Text 1 higher mathematics

There is no distinct boundary between elementary and higher mathematics, the division being conditional. But nevertheless we can point out some characteristic features of higher mathematics.

One of them is the universality, generality of its methods. As an example, let us take the problem of finding volumes of solids. Elementary mathematics gives us dif-ferent formulas for computing the volumes of a prism, pyramid, cone, sphere and some other simple solids. Each formula is obtained on the basis of a special argument which is rather complicated in certain cases. But in higher mathematics we have general formulas expressing the volume of any solid, the length of any curve, the area of any surface and the like.

There is another characteristic feature of higher mathematics. It is the systematic consideration of variable quantities . When investigating various objects and processes by means of elementary mathematics we usually regard such important quantities as velocities, accelerations, masses, forces etc as being invariable, constant. But if these quantities vary considerably (as is often the case) we cannot regard them at being constant. To solve such problems we usually apply higher mathematics. A branch of higher mathematics (called differential calculus) is particularly intended for solving various problems connected with an investigation of the dependence of one quantity upon another. The quantities and their interre¬lations can be of any nature (for instance, we can consider the relation between the acceleration, velocity and path length of a motion). Therefore differential calculus deeply penetrates into various natural sciences and enginee-ring.

The third characteristic feature of higher mathematics is the close relationship between its various divisions and the systematic unification of the computational, analytical (based on formulas) and geometric methods in contrast to elementary mathematics in which the connection between algebra and geometry is more or less accidental. In higher mathematics, the coordinate method reduces geometric problems to solving algebraic equations; graphs are used for representing relations between variable quantities; analytical methods of integral calculus are applied for computing areas and volumes of geometric figures and so on.

The most important divisions of higher mathematics were created in the 17th and 18th centuries. They include the coordinate method, differential and integral calculus etc. These divisions are represented mostly in the form they appeared after the works of L. Euler (1707—1783).

Mathematics was created by scientists of many countries. Among Russian mathematicians we should mention. N. I. Lobachevsky (1792—1856), the creator of a non-Euclidean geometry. An intensive development of mathematics in Petersburg began with the works of Academician M. V. Ostrogradsky (1801—1862) and continued by Academician P. L. Chebyshev (1821—1894). After Chebyshev most prominent representatives of the Petersburg mathematical school were A. A. Markov (1856—1922), the creator of the theory of random processes., and A. M. Lyapunov (1857—1918), the founder of the theory of stability.