Govt. Degree College

Department of Mathematics, Government College Banjar, Kullu, H.P. (175123)

Welcome to the Department of Mathematics! As German mathematician Carl Friedrich Gauss once said, “Mathematics is the queen of sciences.” Mathematics is not only a field of study but a vital part of human thought and reasoning, deeply embedded in our quest to understand the world around us. It builds mental discipline, sharpens logical thinking, and cultivates intellectual rigor – qualities that are valuable in every area of life.

Department of mathematics has two well qualified faculty members. Our department offers a structured and comprehensive program that equips students with essential skills and knowledge that extend beyond mathematics itself, supporting fields like science, economics, engineering, and even the arts. As an undergraduate student here, you will gain a strong foundation in mathematical theory and application, preparing you to excel in diverse pathways and tackle complex challenges with confidence and creativity.

Program Learning Objective (PLO) of the department:

The main objective of this subject is to cultivate a mathematical aptitude and nurture the interests of the students towards problem solving aptitude. Further, it aims at motivating the young minds for research in mathematical sciences and to train computational scientists who can work on real life challenging problems. Moreover, here is the following major program learning objective (PLO) of the department:

• Develop the ability to think critically, logically, and analytically and to use mathematical reasoning in everyday life.
• Create deep interest in learning mathematics.
• Communicate mathematics effectively by written, computational and graphic means.
• Develop broad and balanced knowledge and understanding of definitions, concepts, principles, and theorems.
• The program covers the full range of mathematics from Classical Calculus to modern Number Theory. Thus, it provides learners sufficient knowledge and skills to enable them undertake further studies in mathematics and its allied areas on multiple disciplines.
• Pursuing a degree in mathematics introduces the students to a number of interesting and useful ideas and helps them prepare for job exams in the field of education, research, government sector, business sector and industry.

Also, students majoring in Mathematics attain proficiency in Critical thinking: The ability to identify, reflect upon, evaluate, integrate, and apply different types of in- formation and knowledge to form independent judgments. Analytical and logical thinking and the habit of drawing conclusions based on quantitative information.

Problem Solving : The ability to assess and interpret complex situations, choose among several potentially appropriate mathematical methods of solution, persist in the face of difficulty, and present full and cogent solutions that include appropriate justification for their reasoning.

Effective communication : The ability to communicate and interact effectively with different audiences, developing their ability to collaborate intellectually and creatively in diverse contexts, and to appreciate ambiguity and nuance, while emphasizing the importance of clarity and precision in communication and reasoning.

Program Learning Objective (PLO) of the department:

• Enabling students to develop a positive attitude towards mathematics as an interesting and valuable subject of study.
• A student should get a relational understanding of mathematical concepts and concerned structures, and should be able to follow the patterns involved, mathematical reasoning.
• Ability to analyze a problem, identify and define the computing requirements, which may be appropriate to its solution.
• Introduction to various courses like Real analysis, Linear algebra, complex analysis etc.
• Enhancing students’ overall development and to equip them with mathematical modeling abilities, problem solving skills, creative talent, and power of communication necessary for various kinds of employment.
• Ability to pursue advanced studies and research in pure and applied mathematical science.
• Think in a critical manner.
• Know when there is a need for information, to be able to identify, locate, evaluate, and effectively use that information for the issue or problem at hand.
• Formulate and develop mathematical arguments in a logical manner.
• Acquire good knowledge and understanding in advanced areas of mathematics and statistics, chosen by the student from the given courses.
• Understand, formulate, and use quantitative models arising in social science, Business, and other contexts.

Course Specific Learning Objective (CLO):

DSE (Discipline Specific Elective):

Choices from DSE provide the students with liberty of exploring his interest within the mathematics. They enhance the ability of learners to apply the knowledge and skills acquired during the programme to solve specific theoretical and applied problems in mathematics.

SEC (Skill Enhancement Courses):

These courses enable the student to acquire the skill relevant to Mathematics. They familiarize the students with suitable tools of mathematical analysis to handle issue and problems in mathematics and related sciences.

Let us have a look on different courses of this programme:-

MATH 101 ^th : Differential Calculus

This course will enable the students to assimilate the notions of limits, continuity, and differentiability of functions at a point. Sketch curves in Cartesian and Polar co-ordinate systems, expansion of functions using Maclurin’s and Taylor’s theorems, curvature, asymptotes, and curve tracing.

On Completion of this course the students will be able to:

• Explain the relationship between the derivative of a function as a function and the notion of the derivative as the slope of the tangent line to a function at a point.
• Compare and contrast the ideas of continuity and differentiability.
• To inculcate to solve algebraic equations and inequalities involving the sequence root and modulus function.
• To able to calculate limits in indeterminate forms by a repeated use of L’ Hospital rule. To find the roots of algebraic and transcendental equations by using Rolls theorem and Mean value theorem and solve the Taylor's series and Maclarian series.
• To find maxima and minima, critical points and inflection points of functions of several variables and to determine the concavity and convexity, radius of curvature of curves.
• To able to evaluate integrals of rational functions by partial fractions and also Jacobian of functions.

MATH102^th : Differential Calculus

It will enable learners to learn various techniques of getting exact solutions of solvable first order differential equations and linear differential equations of higher order. They will be introduced to partial differential equations and their solutions.

On successful completion of the course, Students will be able to:

The main aim of the course is to introduce the students to the technique of solving various problems of engineering and science .

