We share the vision of developing our intellectually vigorous community of students and faculty, together engaging in teaching and learning the advance knowledge in diverse areas of mathematics and support current progress in science.

The Mathematics Department offers a broad and challenging academic program that supports the mission of our college. We aim to provide high-quality education in graduate mathematics.

• To discover, mentor and nurture mathematically inclined students and provide them a supportive environment that fosters intellectual growth.

• To prepare our graduate students to develop the attitude and ability to apply mathematical methods and ideas in a wide variety of careers.

• The Mathematics Department provides a range of programs of study in mathematics including: courses, curriculum and instructional practices that support effective student learning.

• We also aim to promote and sustain : 

• An environment that fosters creativity, critical thinking, enquiry and active learning.

• Equity, inclusion and diversity amongst our staff, faculty and students, aiming to increase participation from groups that are historically under-represented in Mathematics.

• The continued growth of faculty as both teachers and scholars of mathematics with professional involvement.

• Opportunities for close interaction between students and faculty both within and beyond the classroom, including student mentoring. 

 

1. Knowledge and understanding:

• know and demonstrate understanding of the concepts from the five branches of mathematics (number theory, real analysis, calculus, trigonometry and complex integration)

• Use appropriate mathematical concepts and skills to solve problems in both familiar and unfamiliar situations.

• Select and apply general rules correctly to solve problems including those in real-life contexts.

2. Investigating patterns

• select and apply appropriate inquiry and mathematical problem-solving techniques

• recognize patterns

• describe patterns as relationships or general rules

• draw conclusions consistent with findings

• Justify or prove mathematical relationships and general rules.

3. Communication in mathematics

• use appropriate mathematical language (notation, symbols, terminology) in both oral and written explanations

• use different forms of mathematical representation (formulae, diagrams, tables, charts, graphs and models)

• Move between different forms of representation.

4. Reflection in mathematics

• use appropriate mathematical language (notation, symbols, terminology) in both oral and written explanations

• use different forms of mathematical representation (formulae, diagrams, tables, charts, graphs and models)

• move between different forms of representation.

A V KANTHAMMA COLLEGE FOR WOMEN, HASSAN

DEPARTMENT OF MATHEMATICS

PROGRAM OUTCOME for the academic year 2022-23

NEP:

PO 1	Disciplinary Knowledge: Bachelor degree in Mathematics is the culmination of in-depth knowledge of Algebra, Calculus, Geometry, differential equations and several other branches of pure and applied mathematics. This also leads to study the related areas such as computer science and other allied subjects
PO 2	Communication Skills: Ability to communicate various mathematical concepts effectively using examples and their geometrical visualization. The skills and knowledge gained in this program will lead to the proficiency in analytical reasoning which can be used for modeling and solving of real life problems.
PO 3	Critical thinking and analytical reasoning: The students undergoing this programmer acquire ability of critical thinking and logical reasoning and capability of recognizing and distinguishing the various aspects of real life problems.
PO 4	Problem Solving: The Mathematical knowledge gained by the students through this programmer develop an ability to analysis the problems, identify and define appropriate computing requirements for its solutions. This programmer enhances students overall development and also equip them with mathematical modeling, ability, problem solving skills.
PO 5	Research related skills: The completing this programmer develop the capability of inquiring about appropriate questions relating to the Mathematical concepts in different areas of Mathematics.
PO 6	Information/digital Literacy: The completion of this programmer will enable the learner to use appropriate software to solve system of algebraic equation and differential equations.
PO 7	Self – directed learning: The student completing this program will develop an ability of working independently and to make an in-depth study of various notions of Mathematics.
PO 8	Moral and ethical awareness/reasoning: The student completing this program will develop an ability to identify unethical behavior such as fabrication, falsification or misinterpretation of data and adopting objectives, unbiased and truthful actions in all aspects of life in general and Mathematical studies in particular.
PO 9	Lifelong learning: This programmer provides self-directed learning and lifelong learning skills. This programmer helps the learner to think independently and develop algorithms and computational skills for solving real word problems.
PO 10	Ability to peruse advanced studies and research in pure and applied Mathematical sciences.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PROGRAM SPECIFIC OUTCOME

2018-19

I SEMESTER: ALGEBRA-I and CALCULUS-I[DSC- MATH-01]

PSO-1 Matrices

· Rank of matrix, Elementary row operations, Inverse of a non-singular matrix.

· Systems of m linear equations in n unknowns

· Eigen values and eigenvectors of a square matrix, Diagonalization of a real symmetric matrix, Cayley Hamilton theorem, Applications to determine the powers of square matrices and inverse of non-singular matrices.

PSO-2 Theory of Equations

· Theory of equations, Euclid’s algorithm, Polynomials with integral coefficients, Remainder theorem, Factors theorem, Fundamental theorem of algebra

· Irrational and complex roots occurring in conjugate pairs, Relation between roots and coefficients of polynomial equation.

· Symmetric function, Transformation, Reciprocal equation.

· Descartes’ rule of signs, Multiple roots, Solving cubic equations by Cordon’s method, Solving quadric equations by Descartes method.

PSO-3 Differential Calculus-I and Integral Calculus-I

· Derivative of a function, Derivative of higher order- n^th derivative of some standard functions, Problems.

· Leibnitz theorem, Maxima and Minima.

· Concavity, Convexity and Point of inflection.

· Reductionformulae for,,,,,, , , ,  with definite limits.

PSO-4 Differential Calculus-II

· Polar coordinates- angle between the radius vector and the tangent at a point on a curve, angel of intersection between two curves.

· Pedal equations, Derivative of arc.

· Coordinates of center of curvature, Radius of curvature, Circle of curvature, Evolutes.

 

 

II SEMESTER: CALCULUS-II and THEORY OF NUMBERS [DSC-MATH-02]

PSO-I Limits, Continuity and Differentiability

· Limit of a function- Properties and Problems.

· Continuity of function- Properties and Problems, Infimum and supremum of a function, Theorems on continuity, Intermediate value theorem.

· Differentiability.

PSO-II Differential Calculus-III

· Rolle’s Theorem, Lagrange’s mean value theorem, Couchy’s mean value theorem, Taylor’s theorem, Maclaurin’s theorem.

· Taylor’s infinite series, power series expansion, Maclaurin’s infinite series.

· Indeterminate forms.

PSO-3 Partial Derivatives

· Function of two or more variables, Explicit and implicit functions, the neighbourhood of appoint.

· Limit of a function, Continuity, Partial derivatives.

· Homogeneous function, Euler’s theorem, Chain rule, Change of variables, Directional derivative.

· Partial derivatives of higher order, Taylor’s theorem for two variables, Derivative of implicit functions, Jacobians.  

PSO-4 Theory of Numbers

· Division Algorithm, Divisibility, Prime and composite numbers, Euclidean algorithm.

· Fundamental theorem of arithmetic, Greatest common divisors and least common multiple.

· Congruences, Linear congruences, Simultaneous Congruences, Wilson’s, Euler’s and Fermat’s theorem and their applications.

II SEMESTER:ALGEBRA-II AND DIFFERENTIAL EQUATIONS-1 [DSC-MATH-03]

PSO-1: GROUP THEORY

· Definition and examples of groups,some general properties of groups, Group of permutations.

