![]() ![]() ![]() ![]() The desired relation is the ordinary integral of this equation.DRPS : Course Catalogue : School of Mathematics : Mathematics He considered (1) as a differential equation, connecting $ x $ Of the fourth degree and posed the problem of the relations between $ x $ ![]() Euler studied an arbitrary polynomial $ f(x) $ Fagnano, in the first half of the 18th century, gave numerous examples of such transformations. From the analytic point of view, this was tantamount to the conversion of an integral into another integral, and in certain cases the result was a transformation of an integral into itself. They found methods for transforming one curve into another of the same arc length, even though the respective arcs could not be brought into correspondence. They studied integrals expressing the arc lengths of certain curves. Elliptic integral).Īt the end of the 17th century, Jakob and Johann Bernoulli noted a new interesting property of elliptic integrals. At first, these were the integrals that received the name "elliptic", and it was only later that the meaning was extended to include functions and curves (cf. The system of concepts and results which one now calls the theory of elliptic curves arose as a part of analysis (rather than geometry) - the theory of integrals of rational functions on an elliptic curve. Historically, the first stage of development of the theory of algebraic curves consisted in the clarification of the fundamental concepts and ideas of this theory, using elliptic curves as examples. Modern algebraic geometry arose as the theory of algebraic curves (cf. However, the achievements of this branch of algebraic geometry still are in the domain proper of projective geometry, but as a result it left to algebraic geometry the traditional study of projective algebraic varieties. The utilization of coordinates in projective geometry created a situation in which algebraic methods could compete with synthetic methods. Nevertheless, its crystallization into an independent branch of science only began in the mid-19th century. The genesis of algebraic geometry dates back to the 17th century, with the introduction of the concept of coordinates into geometry. Conversely, the ideas and methods of these disciplines are utilized in algebraic geometry. Theory and the index of elliptic operators), in the theory of complex spaces, in the theory of categories (topoi, Abelian categories), and in functional analysis (representation theory). The concepts and the results of algebraic geometry are extensively used in number theory (Diophantine equations and the evaluation of trigonometric sums), in differential topology (both with respect to singularities and differentiable structures), in group theory (algebraic groups and simple finite groups connected with Lie groups), in the theory of differential equations ( $ K $. This fact makes it possible to introduce a large number of classical structures which induce such invariants of algebraic varieties that can be obtained by purely algebraic means only with great difficulty or perhaps not all. In the case of algebraic geometry over the field of complex numbers, every algebraic variety is simultaneously a complex-analytic, differentiable and topological space in the ordinary Hausdorff topology. In turn, algebra provides a flexible and powerful apparatus which is just as suitable for the conversion of a tentative reasoning into a proof as for the formulation of such proofs in their most obvious and most general form. This language implies problems, constructions and considerations which do not follow from the point of view of pure algebra. If the coordinates are real numbers, and if the space is two- or three-dimensional, the situation may be clearly visualized however, the language of geometry is also used in more general situations. The geometrical intuition appears when every "set of solutions" is identified with a "set of points in a coordinate space". Scheme Algebraic space).Īlgebraic geometry may be "naively" defined as the study of solutions of algebraic equations. Algebraic variety) and their various generalizations (schemes, algebraic spaces, etc., cf. The branch of mathematics dealing with geometric objects connected with commutative rings: algebraic varieties (cf. ![]()
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