system is as follows. First, solve the top equation for x in terms of y: x = 3 − 2 y. Now substitute this expression for x into the bottom equation: 4 + 9 y = 15. This results in a single equation involving only the variable y. Solving gives y = 1, substituting this back into the equation for x yields x = 3 / 2. One extremely helpful view is that each unknown is a weight for a vector in a linear combination. X 1 + 2 + ⋯ + x n = This allows all the language and theory of vector spaces to be brought to bear. If every vector within that span has exactly one expression as a linear combination of the given left-hand vectors, then any solution is unique.

System of linear equations
–
A linear system in three variables determines a collection of planes. The intersection point is the solution.

4.
System of ordinary differential equations
–
In mathematics, an ordinary differential equation is a differential equation containing one or more functions of one independent variable and its derivatives. The term ordinary is used with the term partial equation which may be with respect to more than one independent variable. ODEs that are linear differential equations have closed-form solutions that can be multiplied by coefficients. Ordinary differential equations arise in many contexts of mathematics and science. Mathematical descriptions of change use differentials and derivatives. Often, quantities are defined as the rate of change of other quantities, or gradients of quantities, how they enter differential equations. Mathematical fields include analytical mechanics. Scientific fields include economics. Many mathematicians have contributed including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d'Alembert, Euler. In general, F is a function of the x of the particle at t. The unknown function x is indicated in the F. In what follows, let y be an independent variable, y = f is an unknown function of x. The notation for differentiation varies depending upon the author and upon which notation is most useful for the task at hand. Given F, a function of x, y, derivatives of y. Then an equation of the F = y is called an ordinary differential equation of order n.

System of ordinary differential equations
–
Navier–Stokes differential equations used to simulate airflow around an obstruction.

5.
Partial differential equation
–
In mathematics, a partial differential equation is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are either solved by hand, or used to create a relevant computer model. PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid dynamics, quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in partial differential equations. Partial differential equations are equations that involve rates of change with respect to continuous variables. The dynamics for the rigid body take place in a finite-dimensional space; the dynamics for the ﬂuid occur in an infinite-dimensional conﬁguration space. Here again, there will be simple solutions for linear problems. Classic domains where PDEs are used include acoustics, fluid dynamics, heat transfer. A partial equation for the function u is an equation of the form f = 0. If f is a linear function of its derivatives, then the PDE is called linear. Common examples of linear PDEs include the heat equation, the wave equation, Laplace's equation, Helmholtz equation, Poisson's equation. A relatively simple PDE is ∂ u ∂ x = 0. This relation implies that the u is independent of x.

Partial differential equation
–
Navier–Stokes differential equations used to simulate airflow around an obstruction.

6.
Elementary algebra
–
Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers, algebra introduces quantities without fixed values, known as variables. This use of variables entails an understanding of the general rules of the operators introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned outside the realm of real and complex numbers. Quantitative relationships in science and mathematics are expressed as algebraic equations. Algebraic notation describes how algebra is written. It has its own terminology. Letters represent constants. They are usually written in italics. Algebraic operations work in the same way as arithmetic operations, such as addition, subtraction, multiplication, division and exponentiation. and are applied to algebraic variables and terms. Multiplication symbols implied when there is no space between two variables or terms, or when a coefficient is used. For example, 3 × x 2 is written as 3 x 2, 2 × x × y may be written 2 x y. Usually terms with the highest power, are written on the left, for example, x 2 is written to the left of x. When a coefficient is one, it is usually omitted.

Elementary algebra
–
A typical algebra problem.

Elementary algebra
–
Two-dimensional plot (magenta curve) of the algebraic equation

7.
Non-linear system
–
In physical sciences, a nonlinear system is a system in which the output is not directly proportional to the input. Nonlinear problems are to engineers, physicists and mathematicians and many other scientists because most systems are inherently nonlinear in nature. Nonlinear systems may appear chaotic, counterintuitive, contrasting with the much simpler linear systems. It does not matter if nonlinear known functions appear in the equations. As nonlinear equations are difficult to solve, nonlinear systems are commonly approximated by linear equations. It follows that some aspects of the behavior of a system appear commonly to be counterintuitive, unpredictable or even chaotic. Although chaotic behavior may resemble random behavior, it is absolutely not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why long-term forecasts are impossible with current technology. Some authors use the term science for the study of nonlinear systems. This is disputed by others: Using a term like science is like referring to the bulk of zoology as the study of non-elephant animals. Additivity implies homogeneity for any rational α, and, for continuous functions, for any real α. For a complex α, homogeneity does not follow from additivity. For example, an map is additive but not homogeneous. The equation is called homogeneous if C = 0.

Non-linear system
–
Illustration of a pendulum