· Powers of an element of a group, Subgroups,cyclic groups problems and theorems.

· Cosets, Index of a group,Lagrange’s theorem,consequences.

PSO-2:NORMAL SUBGROUPS AND HOMOMORPHISM

· Normal subgroups,Quotient groups.

· Homomorphism,Kernel of homomorphism,Isomorphism,Automorphism,Fundamental theorem of homomorphism.

PSO-3:DIFFERENTIAL EQUATIONS

· Recapitulation of Definition,examples of differential equations,formation of differential equations by elimination of arbitrary constants.

· Differential equations of first order.

· The general solution of a linear equation.

PSO-4:ORDINARY DIFFERENTIAL EQUATIONS

· Ordinary Linear differential equations with constant coefficients,Complementary function, Particular integral, Inverse differential operators

· Cauchy-Euler differential equations, Simultaneous differential equations.

IV SEMESTER:DIFFERENTIAL EQUATIONS-II AND REAL ANALYSIS-I [DSC-MATH-04]

PSO-1: LINEAR DIFFERENTIAL EQUATIONS

· Solution of ordinary second order linear differential equations with variable coefficient by various methods: Changing the independent variable, changing the dependent variable,by method of variation of parameters,exact equations.

· Total differential equations.

· Simultaneous equations.

PSO-2:PARTIAL DIFFERENTIAL EQUATIONS

· Basic Concepts-formation of partial differential equations by elimination of arbitrary constants and functions.

· Solution of partial differential equations.

· Standard types of first order non-linear partial differential equations.

· Rules for finding the Complementary function,Rules for finding theparticular integral, Method of separation of variables.

PSO-3:RIEMANN INTEGRATION AND LINE INTEGRAL

· The Riemann integral-Upper and lower sums,criterion for integrability.

· Properties of Riemann integrals,Fundamental theorem of calculus.

· Integration as a limit of sum.

· Definition of line integral and basic properties-examples on evaluation of line integrals.

PSO-4:MULTIPLE INTEGRALS

· Definition of a double integral,Evaluation of double integrals,Surface areas.

· Definition of a triple integral,Evaluation of triple integrals,Volume as atriple integral.

V SEMESTER: REAL ANALYSIS –II AND ALGEBRA-III [DSE-MATH-01]

PSO-1: Sequences

· Sequence   of real numbers, Bounded and unbounded sequences, infimum and supremum of a sequence.

· Limit of a sequence.

· Convergent, divergent and oscillatory sequences, Standard properties.

· Monotonic sequences and their properties, Cauchy’s general principle of convergence.

PSO-2:INFINITE SERIES

· Infinite series of real numbers, convergence, divergence and oscillation of series.

· Series of positive terms, geometric series, p- series, comparison tests.

· D’ Alembert’s ratio test, Raabe’s test, Cauchy’s root test, Leibnitz’s test for alternating series.

· Summation of Binomial, Exponential and Logarithmic series.

PSO-3: RINGS AND FIELDS

· Rings, Integral Domains, Division rings, sub rings.

· Fields, Subfields.

· Characteristic of a ring.

· Ideals, Algebra of Ideals,Principal ideal ring, Divisibility in integral domain, Units and Associates.

PSO-4: POLYNOMIAL RINGS AND HOMOMORPHISMS

· Polynomial rings, Divisibility, reducible polynomials, DivisionAlgorithm, Greatest common divisors, Euclidean Algorithm, unique factorization theorem.

· Quotient rings.

· Homomorphism of rings, Kernel of a ring homomorphism, Fundamental theorem of homomorphism.

· Maximal ideals, Prime ideals, Eisenstein’s criterion of irreducibility.

V SEMESTER: NUMERICAL ANALYSIS [SEC-MATH-02]

 PSO-1: NUMERICAL ANALYSIS

· Numerical solutions of Algebraic and transcendental equations, Bisection method, the method of false position (Regula-falsi method), Newton –Raphson method.

· Numerical solutions of first order linear differential equations: Euler-Cauchy method, Euler’s modified method, Runge-Kutta fourth order method, Picard’s method.

PSO-2: FINITE DIFFERENCES AND NUMERICAL INTEGRATION

· Forward and Backward difference.

· Newton –Gregory forward and backward interpolation formulae.

· Lagrange’s interpolation formulae.

· General Quadrature formula: Trapezoidal rule, Simpson’s 1/3^rd rule, Simpson’s 3/8^thrule, Weddle’s rule.

 VI SEMESTER: ALGEBRA-IV AND COMPLEX ANALYSIS- [DSE-MATH-02]

PSO-1: VECTOR SPACES

· Vector spaces, vectorsubspaces, Algebra of subspaces.

· Liner Combination, linear span, linear dependence and independence of vectors.

· Basis of a vector space, dimension of vector space.

· Quotient spaces, Homomorphism of vector spaces, isomorphism of vector spaces

· Direct sums.

PSO-2: LINEAR TRANSFORMATIONS

· Linear transformation, Change of basis and effect of associated matrices.

· Kernel and image of a linear transformation.

· Rank and Nullity theorem.

· Eigen values and Eigen vectors of a linear transformation.

PSO-3: FUNCTIONS OF A COMPLEX VARIABLE

· Equation to a circle and straight line in complex form.

· Continuity and Differentiability, Analytic functions, Singular points.

· Cauchy Riemann equations in Cartesian and Polar form.

· Necessary and sufficient condition for function to be analytic.

· Harmonic functions.

· Construction of analytic function: Milne Thomson method, using the concept of Harmonic function.

PSO-4: TRANSFORMATIONS

· Definition, Jacobean of transformation, Identity transformation.

· Linear transformation.

· Bilinear transformations, Cross Ratio of Four points.

· Conformal mappings, Discussion of the transformations w=z², w=sinz, w=e^z, w=.

VI SEMESTER: COMPLEX ANALYSIS AND IMPROPER INTEGRALS [SEC-MATH-03]

  PSO-1: COMPLEX INTEGRATION

· The complex line integral, proof of Cauchy’s integral theorem using Green’s theorem.

· Cauchy’s integral formula for the function and the derivatives.

· Cauchy’s inequality, Liouville’s theorem.

· Fundamental theorem of Algebra.

PSO-2: IMPROPER INTEGRALS

· Definition, Gamma and Beta functions.

· Relation between Beta and Gamma functions.

· Application to evaluation of integrals.

· Duplication formula.

PROGRAM SPECIFIC OUTCOME

2019-20

I SEMESTER: ALGEBRA-I and CALCULUS-I [DSC- MATH-01]

PSO-1 Matrices

· Rank of matrix, Elementary row operations, Inverse of a non-singular matrix.

· Systems of m linear equations in n unknowns

· Eigen values and eigenvectors of a square matrix, Diagonalization of a real symmetric matrix, Cayley Hamilton theorem, Applications to determine the powers of square matrices and inverse of non-singular matrices.

PSO-2 Theory of Equations

· Theory of equations, Euclid’s algorithm, Polynomials with integral coefficients, Remainder theorem, Factors theorem, Fundamental theorem of algebra

· Irrational and complex roots occurring in conjugate pairs, Relation between roots and coefficients of polynomial equation.