• Distinguish between linear, nonlinear, partial and ordinary differential equations.
• Solve basic application problems described by second order linear differential equations with constant coefficients and also with variable coefficient.
• Find the solutions of first order first degree differential equations, wronskian and its properties.
• Find the transforms of derivatives and integrals.
• Obtain the solution by Variation of parameters method, Cauchy- Euler equation and Legendre differential equations.
• Find the solutions of simultaneous differential equations and Total differential equations.
• To find the solutions of partial differential equations, Linear partial differential equation of first order, Lagrange’s method. Classification of second order partial differential equations into elliptical, parabolic and hyperbolic.

MATH102^th : Real Analysis

Learners will get concept of real line, completeness property, real sequences, infinite series, uniform convergence, and power series.

After completing the course students are expected to be able to:

• Describe the basic difference between the rational and real numbers. Give the definition of concepts related to metric spaces such as countability, compactness, convergent etc.
• Give the essence of the proof of Bolzano-Weistrass theorem the contraction theorem as well as existence of convergent subsequence using Cauchy ‘s Criteria.
• Evaluate the limits of wide class of real sequences.
• Determine whether or not real series are convergent by comparison test, p- test, root test and ratio test. We can also discuss the convergence of alternating series by using Leibnitz rule.
• To find solutions of sequences and series of functions. We can understand the concept of pointwise, and uniform convergence with the help of Mn -test and M test. Results about uniform convergence, power series and radius of convergence.
• Students will be able to demonstrate basic knowledge of key topics in classical real analysis.
• The course pervious the basic for further studies with in function analysis, topology & function Theory.

MATH102^th : Algebra

Students will learn about algebraic structures, groups, subgroups, their types; They will be also introduced to rings, sub rings, integral domains and fields.

On successful completion of the course, students will be able to:

• Students will be able to understand definition of group, abelian and non-abelian groups, the groups 𝑍𝑛 of integers under addition modulo n and the group U(n), Cyclic group, Normal subgroups, quotient groups.
• Understand group homomorphism.
• Understand basic theory of Rings, Commutative and Non-Commutative rings, Polynomial rings, rings of matrices, subring, Ideals, Integral domain, and fields in detail.

MATH309^th : Integral Calculus

This SEC will lead expertise in integration, rectification, and quadrature. Students will learn to handle double and triple integrals.

On successful completion of the course:

• This course will provide understanding of integration by partial fraction, integration of rational and irrational functions, properties of definite integrals, reduction formulae. Areas and lengths of curves in the plane, volumes, and surfaces of solids of revolution, Cartesian, and parametric forms. Double and triple integrals.

MATH310^th : Vector Calculus

It will teach students to find gradient divergence, curl, and vector integration and provides a peep into the beautiful world of Gauss, Green and Stokes theorem.

On successful completion of the course, students will be able to:

• Vector calculus motivates the study of vector differentiation and integration in two- and three-dimensional spaces.
• It helps to understand the students about Scalar and vector product of three and product of four vectors. Reciprocal vectors. Vector differentiation, scalar point function and vector point function. Derivative along a curve, directional derivatives. To understand the concept of orthogonal curvilinear coordinates. Gradient, Divergence, Curl, and Laplacian operator in terms of orthogonal curvilinear coordinates system.
• To understand the concept of vector integration: line integral, surface integral, volume integral. Theorem of Gauss, Green and Stokes and its applications.
• It is widely accepted as a prerequisite in various fields of science and engineering. It offers important tools for understanding functions (both real & complex) non-Euclidean geometry and topology.
• These tools are employed successfully in different branches of engineering and physics (such as electromagnetic fields, fluid flow and gravitational fields).

MATH313^th : Probability and Statistics

This course will change one’s sight of viewing different experiments by defining sample space, probability, providing in depth knowledge of different probability distributions and joint probability distribution functions.

On successful completion of the course, students will be able to:

• Understand basic theoretical and applied principles of statistics needed to enter the job force.
• They will have a better informative view on sample space, probability axioms, cumulative distribution functions, probability density function, mathematical expectations, moments, moment generating functions, Binomial, Poisson, continuous distribution. Joint cumulative distribution function & its properties, joint probability density functions, marginal and conditional distributions, expectation of function of two random variables.

MATH317^th : Transportation and Game Theory

No mathematics programme is complete without operations research. Operations research is that branch of mathematics which helps in optical decision making in various business situations. So, this section deals with solution of transportation problems, assignment problems and game theory.

The game theory provides powerful tools for analyzing transport systems and making decisions in situations. The students will be able to thoroughly grasp the topics like transportation problem and its mathematical formulation, optimal solution, Hungarian method for solving assignment problem. In game theory, solving 2-person zero sum games, games with mixed strategies, graphical solution.

MATH303^th : Linear Algebra

The main aim of the course is to introduce the students:

• Vector space, subspace sum and Direct sum of subspaces, Linear dependent, Linear independent subset of a vector space, spanning set and basis of a vector space.
• The homomorphism and isomorphism of a vector space also they can understand the concept of Linear transformation and its matrix representation. Null space and range space of Linear transformation, rank, and nullity with its applications.
• The algebra of Linear transformations minimal polynomials of Linear transformation and singular and non-singular transformations.
• The inner product space, Cauchy- Schwartz inequality. Orthogonal basis and orthonormal sets. Bessel’s inequality for finite dimensional vector space. Gram Schmidt orthogonalization process.

MATH305^th : Complex Analysis

The main aim of the course is to introduce the students:

• Technique of solving various problems of engineering and science
• Limits, limits involving the point at infinity continuity, properties of complex numbers, region in a complex plane function of complex variables and their mapping. Cauchy- Reimann equations
• The analytic functions, examples of analytic function derivatives and integrals of analytic function
• Understand the concept of contours integration and Cauchy integral formula.
• Liouville's theorem and the fundamental theorem of algebra. Convergence of series, Taylor series and Laurent series.