Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries.
In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign =. The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation.
The GRE physics test is an examination administered by the Educational Testing Service (ETS). The test attempts to determine the extent of the examinees' understanding of fundamental principles of physics and their ability to apply them to problem solving. Many graduate schools require applicants to take the exam and base admission decisions in part on the results.
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow.
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.
The Graduate Record Examinations (GRE) is a standardized test that is part of the admissions process for many graduate schools in the United States and Canada and a few other countries. The GRE is owned and administered by Educational Testing Service (ETS). The test was established in 1936 by the Carnegie Foundation for the Advancement of Teaching.
In mathematics, differential refers to several related notions derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives of functions.
Further Mathematics is the title given to a number of advanced secondary mathematics courses. The term "Higher and Further Mathematics", and the term "Advanced Level Mathematics", may also refer to any of several advanced mathematics courses at many institutions.
In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, geometry, etc.
The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme that has collaboratively been produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH. The MSC is used by many mathematics journals, which ask authors of research papers and expository articles to list subject codes from the Mathematics Subject Classification in their papers. The current version is MSC2020.
The GRE Literature in English Test was a standardized test administered by the Educational Testing Service. It was intended to evaluate applicants seeking admission to a graduate program in English Studies. The test surveyed a wide range of topics related to literature in English, but the focus was mainly on works long accepted as part of the canon.
GRE Subject Biochemistry, Cell and Molecular Biology was a standardized exam provided by ETS that was discontinued in December 2016. It is a paper-based exam and there are no computer-based versions of it. ETS places this exam three times per year: once in April, once in October and once in November. Some graduate programs in the United States recommend taking this exam, while others require this exam score as a part of the application to their graduate programs. ETS sends a bulletin with a sample practice test to each candidate after registration for the exam. There are 180 questions within the biochemistry subject test.
In mathematics, secondary calculus is a proposed expansion of classical differential calculus on manifolds, to the "space" of solutions of a (nonlinear) partial differential equation. It is a sophisticated theory at the level of jet spaces and employing algebraic methods.
In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms. Instances of this are:
- The whole topological information of a smooth manifold is encoded in the algebraic properties of its -algebra of smooth functions as in the Banach–Stone theorem.
- Vector bundles over correspond to projective finitely generated modules over via the functor which associates to a vector bundle its module of sections.
- Vector fields on are naturally identified with derivations of the algebra .
- More generally, a linear differential operator of order k, sending sections of a vector bundle to sections of another bundle is seen to be an -linear map :\Gamma (E)\to \Gamma (F)} between the associated modules, such that for any elements :
Mathematics education in the United States varies considerably from one state to the next, and even within a single state. However, with the adoption of the Common Core Standards in most states and the District of Columbia beginning in 2010, mathematics content across the country has moved into closer agreement for each grade level. The SAT, a standardized university entrance exam, has been reformed to better reflect the contents of the Common Core. However, many students take alternatives to the traditional pathways, including accelerated tracks. As of 2023, twenty-seven states require students to pass three math courses before graduation from high school, while seventeen states and the District of Columbia require four. A typical sequence of secondary-school courses in mathematics reads: Pre-Algebra, Algebra I, Geometry, Algebra II, Pre-calculus, and Calculus or Statistics. However, some students enroll in integrated programs while many complete high school without passing Calculus or Statistics. At the other end, counselors at competitive public or private high schools usually encourage talented and ambitious students to take Calculus regardless of future plans in order to increase their chances of getting admitted to a prestigious university and their parents enroll them in enrichment programs in mathematics.
Additional Mathematics is a qualification in mathematics, commonly taken by students in high-school. It features a range of problems set out in a different format and wider content to the standard Mathematics at the same level.
Math in Moscow (MiM) is a one-semester study abroad program for North American and European undergraduates held at the Independent University of Moscow (IUM) in Moscow, Russia. The program consists mainly of math courses that are taught in English. The program was first offered in 2001, and since 2008 has been run jointly by the Independent University of Moscow, Moscow Center for Continuous Mathematical Education, and the Higher School of Economics (HSE).
The GRE subject test in chemistry is a standardized test in the United States created by the Educational Testing Service, and is designed to assess a candidate's potential for graduate or post-graduate study in the field of chemistry. It contains questions from many fields of chemistry. 15% of the questions will come from analytical chemistry, 25% will come from inorganic chemistry, 30% will come from organic chemistry and 30% will come from physical chemistry.
The GRE subject test in biology was a standardized test in the United States created by the Educational Testing Service, and is designed to assess a candidate's potential for graduate or post-graduate study in the field of biology. The test is comprehensive and covers—in equal proportions—molecular biology, organismal biology, and ecology and evolution.
Alexandre Mikhailovich Vinogradov was a Russian and Italian mathematician. He made important contributions to the areas of differential calculus over commutative algebras, the algebraic theory of differential operators, homological algebra, differential geometry and algebraic topology, mechanics and mathematical physics, the geometrical theory of nonlinear partial differential equations and secondary calculus.