· Symmetric function, Transformation, Reciprocal equation.

· Descartes’ rule of signs, Multiple roots, Solving cubic equations by Cordon’s method, Solving quadric equations by Descartes method.  

PSO-3 Differential Calculus-I and Integral Calculus-I

· Derivative of a function, Derivative of higher order- n^th derivative of some standard functions, Problems.

· Leibnitz theorem, Maxima and Minima.

· Concavity, Convexity and Point of inflection.

· Reduction formulae for,,,,,, , , ,  with definite limits.

PSO-4 Differential Calculus-II

· Polar coordinates- angle between the radius vector and the tangent at a point on a curve, angel of intersection between two curves.

· Pedal equations, Derivative of arc.

· Coordinates of center of curvature, Radius of curvature, Circle of curvature, Evolutes.

 

II SEMESTER: CALCULUS-II and THEORY OF NUMBERS [DSC-MATH-02]

PSO-I Limits, Continuity and Differentiability

· Limit of a function- Properties and Problems.

· Continuity of function- Properties and Problems, Infimum and supremum of a function, Theorems on continuity, Intermediate value theorem.

· Differentiability.

PSO-II Differential Calculus-III

· Rolle’s Theorem, Lagrange’s mean value theorem, Couchy’s mean value theorem, Taylor’s theorem, Maclaurin’s theorem.

· Taylor’s infinite series, power series expansion, Maclaurin’s infinite series.

· Indeterminate forms.

PSO-3 Partial Derivatives

· Function of two or more variables, Explicit and implicit functions, the neighbourhood of appoint.

· Limit of a function, Continuity, Partial derivatives.

· Homogeneous function, Euler’s theorem, Chain rule, Change of variables, Directional derivative.

· Partial derivatives of higher order, Taylor’s theorem for two variables, Derivative of implicit functions, Jacobians.  

PSO-4 Theory of Numbers

· Division Algorithm, Divisibility, Prime and composite numbers, Euclidean algorithm.

· Fundamental theorem of arithmetic, Greatest common divisors and least common multiple.

· Congruences, Linear congruences, Simultaneous Congruences, Wilson’s, Euler’s and Fermat’s theorem and their applications.

 III SEMESTER: ALGEBRA-II AND DIFFERENTIAL EQUATIONS-1 [DSC-MATH-03]

PSO-1: GROUP THEORY

· Definition and examples of groups, some general properties of groups, Group of permutations.

· Powers of an element of a group, Subgroups, cyclic groups problems and theorems.

· Cosets, Index of a group, Lagrange’s theorem, consequences.

PSO-2: NORMAL SUBGROUPS AND HOMOMORPHISM

· Normal subgroups, Quotient groups.

· Homomorphism, Kernel of homomorphism, Isomorphism, Automorphism, Fundamental theorem of homomorphism.

PSO-3: DIFFERENTIAL EQUATIONS

· Recapitulation of Definition, examples of differential equations, formation of differential equations by elimination of arbitrary constants.

· Differential equations of first order.

· The general solution of a linear equation.

PSO-4: ORDINARY DIFFERENTIAL EQUATIONS

· Ordinary Linear differential equations with constant coefficients, Complementary function, Particular integral, Inverse differential operators

· Cauchy-Euler differential equations, Simultaneous differential equations.

IV SEMESTER: DIFFERENTIAL EQUATIONS-II AND REAL ANALYSIS-I [DSC-MATH-04]

PSO-1: LINEAR DIFFERENTIAL EQUATIONS

· Solution of ordinary second order linear differential equations with variable coefficient by various methods: Changing the independent variable, changing the dependent variable, by method of variation of parameters, exact equations.

· Total differential equations.

· Simultaneous equations.

PSO-2: PARTIAL DIFFERENTIAL EQUATIONS

· Basic Concepts-formation of partial differential equations by elimination of arbitrary constants and functions.

· Solution of partial differential equations.

· Standard types of first order non-linear partial differential equations.

· Rules for finding the Complementary function, Rules for finding the particular integral, Method of separation of variables.

PSO-3: RIEMANN INTEGRATION AND LINE INTEGRAL

· The Riemann integral-Upper and lower sums, criterion for integrability.

· Properties of Riemann integrals, Fundamental theorem of calculus.

· Integration as a limit of sum.

· Definition of line integral and basic properties-examples on evaluation of line integrals.

PSO-4: MULTIPLE INTEGRALS

· Definition of a double integral, Evaluation of double integrals, Surface areas.

· Definition of a triple integral, Evaluation of triple integrals, Volume as a triple integral.

V SEMESTER: REAL ANALYSIS –II AND ALGEBRA-III [DSE-MATH-01]

PSO-1: Sequences

· Sequence   of real numbers, Bounded and unbounded sequences, infimum and supremum of a sequence.

· Limit of a sequence.

· Convergent, divergent and oscillatory sequences, Standard properties.

· Monotonic sequences and their properties, Cauchy’s general principle of convergence.

PSO-2: INFINITE SERIES

· Infinite series of real numbers, convergence, divergence and oscillation of series.

· Series of positive terms, geometric series, p- series, comparison tests.

· D’ Alembert’s ratio test, Raabe’s test, Cauchy’s root test, Leibnitz’s test for alternating series.

· Summation of Binomial, Exponential and Logarithmic series.

PSO-3: RINGS AND FIELDS

· Rings, Integral Domains, Division rings, sub rings.

· Fields, Subfields.

· Characteristic of a ring.

· Ideals, Algebra of Ideals, Principal ideal ring, Divisibility in integral domain, Units and Associates.    

PSO-4: POLYNOMIAL RINGS AND HOMOMORPHISMS

· Polynomial rings, Divisibility, reducible polynomials, Division Algorithm, Greatest common divisors, Euclidean Algorithm, unique factorization theorem.

· Quotient rings.

· Homomorphism of rings, Kernel of a ring homomorphism, Fundamental theorem of homomorphism.

· Maximal ideals, Prime ideals, Eisenstein’s criterion of irreducibility.

V SEMESTER: NUMERICAL ANALYSIS [SEC-MATH-02]

 PSO-1: NUMERICAL ANALYSIS

· Numerical solutions of Algebraic and transcendental equations, Bisection method, the method of false position (Regula-falsi method), Newton –Raphson method.

· Numerical solutions of first order linear differential equations: Euler-Cauchy method, Euler’s modified method, Runge-Kutta fourth order method, Picard’s method.

PSO-2: FINITE DIFFERENCES AND NUMERICAL INTEGRATION

· Forward and Backward difference.

· Newton –Gregory forward and backward interpolation formulae.

· Lagrange’s interpolation formulae.

· General Quadrature formula: Trapezoidal rule, Simpson’s 1/3^rd rule, Simpson’s 3/8^th rule, Weddle’s rule.

VI SEMESTER: ALGEBRA-IV AND COMPLEX ANALYSIS- [DSE-MATH-02]

PSO-1: VECTOR SPACES

· Vector spaces, vector subspaces, Algebra of subspaces.

