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Calculus

Calculus
History[edit] Modern calculus was developed in 17th century Europe by Isaac Newton and Gottfried Wilhelm Leibniz (see the Leibniz–Newton calculus controversy), but elements of it have appeared in ancient Greece, China, medieval Europe, India, and the Middle East. Ancient[edit] The ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Medieval[edit] Modern[edit] In Europe, the foundational work was a treatise due to Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. These ideas were arranged into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton.[13] He is now regarded as an independent inventor of and contributor to calculus. Leibniz and Newton are usually both credited with the invention of calculus. Foundations[edit] Related:  System Theory

Linear algebra Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces. Linear algebra is central to both pure and applied mathematics. Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, computer animation, and the social sciences (particularly in economics). History[edit] The study of linear algebra first emerged from the study of determinants, which were used to solve systems of linear equations. The study of matrix algebra first emerged in England in the mid-1800s. In 1882, Hüseyin Tevfik Pasha wrote the book titled "Linear Algebra".[4][5] The first modern and more precise definition of a vector space was introduced by Peano in 1888;[3] by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Educational history[edit]

Dynamical system The Lorenz attractor arises in the study of the Lorenz Oscillator, a dynamical system. Overview[edit] Before the advent of computers, finding an orbit required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system. For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The difficulties arise because: The systems studied may only be known approximately—the parameters of the system may not be known precisely or terms may be missing from the equations. History[edit] Many people regard Henri Poincaré as the founder of dynamical systems.[3] Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). .

Pure mathematics Mathematical formulæ Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics,[1] and at variance with the trend towards meeting the needs of navigation, astronomy, physics, economics, engineering, and so on. Another insightful view put forth is that pure mathematics is not necessarily applied mathematics: it is possible to study abstract entities with respect to their intrinsic nature, and not be concerned with how they manifest in the real world.[2] Even though the pure and applied viewpoints are distinct philosophical positions, in practice there is much overlap in the activity of pure and applied mathematicians. To develop accurate models for describing the real world, many applied mathematicians draw on tools and techniques that are often considered to be "pure" mathematics. History[edit] 19th century[edit]

Derivative The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point. The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. In fact, the derivative at a point of a function of a single variable is the slope of the tangent line to the graph of the function at that point. The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. Differentiation and the derivative[edit] The simplest case, apart from the trivial case of a constant function, is when y is a linear function of x, meaning that the graph of y divided by x is a line. y + Δy = f(x + Δx) = m (x + Δx) + b = m x + m Δx + b = y + m Δx. It follows that Δy = m Δx. This gives an exact value for the slope of a line. Rate of change as a limit value Figure 1. Figure 2. Figure 3. Figure 4.

Topology Möbius strips, which have only one surface and one edge, are a kind of object studied in topology. Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation. Such ideas go back to Leibniz, who in the 17th century envisioned the geometria situs (Latin for "geometry of place") and analysis situs (Greek-Latin for "picking apart of place"). The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. Topology has many subfields: See also: topology glossary for definitions of some of the terms used in topology, and topological space for a more technical treatment of the subject. History[edit] Topology began with the investigation of certain questions in geometry. For further developments, see point-set topology and algebraic topology. Elementary introduction[edit]

Feedback "...'feedback' exists between two parts when each affects the other. A feedback loop where all outputs of a process are available as causal inputs to that process "Simple causal reasoning about a feedback system is difficult because the first system influences the second and second system influences the first, leading to a circular argument. In this context, the term "feedback" has also been used as an abbreviation for: Feedback signal – the conveyance of information fed back from an output, or measurement, to an input, or effector, that affects the system.Feedback loop – the closed path made up of the system itself and the path that transmits the feedback about the system from its origin (for example, a sensor) to its destination (for example, an actuator).Negative feedback – the case where the fed-back information acts to control or regulate a system by opposing changes in the output or measurement. History[edit] Types[edit] Positive and negative feedback[edit] Terminology[edit]

Logarithmic scale A simple example is a chart whose vertical or horizontal axis has equally spaced increments that are labeled 1, 10, 100, 1000, instead of 0, 1, 2, 3. Each unit increase on the logarithmic scale thus represents an exponential increase in the underlying quantity for the given base (10, in this case). Definition and base[edit] Logarithmic scales are either defined for ratios of the underlying quantity, or one has to agree to measure the quantity in fixed units. Example scales[edit] On most logarithmic scales, small values (or ratios) of the underlying quantity correspond to negative values of the logarithmic measure. Some logarithmic scales were designed such that large values (or ratios) of the underlying quantity correspond to small values of the logarithmic measure. Logarithmic units[edit] Examples[edit] Motivation[edit] the logarithms of any given number a to two different bases (here b and c) differ only by the constant factor logc b. Graphic representation[edit] Comparing the scales[edit]

Integral A definite integral of a function can be represented as the signed area of the region bounded by its graph. The term integral may also refer to the related notion of the antiderivative, a function F whose derivative is the given function f. In this case, it is called an indefinite integral and is written: However, the integrals discussed in this article are termed definite integrals. The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century. Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation: if f is a continuous real-valued function defined on a closed interval [a, b], then, once an antiderivative F of f is known, the definite integral of f over that interval is given by Integrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering. History[edit] Pre-calculus integration[edit] or is used (W3C 2006).

Number theory Number theory (or arithmetic[note 1]) is a branch of pure mathematics devoted primarily to the study of the integers, sometimes called "The Queen of Mathematics" because of its foundational place in the discipline.[1] Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (e.g., the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, e.g., as approximated by the latter (Diophantine approximation). The older term for number theory is arithmetic. History[edit] Origins[edit] Dawn of arithmetic[edit] such that .

State-space representation In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. "State space" refers to the space whose axes are the state variables. The state of the system can be represented as a vector within that space. To abstract from the number of inputs, outputs and states, these variables are expressed as vectors. Additionally, if the dynamical system is linear, time-invariant, and finite-dimensional, then the differential and algebraic equations may be written in matrix form.[1][2] The state-space representation (also known as the "time-domain approach") provides a convenient and compact way to model and analyze systems with multiple inputs and outputs. inputs and outputs, we would otherwise have to write down Laplace transforms to encode all the information about a system. State variables[edit] Linear systems[edit] inputs, outputs and where: is called the "state vector", ).

Leonhard Euler Leonhard Euler (/ˈɔɪlər/ OY-lər;[2] Swiss Standard German: [ˈɔɪlər] ( listen); German Standard German: [ˈɔʏlɐ] ( A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all. Life[edit] Early years[edit] Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastor's daughter. Euler's formal education started in Basel, where he was sent to live with his maternal grandmother. Saint Petersburg[edit] Around this time Johann Bernoulli's two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. 1957 Soviet Union stamp commemorating the 250th birthday of Euler. Euler arrived in Saint Petersburg on 17 May 1727. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia and to close the scientific gap with Western Europe. In St.

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