  ### B.Sc Mathematics:

#### B Sc (Mathematics)

Logic, the study of principles of techniques and reasoning, is fundamental to every branch of learning. Besides, being the basis of all mathematical reasoning, it is required in the field of computer science for developing programming languages and also to check the correctness of the programs. Electronic engineers apply logic in the design of computer chips.

The mathematics required for viewing and analyzing the physical world around us is contained in calculus. While Algebra and Geometry provide us very useful tools for expressing the relationship between static quantities, the concepts necessary to explore the relationship between moving/changing objects are provided in calculus.

Exponential functions model a wide variety of phenomena of interest in science, engineering, mathematics, and economics. They arise naturally when we model the growth of a biological population, the spread of a disease, the radioactive decay of atoms, and the study of heat transfer problems, and so on.

Linear algebra is the study of linear systems of equations, vector spaces, and linear transformations. Virtually every area of mathematics relies on or extends the tools of linear algebra. Solving systems of linear equations is a basic tool of many mathematical procedures used for solving problems in science and engineering.

In this course, basic ideas and methods of real and complex analysis are taught. Real analysis is a theoretical version of single variable calculus. So many familiar concepts of calculus are reintroduced but at a much deeper and more rigorous level than in a calculus course. At the same time there are concepts and results that are new and not studied in the calculus course but very much needed in more advanced courses. The aim is to provide students with a level of mathematical sophistication that will prepare them for further work in mathematical analysis and other fields of knowledge and also to develop their ability to analyze and prove statements of mathematics using logical arguments.

The goal of numerical analysis is to provide techniques and algorithms to find an approximate numerical solution to problems in several areas of mathematics where it is impossible or hard to find the actual/closed-form solution by analytical methods and also to make an error analysis to ascertain the accuracy of the approximate solution. The subject addresses a variety of questions ranging from the approximation of functions and integrals to the approximate solution of algebraic, transcendental, differential, and integral equations, with particular emphasis on the stability, accuracy, efficiency, and reliability of numerical algorithms.

Linear programming problems are having wide applications in mathematics, statistics, computer science, economics, and many social and managerial sciences. For mathematicians it is a sort of mathematical modeling process, for statisticians and economists, it is useful for planning many economic activities such as transport of raw materials and finished products from one place to another with minimum cost, and for military heads, it is useful for scheduling the training activities and deployment of army personnel.

The course thoroughly exposes one to the rigor and methods of an analysis course. One has to understand the definitions and theorems of text and study examples well to acquire skills in various problem-solving techniques. The course will teach one how to combine different definitions, theorems and techniques to solve problems one has never seen before. One shall acquire the ability to realize when and how to apply a particular theorem and how to avoid common errors and pitfalls. The course will prepare students to formulate and present the ideas of mathematics and to communicate them elegantly.

The intention of the course is to extend the immensely useful ideas and notions such as limit, continuity, derivative,s and integral seen in the context of the function of a single variable to function of several variables. The corresponding results will be the higher-dimensional analogs of what we learned in the case of single-variable functions. The results we develop in the course of the calculus of multivariable are extremely useful in several areas of science and technology as many functions that arise in real-life situations are functions of multivariable.

The course is intended to find out ways and means for solving differential equations and the topic has a wide range of applications in physics, chemistry, biology, medicine, economics, and engineering.

By the end of program Bachelor of Science in Mathematics, students will be able:

• To enjoy and master several techniques of problem-solving such as recursion, induction, etc., the importance of pattern recognition in mathematics, the art of conjecturing, and a few applications of number theory. Enthusiastic students will have acquired knowledge to read and enjoy on their own a few applications of number theory in the field of art, geometry, and coding theory.
• To the fundamental ideas of limit, continuity, and differentiability and also to some basic theorems of differential calculus. It is also shown how these ideas can be applied in the problem of sketching curves and in the solution of some optimization problems of interest in real life.
• To study a related notion of convergence of a series, which is practically done by applying several different tests such as integral test, comparison test and so on.
• To learn the fundamentals of linear algebra by capturing the ideas geometrically, by justifying them algebraically, and by preparing them to apply it in several different fields such as data communication, computer graphics, modeling, etc.
• To understand the abstract notion of a group, learn several examples, are taught to check whether an algebraic system forms a group or not, and are introduced to some fundamental results of group theory. The idea of structural similarity, the notion of cyclic group, permutation group, various examples, and very fundamental results in the areas are also explored.
• To learn and deduce rigorously many properties of a real number system by assuming a few fundamental facts about it as axioms.
• To know about sequences, their limits, several basic and important theorems involving sequences, and their applications.
• To understand some basic topological properties of real number systems such as the concept of open and closed sets, their properties, their characterization, and so on.
• To get a rigorous introduction to algebraic, geometric, and topological structures of the complex number systems, functions of a complex variable, their limit and continuity, and so on.
• To solve linear programming problems geometrically, understand game theory, solve transportation and assignment problems by algorithms that take advantage of the simpler nature of these problems
• To understand several basic facts about parabola, hyperbola, and ellipse (conics) such as their equation in standard form, focal length properties, and reflection properties, their tangents, and normal.
• To recognize and classify conics.
• To identify a number of areas where the modeling process results in a differential equation.

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