· Liner Combination, linear span, linear dependence and independence of vectors.

· Basis of a vector space, dimension of vector space.

· Quotient spaces, Homomorphism of vector spaces, isomorphism of vector spaces

· Direct sums.

PSO-2: LINEAR TRANSFORMATIONS

· Linear transformation, Change of basis and effect of associated matrices.

· Kernel and image of a linear transformation.

· Rank and Nullity theorem.

· Eigen values and Eigen vectors of a linear transformation.

PSO-3: FUNCTIONS OF A COMPLEX VARIABLE

· Equation to a circle and straight line in complex form.

· Continuity and Differentiability, Analytic functions, Singular points.

· Cauchy Riemann equations in Cartesian and Polar form.

· Necessary and sufficient condition for function to be analytic.

· Harmonic functions.

· Construction of analytic function: Milne Thomson method, using the concept of Harmonic function.

PSO-4: TRANSFORMATIONS

· Definition, Jacobean of transformation, Identity transformation.

· Linear transformation.

· Bilinear transformations, Cross Ratio of Four points.

· Conformal mappings, Discussion of the transformations w=z², w=sinz, w=e^z, w=.

VI SEMESTER: COMPLEX ANALYSIS AND IMPROPER INTEGRALS [SEC-MATH-03]

  PSO-1: COMPLEX INTEGRATION

· The complex line integral, proof of Cauchy’s integral theorem using Green’s theorem.

· Cauchy’s integral formula for the function and the derivatives.

· Cauchy’s inequality, Liouville’s theorem.

· Fundamental theorem of Algebra.

PSO-2: IMPROPER INTEGRALS

· Definition, Gamma and Beta functions.

· Relation between Beta and Gamma functions.

· Application to evaluation of integrals.

· Duplication formula.

PROGRAM SPECIFIC OUTCOME

2020-21

I SEMESTER: ALGEBRA-I and CALCULUS-I [DSC- MATH-01]

PSO-1 Matrices

· Rank of matrix, Elementary row operations, Inverse of a non-singular matrix.

· Systems of m linear equations in n unknowns

· Eigen values and eigenvectors of a square matrix, Diagonalization of a real symmetric matrix, Cayley Hamilton theorem, Applications to determine the powers of square matrices and inverse of non-singular matrices.

PSO-2 Theory of Equations

· Theory of equations, Euclid’s algorithm, Polynomials with integral coefficients, Remainder theorem, Factors theorem, Fundamental theorem of algebra

· Irrational and complex roots occurring in conjugate pairs, Relation between roots and coefficients of polynomial equation.

· Symmetric function, Transformation, Reciprocal equation.

· Descartes’ rule of signs, Multiple roots, Solving cubic equations by Cordon’s method, Solving quadric equations by Descartes method.  

PSO-3 Differential Calculus-I and Integral Calculus-I

· Derivative of a function, Derivative of higher order- n^th derivative of some standard functions, Problems.

· Leibnitz theorem, Maxima and Minima.

· Concavity, Convexity and Point of inflection.

· Reduction formulae for,,,,,, , , ,  with definite limits.

PSO-4 Differential Calculus-II

· Polar coordinates- angle between the radius vector and the tangent at a point on a curve, angel of intersection between two curves.

· Pedal equations, Derivative of arc.

· Coordinates of center of curvature, Radius of curvature, Circle of curvature, Evolutes.

 

II SEMESTER: CALCULUS-II and THEORY OF NUMBERS [DSC-MATH-02]

PSO-I Limits, Continuity and Differentiability

· Limit of a function- Properties and Problems.

· Continuity of function- Properties and Problems, Infimum and supremum of a function, Theorems on continuity, Intermediate value theorem.

· Differentiability.

PSO-II Differential Calculus-III

· Rolle’s Theorem, Lagrange’s mean value theorem, Couchy’s mean value theorem, Taylor’s theorem, Maclaurin’s theorem.

· Taylor’s infinite series, power series expansion, Maclaurin’s infinite series.

· Indeterminate forms.

PSO-3 Partial Derivatives

· Function of two or more variables, Explicit and implicit functions, the neighbourhood of appoint.

· Limit of a function, Continuity, Partial derivatives.

· Homogeneous function, Euler’s theorem, Chain rule, Change of variables, Directional derivative.

· Partial derivatives of higher order, Taylor’s theorem for two variables, Derivative of implicit functions, Jacobians.  

PSO-4 Theory of Numbers

· Division Algorithm, Divisibility, Prime and composite numbers, Euclidean algorithm.

· Fundamental theorem of arithmetic, Greatest common divisors and least common multiple.

· Congruences, Linear congruences, Simultaneous Congruences, Wilson’s, Euler’s and Fermat’s theorem and their applications.

III SEMESTER: ALGEBRA-II AND DIFFERENTIAL EQUATIONS-1 [DSC-MATH-03]

PSO-1: GROUP THEORY

· Definition and examples of groups, some general properties of groups, Group of permutations.

· Powers of an element of a group, Subgroups, cyclic groups problems and theorems.

· Cosets, Index of a group, Lagrange’s theorem, consequences.

PSO-2: NORMAL SUBGROUPS AND HOMOMORPHISM

· Normal subgroups, Quotient groups.

· Homomorphism, Kernel of homomorphism, Isomorphism, Automorphism, Fundamental theorem of homomorphism.

PSO-3: DIFFERENTIAL EQUATIONS

· Recapitulation of Definition, examples of differential equations, formation of differential equations by elimination of arbitrary constants.

· Differential equations of first order.

· The general solution of a linear equation.

PSO-4: ORDINARY DIFFERENTIAL EQUATIONS

· Ordinary Linear differential equations with constant coefficients, Complementary function, Particular integral, Inverse differential operators

· Cauchy-Euler differential equations, Simultaneous differential equations.

IV SEMESTER: DIFFERENTIAL EQUATIONS-II AND REAL ANALYSIS-I [DSC-MATH-04]

PSO-1: LINEAR DIFFERENTIAL EQUATIONS

· Solution of ordinary second order linear differential equations with variable coefficient by various methods: Changing the independent variable, changing the dependent variable, by method of variation of parameters, exact equations.

· Total differential equations.

· Simultaneous equations.

PSO-2: PARTIAL DIFFERENTIAL EQUATIONS

· Basic Concepts-formation of partial differential equations by elimination of arbitrary constants and functions.

· Solution of partial differential equations.

· Standard types of first order non-linear partial differential equations.

· Rules for finding the Complementary function, Rules for finding the particular integral, Method of separation of variables.

PSO-3: RIEMANN INTEGRATION AND LINE INTEGRAL

· The Riemann integral-Upper and lower sums, criterion for integrability.

· Properties of Riemann integrals, Fundamental theorem of calculus.

· Integration as a limit of sum.

· Definition of line integral and basic properties-examples on evaluation of line integrals.

PSO-4: MULTIPLE INTEGRALS

· Definition of a double integral, Evaluation of double integrals, Surface areas.

· Definition of a triple integral, Evaluation of triple integrals, Volume as a triple integral.

