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Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathematics), matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as line (geometry), lines, plane (geometry), planes and rotation (mathematics), rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to Space of functions, function spaces. Linear algebra is also used in most sciences and fields of engineering because it allows mathematical model, modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Approximation
In science, engineering, and other quantitative disciplines, order of approximation refers to formal or informal expressions for how accurate an approximation is. Usage in science and engineering In formal expressions, the ordinal number used before the word order refers to the highest power in the series expansion used in the approximation. The expressions: a ''zeroth-order approximation'', a ''first-order approximation'', a ''second-order approximation'', and so forth are used as fixed phrases. The expression a ''zero-order approximation'' is also common. Cardinal numerals are occasionally used in expressions like an ''order-zero approximation'', an ''order-one approximation'', etc. The omission of the word ''order'' leads to phrases that have less formal meaning. Phrases like first approximation or to a first approximation may refer to ''a roughly approximate value of a quantity''. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gabriel Cramer
Gabriel Cramer (; 31 July 1704 – 4 January 1752) was a Genevan mathematician. Biography Cramer was born on 31 July 1704 in Geneva, Republic of Geneva to Jean-Isaac Cramer, a physician, and Anne Mallet. The progenitor of the Cramer family in Geneva was Jean-Ulrich Cramer, Gabriel's great-grandfather, who immigrated from Strasbourg in 1634. Cramer's mother, a member of the Mallet family, was of Huguenot origin. Cramer showed promise in mathematics from an early age. In 1722, aged 18, he received his doctorate from the Academy of Geneva, and at 20 he was made co-chair (along with Jean-Louis Calandrini) of mathematics at the Academy. He became the sole professor of mathematics in 1734 and was appointed professor of philosophy at the Academy in 1750. Cramer was also involved in the politics of the Republic of Geneva, entering first the Council of Two Hundred in 1734 then the Council of Sixty in 1750. He was a member of the science academies of Bologna, Lyon, and Montpellier ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gottfried Wilhelm Leibniz
Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to many other branches of mathematics, such as binary arithmetic and statistics. Leibniz has been called the "last universal genius" due to his vast expertise across fields, which became a rarity after his lifetime with the coming of the Industrial Revolution and the spread of specialized labor. He is a prominent figure in both the history of philosophy and the history of mathematics. He wrote works on philosophy, theology, ethics, politics, law, history, philology, games, music, and other studies. Leibniz also made major contributions to physics and technology, and anticipated notions that surfaced much later in probability theory, biology, medicine, geology, psychology, linguistics and computer science. Leibniz contributed to the field ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the matrix and the linear map represented, on a given basis (linear algebra), basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible matrix, invertible and the corresponding linear map is an linear isomorphism, isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse. The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries. The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Geometry
In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry. Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space. As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometric shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. That the algebra of the real numbers can be employed to yield r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordinates
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the Position (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as Euclidean space. The coordinates are not interchangeable; they are commonly distinguished by their position in an ordered tuple, or by a label, such as in "the ''x''-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and ''vice versa''; this is the basis of analytic geometry. Common coordinate systems Number line The simplest example of a coordinate system is the identification of points on a line (geometry), line with real numbers using the ''number line''. In this system, an arbitrary point ''O'' (the ''ori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
René Descartes
René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramount to his method of inquiry, and he connected the previously separate fields of geometry and algebra into analytic geometry. Descartes spent much of his working life in the Dutch Republic, initially serving the Dutch States Army, and later becoming a central intellectual of the Dutch Golden Age. Although he served a Dutch Reformed Church, Protestant state and was later counted as a Deism, deist by critics, Descartes was Roman Catholicism, Roman Catholic. Many elements of Descartes's philosophy have precedents in late Aristotelianism, the Neostoicism, revived Stoicism of the 16th century, or in earlier philosophers like Augustine of Hippo, Augustine. In his natural philosophy, he differed from the Scholasticism, schools on two major point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Systems Of Linear Equations
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables. For example, : \begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in the three variables . A '' solution'' to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. In the example above, a solution is given by the ordered triple (x,y,z)=(1,-2,-2), since it makes all three equations valid. Linear systems are a fundamental part of linear algebra, a subject used in most modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A system of non-linear equations can often be approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
JHU Press
Johns Hopkins University Press (also referred to as JHU Press or JHUP) is the publishing division of Johns Hopkins University. It was founded in 1878 and is the oldest continuously running university press in the United States. The press publishes books and journals, and operates other divisions including fulfillment and electronic databases. Its headquarters are in Charles Village section of Baltimore, Maryland. In 2017, after the retirement of Kathleen Keane, who is credited with modernizing JHU Press for the digital age, the university appointed new director Barbara Pope. Overview Daniel Coit Gilman, the first president of Johns Hopkins University, inaugurated the press in 1878. The press began as the university's Publication Agency, publishing the '' American Journal of Mathematics'' in its first year and the '' American Chemical Journal'' in its second. It published its first book, ''Sidney Lanier: A Memorial Tribute'', in 1881 to honor the poet who was one of the univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Nine Chapters On The Mathematical Art
''The Nine Chapters on the Mathematical Art'' is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 1st century CE. This book is one of the earliest surviving mathematical texts from China, the others being the '' Suan shu shu'' (202 BCE – 186 BCE) and ''Zhoubi Suanjing'' (compiled throughout the Han until the late 2nd century CE). It lays out an approach to mathematics that centres on finding the most general methods of solving problems, which may be contrasted with the approach common to ancient Greek mathematicians, who tended to deduce propositions from an initial set of axioms. Entries in the book usually take the form of a statement of a problem, followed by the statement of the solution and an explanation of the procedure that led to the solution. These were commented on by Liu Hui in the 3rd century. The book was later included in the early Tang collection, the '' Ten Computational C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