V SEMESTER: REAL ANALYSIS –II AND ALGEBRA-III [DSE-MATH-01]

PSO-1: Sequences

· Sequence   of real numbers, Bounded and unbounded sequences, infimum and supremum of a sequence.

· Limit of a sequence.

· Convergent, divergent and oscillatory sequences, Standard properties.

· Monotonic sequences and their properties, Cauchy’s general principle of convergence.

PSO-2: INFINITE SERIES

· Infinite series of real numbers, convergence, divergence and oscillation of series.

· Series of positive terms, geometric series, p- series, comparison tests.

· D’ Alembert’s ratio test, Raabe’s test, Cauchy’s root test, Leibnitz’s test for alternating series.

· Summation of Binomial, Exponential and Logarithmic series.

PSO-3: RINGS AND FIELDS

· Rings, Integral Domains, Division rings, sub rings.

· Fields, Subfields.

· Characteristic of a ring.

· Ideals, Algebra of Ideals, Principal ideal ring, Divisibility in integral domain, Units and Associates.    

PSO-4: POLYNOMIAL RINGS AND HOMOMORPHISMS

· Polynomial rings, Divisibility, reducible polynomials, Division Algorithm, Greatest common divisors, Euclidean Algorithm, unique factorization theorem.

· Quotient rings.

· Homomorphism of rings, Kernel of a ring homomorphism, Fundamental theorem of homomorphism.

· Maximal ideals, Prime ideals, Eisenstein’s criterion of irreducibility.

 

 

 

 

 

V SEMESTER: NUMERICAL ANALYSIS [SEC-MATH-02]

 PSO-1: NUMERICAL ANALYSIS

· Numerical solutions of Algebraic and transcendental equations, Bisection method, the method of false position (Regula-falsi method), Newton –Raphson method.

· Numerical solutions of first order linear differential equations: Euler-Cauchy method, Euler’s modified method, Runge-Kutta fourth order method, Picard’s method.

PSO-2: FINITE DIFFERENCES AND NUMERICAL INTEGRATION

· Forward and Backward difference.

· Newton –Gregory forward and backward interpolation formulae.

· Lagrange’s interpolation formulae.

· General Quadrature formula: Trapezoidal rule, Simpson’s 1/3^rd rule, Simpson’s 3/8^th rule, Weddle’s rule.

VI SEMESTER: ALGEBRA-IV AND COMPLEX ANALYSIS- [DSE-MATH-02]

PSO-1: VECTOR SPACES

· Vector spaces, vector subspaces, Algebra of subspaces.

· Liner Combination, linear span, linear dependence and independence of vectors.

· Basis of a vector space, dimension of vector space.

· Quotient spaces, Homomorphism of vector spaces, isomorphism of vector spaces

· Direct sums.

PSO-2: LINEAR TRANSFORMATIONS

· Linear transformation, Change of basis and effect of associated matrices.

· Kernel and image of a linear transformation.

· Rank and Nullity theorem.

· Eigen values and Eigen vectors of a linear transformation.

PSO-3: FUNCTIONS OF A COMPLEX VARIABLE

· Equation to a circle and straight line in complex form.

· Continuity and Differentiability, Analytic functions, Singular points.

· Cauchy Riemann equations in Cartesian and Polar form.

· Necessary and sufficient condition for function to be analytic.

· Harmonic functions.

· Construction of analytic function: Milne Thomson method, using the concept of Harmonic function.

PSO-4: TRANSFORMATIONS

· Definition, Jacobean of transformation, Identity transformation.

· Linear transformation.

· Bilinear transformations, Cross Ratio of Four points.

· Conformal mappings, Discussion of the transformations w=z², w=sinz, w=e^z, w=.

VI SEMESTER: COMPLEX ANALYSIS AND IMPROPER INTEGRALS [SEC-MATH-03]

  PSO-1: COMPLEX INTEGRATION

· The complex line integral, proof of Cauchy’s integral theorem using Green’s theorem.

· Cauchy’s integral formula for the function and the derivatives.

· Cauchy’s inequality, Liouville’s theorem.

· Fundamental theorem of Algebra.

PSO-2: IMPROPER INTEGRALS

· Definition, Gamma and Beta functions.

· Relation between Beta and Gamma functions.

· Application to evaluation of integrals.

· Duplication formula.

PROGRAM SPECIFIC OUTCOME 2021-22

 

I SEMESTER: ALGEBRA-I and CALCULUS-I [MATDSCT 1.1]

 

PSO-1 MATRICES

· Rank of matrix, Elementary row operations, Inverse of a non-singular matrix.

· Systems of m linear equations in n unknowns

· Eigen values and eigenvectors of a square matrix, Diagonalization of a real symmetric matrix, Cayley Hamilton theorem, Applications to determine the powers of square matrices and inverse of non-singular matrices.

PSO-2 THEORY OF EQUATIONS

· Theory of equations, Euclid’s algorithm, Polynomials with integral coefficients, Remainder theorem, Factors theorem, Fundamental theorem of algebra

· Irrational and complex roots occurring in conjugate pairs, Relation between roots and coefficients of polynomial equation.

· Symmetric function, Transformation, Reciprocal equation.

· Descartes’ rule of signs, multiple roots, solving cubic equations by Cordon’s method, solving quadric equations by Descartes method.  

PSO-3POLAR CO-ORDINATES

· Polar coordinates- angle between the radius vector and the tangent at a point on a curve, angel of intersection between two curves.

· Pedal equations, Derivative of arc.

Coordinates of center of curvature, Radius of curvature, Circle of curvature.

PSO-4 SUCCESSIVE DIFFERENTIATION AND INTEGRAL CALCULUS-I

· Derivative of a function, Derivative of higher order- n^th derivative of some standard functions, Problems.

· Leibnitz theorem, Maxima and Minima.

· Concavity, Convexity and Point of inflection.

· Reduction formulae for,,,,,, , , ,  with definite limits.

 

II SEMESTER: ALGEBRA-II and CALCULUS-II [MATDSCT 2.1]

PSO-I NUMBER THEORY

· Division Algorithm, Divisibility, Prime and composite numbers, Euclidean algorithm.

· Fundamental theorem of arithmetic, Greatest common divisors and least common multiple.

· Congruences, Linear congruences, Simultaneous Congruences, Wilson’s, Euler’s and Fermat’s theorem and their applications.

PSO-II DIFFERENTIAL CALCULUS-I

· Limit of a function- Properties and Problems.

· Continuity of function- Properties and Problems, Infimum and supremum of a function, Theorems on continuity, Intermediate value theorem.

· Differentiability.

· Rolle’s Theorem, Lagrange’s mean value theorem, Couchy’s mean value theorem, Taylor’s theorem, Maclaurin’s theorem.

· Taylor’s infinite series, power series expansion, Maclaurin’s infinite series.

· Indeterminate forms.

PSO-IIIPARTIAL DERIVATIVES

· Function of two or more variables, Explicit and implicit functions, the neighbourhood of appoint.

· Limit of a function, Continuity, Partial derivatives.

· Homogeneous function, Euler’s theorem, Chain rule, Change of variables, Directional derivative.

· Partial derivatives of higher order, Taylor’s theorem for two variables, Derivative of implicit functions, Jacobians.  

PSO-IVINTEGRAL CALCULUS-II

· Definition of line integral and basic properties-examples on evaluation of line integrals.

· Definition of a double integral, Evaluation of double integrals, Surface areas.

· Definition of a triple integral, Evaluation of triple integrals, Volume as a triple integral.

III SEMESTER: ALGEBRA-II AND DIFFERENTIAL EQUATIONS-1 [DSC-MATH-03]

PSO-1: GROUP THEORY

· Definition and examples of groups, some general properties of groups, Group of permutations.

· Powers of an element of a group, Subgroups, cyclic groups problems and theorems.

· Cosets, Index of a group, Lagrange’s theorem, consequences.

PSO-2: NORMAL SUBGROUPS AND HOMOMORPHISM

· Normal subgroups, Quotient groups.

· Homomorphism, Kernel of homomorphism, Isomorphism, Automorphism, Fundamental theorem of homomorphism.

PSO-3: DIFFERENTIAL EQUATIONS

· Recapitulation of Definition, examples of differential equations, formation of differential equations by elimination of arbitrary constants.

· Differential equations of first order.

· The general solution of a linear equation.

PSO-4: ORDINARY DIFFERENTIAL EQUATIONS

· Ordinary Linear differential equations with constant coefficients, Complementary function, Particular integral, Inverse differential operators

· Cauchy-Euler differential equations, Simultaneous differential equations.

IV SEMESTER: DIFFERENTIAL EQUATIONS-II AND REAL ANALYSIS-I [DSC-MATH-04]

PSO-1: LINEAR DIFFERENTIAL EQUATIONS

· Solution of ordinary second order linear differential equations with variable coefficient by various methods: Changing the independent variable, changing the dependent variable, by method of variation of parameters, exact equations.

· Total differential equations.

· Simultaneous equations.

PSO-2: PARTIAL DIFFERENTIAL EQUATIONS

· Basic Concepts-formation of partial differential equations by elimination of arbitrary constants and functions.

· Solution of partial differential equations.

· Standard types of first order non-linear partial differential equations.

· Rules for finding the Complementary function, Rules for finding the particular integral, Method of separation of variables.

PSO-3: RIEMANN INTEGRATION AND LINE INTEGRAL

· The Riemann integral-Upper and lower sums, criterion for integrability.

· Properties of Riemann integrals, Fundamental theorem of calculus.

· Integration as a limit of sum.

· Definition of line integral and basic properties-examples on evaluation of line integrals.

PSO-4: MULTIPLE INTEGRALS

· Definition of a double integral, Evaluation of double integrals, Surface areas.

· Definition of a triple integral, Evaluation of triple integrals, Volume as a triple integral.

V SEMESTER: REAL ANALYSIS –II AND ALGEBRA-III [DSE-MATH-01]

PSO-1: Sequences

· Sequence   of real numbers, Bounded and unbounded sequences, infimum and supremum of a sequence.

· Limit of a sequence.

· Convergent, divergent and oscillatory sequences, Standard properties.

· Monotonic sequences and their properties, Cauchy’s general principle of convergence.

PSO-2: INFINITE SERIES

· Infinite series of real numbers, convergence, divergence and oscillation of series.

· Series of positive terms, geometric series, p- series, comparison tests.

· D’ Alembert’s ratio test, Raabe’s test, Cauchy’s root test, Leibnitz’s test for alternating series.

· Summation of Binomial, Exponential and Logarithmic series.

PSO-3: RINGS AND FIELDS

· Rings, Integral Domains, Division rings, sub rings.

· Fields, Subfields.

· Characteristic of a ring.

· Ideals, Algebra of Ideals, Principal ideal ring, Divisibility in integral domain, Units and Associates.    

PSO-4: POLYNOMIAL RINGS AND HOMOMORPHISMS

· Polynomial rings, Divisibility, reducible polynomials, Division Algorithm, Greatest common divisors, Euclidean Algorithm, unique factorization theorem.

· Quotient rings.

· Homomorphism of rings, Kernel of a ring homomorphism, Fundamental theorem of homomorphism.

· Maximal ideals, Prime ideals, Eisenstein’s criterion of irreducibility.

V SEMESTER: NUMERICAL ANALYSIS [SEC-MATH-02]

 PSO-1: NUMERICAL ANALYSIS

· Numerical solutions of Algebraic and transcendental equations, Bisection method, the method of false position (Regula-falsi method), Newton –Raphson method.

· Numerical solutions of first order linear differential equations: Euler-Cauchy method, Euler’s modified method, Runge-Kutta fourth order method, Picard’s method.

PSO-2: FINITE DIFFERENCES AND NUMERICAL INTEGRATION

· Forward and Backward difference.

· Newton –Gregory forward and backward interpolation formulae.

· Lagrange’s interpolation formulae.

· General Quadrature formula: Trapezoidal rule, Simpson’s 1/3^rd rule, Simpson’s 3/8^th rule, Weddle’s rule.

VI SEMESTER: ALGEBRA-IV AND COMPLEX ANALYSIS- [DSE-MATH-02]

PSO-1: VECTOR SPACES

· Vector spaces, vector subspaces, Algebra of subspaces.

· Liner Combination, linear span, linear dependence and independence of vectors.

· Basis of a vector space, dimension of vector space.

· Quotient spaces, Homomorphism of vector spaces, isomorphism of vector spaces

· Direct sums.

PSO-2: LINEAR TRANSFORMATIONS

· Linear transformation, Change of basis and effect of associated matrices.

· Kernel and image of a linear transformation.

· Rank and Nullity theorem.

· Eigen values and Eigen vectors of a linear transformation.

PSO-3: FUNCTIONS OF A COMPLEX VARIABLE

· Equation to a circle and straight line in complex form.

· Continuity and Differentiability, Analytic functions, Singular points.

· Cauchy Riemann equations in Cartesian and Polar form.

· Necessary and sufficient condition for function to be analytic.

· Harmonic functions.

· Construction of analytic function: Milne Thomson method, using the concept of Harmonic function.

PSO-4: TRANSFORMATIONS

· Definition, Jacobean of transformation, Identity transformation.

· Linear transformation.

· Bilinear transformations, Cross Ratio of Four points.

· Conformal mappings, Discussion of the transformations w=z², w=sinz, w=e^z, w=.

VI SEMESTER: COMPLEX ANALYSIS AND IMPROPER INTEGRALS [SEC-MATH-03]

  PSO-1: COMPLEX INTEGRATION

· The complex line integral, proof of Cauchy’s integral theorem using Green’s theorem.

· Cauchy’s integral formula for the function and the derivatives.

· Cauchy’s inequality, Liouville’s theorem.

· Fundamental theorem of Algebra.

PSO-2: IMPROPER INTEGRALS

· Definition, Gamma and Beta functions.

· Relation between Beta and Gamma functions.

· Application to evaluation of integrals.

· Duplication formula.

PROGRAM SPECIFIC OUTCOME 2022-23

 

I SEMESTER: ALGEBRA-I and CALCULUS-I [MATDSCT 1.1]

 

PSO-1 MATRICES

· Rank of matrix, Elementary row operations, Inverse of a non-singular matrix.

· Systems of m linear equations in n unknowns

· Eigen values and eigenvectors of a square matrix, Diagonalization of a real symmetric matrix, Cayley Hamilton theorem, Applications to determine the powers of square matrices and inverse of non-singular matrices.

PSO-2 THEORY OF EQUATIONS

· Theory of equations, Euclid’s algorithm, Polynomials with integral coefficients, Remainder theorem, Factors theorem, Fundamental theorem of algebra.

· Irrational and complex roots occurring in conjugate pairs, Relation between roots and coefficients of polynomial equation.

· Symmetric function.

· Descartes’ rule of signs, multiple roots, solving cubic equations by Cordon’s method, solving quadric equations by Descarte’s method.  

PSO-3POLAR CO-ORDINATES

· Polar coordinates- angle between the radius vector and the tangent at a point on a curve, angel of intersection between two curves.

· Pedal equations, Derivative of arc.

Coordinates of center of curvature, Radius of curvature, Circle of curvature.

PSO-4 SUCCESSIVE DIFFERENTIATION AND INTEGRAL CALCULUS-I

· Derivative of a function, Derivative of higher order- n^th derivative of some standard functions, Problems.

· Leibnitz theorem, Problems.

· Reduction formulae for,,,,,, , , ,  with definite limits.

 

 

II SEMESTER: ALGEBRA-II and CALCULUS-II [MATDSCT 2.1]

PSO-I NUMBER THEORY

· Division Algorithm, Divisibility, Prime and composite numbers, Euclidean algorithm.

· Fundamental theorem of arithmetic, Greatest common divisors and least common multiple.

· Congruences, Linear congruences, Simultaneous Congruences, Wilson’s, Euler’s and Fermat’s theorem and their applications.

PSO-II DIFFERENTIAL CALCULUS-I

· Limit of a function- Properties and Problems.

· Continuity of function- Properties and Problems, Infimum and supremum of a function, Theorems on continuity, Intermediate value theorem.

· Differentiability.

· Rolle’s Theorem, Lagrange’s mean value theorem, Cauchy’s mean value theorem, Taylor’s theorem, Maclaurin’s theorem.

· Taylor’s infinite series, Maclaurin’s infinite series.

· Indeterminate forms.

PSO-IIIPARTIAL DERIVATIVES

· Function of two or more variables, Explicit and implicit functions, the neighbourhood of appoint.

· Limit of a function, Continuity, Partial derivatives.

· Homogeneous function, Euler’s theorem, Chain rule, Change of variables, Directional derivative.

· Partial derivatives of higher order, Taylor’s theorem for two variables, Derivative of implicit functions, Jacobians.  

PSO-IVINTEGRAL CALCULUS-II

· Definition of line integral and basic properties-examples on evaluation of line integrals.

· Definition of a double integral, Evaluation of double integrals, Surface areas.

· Definition of a triple integral, Evaluation of triple integrals, Volume as a triple integral.

III SEMESTER: ALGEBRA-III AND DIFFERENTIAL EQUATIONS-I [MATDSCT 3.1]

PSO-1: GROUP THEORY-I

· Definition and examples of groups, some general properties of groups, Group of permutations.

· Powers of an element of a group, Subgroups, cyclic groups problems and theorems.

PSO-2: GROUP THEORY-II

· Cosets, Index of a group, Lagrange’s theorem, consequences.

· Normal subgroups, Quotient groups.

· Homomorphism, Kernel of homomorphism, Isomorphism, Automorphism, Fundamental theorem of homomorphism.

PSO-3: DIFFERENTIAL EQUATIONS-I

· Recapitulation of Definition, examples of differential equations, formation of differential equations by elimination of arbitrary constants.

· Differential equations of first order.

· The general solution of a linear equation.

PSO-4: DIFFERENTIAL EQUATIONS-II

· Differential equations of first order and higher degree.

· Clairaut’s equations.

· Ordinary Linear differential equations with constant coefficients, Complementary function, Particular integral, Inverse differential operators.

Simultaneous differential equations.

IV SEMESTER: REAL ANALYSIS-I AND DIFFERENTIAL EQUATIONS-II [MATDSCT 4.1]

PSO-1: SEQUENCE

· Sequence of real numbers, Bounded and unbounded sequences, infimum and supremum of a sequence.

· Limit of a sequence.

· Convergent, divergent and oscillatory sequences, Standard properties.

· Monotonic sequences and their properties, Cauchy’s general principle of convergence.

PSO-2: INFINITE SERIES

· Infinite series of real numbers, convergence, divergence and oscillation of series.

· Series of positive terms, geometric series, p- series, comparison tests.

· D’ Alembert’s ratio test, Raabe’s test, Cauchy’s root test, Leibnitz’s test for alternating series.

PSO-3: LINEAR DIFFERENTIAL EQUATIONS

· Cauchy-Euler differential equations.

· Solution of ordinary second order linear differential equations with variable coefficient by various methods: Changing the independent variable, changing the dependent variable, by method of variation of parameters, exact equations.

· Total differential equations.

· Simultaneous equations.

PSO-4: PARTIAL DIFFERENTIAL EQUATIONS

· Basic Concepts-formation of partial differential equations by elimination of arbitrary constants and functions.

· Solution of partial differential equations.

· Standard types of first order non-linear partial differential equations.

· Rules for finding the Complementary function, Rules for finding the particular integral, Method of separation of variables.

V SEMESTER: REAL ANALYSIS –II AND ALGEBRA-III [DSE-MATH-01]

PSO-1: SEQUENCE

· Sequence   of real numbers, Bounded and unbounded sequences, infimum and supremum of a sequence.

· Limit of a sequence.

· Convergent, divergent and oscillatory sequences, Standard properties.

· Monotonic sequences and their properties, Cauchy’s general principle of convergence.

PSO-2: INFINITE SERIES

· Infinite series of real numbers, convergence, divergence and oscillation of series.

· Series of positive terms, geometric series, p- series, comparison tests.

· D’ Alembert’s ratio test, Raabe’s test, Cauchy’s root test, Leibnitz’s test for alternating series.

· Summation of Binomial, Exponential and Logarithmic series.

PSO-3: RINGS AND FIELDS

· Rings, Integral Domains, Division rings, sub rings.

· Fields, Subfields.

· Characteristic of a ring.

· Ideals, Algebra of Ideals, Principal ideal ring, Divisibility in integral domain, Units and Associates.    

PSO-4: POLYNOMIAL RINGS AND HOMOMORPHISMS

· Polynomial rings, Divisibility, reducible polynomials, Division Algorithm, Greatest common divisors, Euclidean Algorithm, unique factorization theorem.

· Quotient rings.

· Homomorphism of rings, Kernel of a ring homomorphism, Fundamental theorem of homomorphism.

· Maximal ideals, Prime ideals, Eisenstein’s criterion of irreducibility.

V SEMESTER: NUMERICAL ANALYSIS [SEC-MATH-02]

 PSO-1: NUMERICAL ANALYSIS

· Numerical solutions of Algebraic and transcendental equations, Bisection method, the method of false position (Regula-falsi method), Newton –Raphson method.

· Numerical solutions of first order linear differential equations: Euler-Cauchy method, Euler’s modified method, Runge-Kutta fourth order method, Picard’s method.

PSO-2: FINITE DIFFERENCES AND NUMERICAL INTEGRATION

· Forward and Backward difference.

· Newton –Gregory forward and backward interpolation formulae.

· Lagrange’s interpolation formulae.

· General Quadrature formula: Trapezoidal rule, Simpson’s 1/3^rd rule, Simpson’s 3/8^th rule, Weddle’s rule.

VI SEMESTER: ALGEBRA-IV AND COMPLEX ANALYSIS- [DSE-MATH-02]

PSO-1: VECTOR SPACES

· Vector spaces, vector subspaces, Algebra of subspaces.

· Liner Combination, linear span, linear dependence and independence of vectors.

· Basis of a vector space, dimension of vector space.

· Quotient spaces, Homomorphism of vector spaces, isomorphism of vector spaces

· Direct sums.

PSO-2: LINEAR TRANSFORMATIONS

· Linear transformation, Change of basis and effect of associated matrices.

· Kernel and image of a linear transformation.

· Rank and Nullity theorem.

· Eigen values and Eigen vectors of a linear transformation.

PSO-3: FUNCTIONS OF A COMPLEX VARIABLE

· Equation to a circle and straight line in complex form.

· Continuity and Differentiability, Analytic functions, Singular points.

· Cauchy Riemann equations in Cartesian and Polar form.

· Necessary and sufficient condition for function to be analytic.

· Harmonic functions.

· Construction of analytic function: Milne Thomson method, using the concept of Harmonic function.

PSO-4: TRANSFORMATIONS

· Definition, Jacobean of transformation, Identity transformation.

· Linear transformation.

· Bilinear transformations, Cross Ratio of Four points.

· Conformal mappings, Discussion of the transformations w=z², w=sinz, w=e^z, w=.

VI SEMESTER: COMPLEX ANALYSIS AND IMPROPER INTEGRALS [SEC-MATH-03]

  PSO-1: COMPLEX INTEGRATION

· The complex line integral, proof of Cauchy’s integral theorem using Green’s theorem.

· Cauchy’s integral formula for the function and the derivatives.

· Cauchy’s inequality, Liouville’s theorem.

· Fundamental theorem of Algebra.

PSO-2: IMPROPER INTEGRALS

· Definition, Gamma and Beta functions.

· Relation between Beta and Gamma functions.

· Application to evaluation of integrals.

· Duplication formula.

 

A V KANTHAMMA COLLEGE FOR WOMEN, HASSAN

DEPARTMENT OF MATHEMATICS

COURSE OUTCOME

2018-19

CO-1

· Find the higher order derivative of the product of two functions and maxima, minima, concavity, convexity and point of inflection.

· Solve a system of linear equations using rank of a matrix.

· Familiarize characteristic roots and characters vectors.

· To find inverse of a matrix by Cayley-Hamilton theorem.

· Analyze different form of equations, finding their roots and understand relation between roots and co-efficients.

· Learn about properties of integrals and Reduction formulae for some standard functions.

· Find the angle of intersection of two curves, Find the radius of curvature, circle of curvature and evolutes.

CO-2

· Explain the definition of limits, continuity, and differentiability as related to functions.

· Understand the mean value theorem.

· Expand the functions using Taylor’s and Maclaurin’s theorem.

· Understand the concepts of partial derivatives and functions of several variables.

· Learn the Divisibility, Prime Numbers, Congruences, Wilson’s, Euler’s and Fermat’s theorem and their applications.

CO-3

· Assess properties implied by the definitions of groups.

· Use various canonical types of groups(including cyclic groups and groups of permutation)

· Analyze and demonstrate examples of subgroups, Normal Subgroups and quotient groups.

· Obtain the solution of differential equations by the method of separation of variables, homogeneous, Linear and exact differential equations.

· Obtain an integrating factor which may reduce a given differential equation into an exact one and provide its solution.

· Find the complementary function and particular integrals of linear differential equations.

 

CO-4

· Method of Solution of the differential equation of the form dx/P=dy/Q=dz/R.

· Use Lagrange’s method for solving the first order linear partial differential equations, Learn the definition and concept of line integral.

· Evaluations of double integral and triple integrals.

· Find the volume of given surface by using triple integrals.

· Learn the definition of Riemann integral,upper sums and lower sums.

· Criterion for integrability, Fundamental theorem of integral calculus.

· Learn First and Second Mean Value theorems of integral calculus.

CO-5: DSE

· Understand the term convergence.

· Applies this term in to problems.

· Illustrate the convergence properties of infinite series.

· Test the convergence of infinite series by comparision tests. D’ Alembert’s, Raabe’s test, Cauchy’s root test.

· Write precise and accurate Mathematical definitions of ring theory.

· Analyze and Demonstrate examples of ideals and quotient ring.

· Use the concept of isomorphism and homomorphism for rings.

·  Finding the greatest common divisor of polynomials.

CO-5: SEC

· Solve the problems using numerical Differentiation and integration.

· Solve the system of linear equations by using numerical methods.

CO-6: DSE

· Understand the idea about vectors.

· Analyze finite and infinite dimensional vectors space and subspace over a field and properties, including basis structure of vector spaces.

· Use the definition and properties of linear transformation and matrices of linear transformations and change of basis including kernel, range and isomorphism.

· Compute with the characteristic polynomials eigenvectors,eigen spaces.

· Represent complex numbers algebraically and geometrically.

· Apply the concept and consequences of analyticity and Cauchy- Riemann equation and results on harmonic functions.

· Study the concept of Transformation of functions of complex variable from z- plane to w-plane.

· Types of Transformations, Jacobian of a linear transformation and to find Jacobian of some functions.

·  Linear and Bilinear transformations cross ratio of four points with properties, conformal mappings and transformation of some standard functions.

CO-6:SEC

· Evaluate complex contour integrals directly and by the fundamental theorem, apply the Cauchy integral theorem.

· Understand the definition of improper integrals.

· Evaluation of improper integrals using Beta and gamma functions.

COURSE OUTCOME

2019-20

CO-1

· Find the higher order derivative of the product of two functions and maxima, minima, concavity, convexity and point of inflection.

· Solve a system of linear equations using rank of a matrix.

· Familiarize characteristic roots and characters vectors.

· To find inverse of a matrix by Cayley-Hamilton theorem.

· Analyze different form of equations, finding their roots and understand relation between roots and co-efficients.

· Learn about properties of integrals and Reduction formulae for some standard functions.

· Find the angle of intersection of two curves, Find the radius of curvature, circle of curvature and evolutes.

CO-2

· Explain the definition of limits, continuity, and differentiability as related to functions.

· Understand the mean value theorem.

· Expand the functions using Taylor’s and Maclaurin’s theorem.

· Understand the concepts of partial derivatives and functions of several variables.

· Learn the Divisibility, Prime Numbers, Congruences, Wilson’s, Euler’s and Fermat’s theorem and their applications.

CO-3

· Assess properties implied by the definitions of groups.

· Use various canonical types of groups(including cyclic groups and groups of permutation)

· Analyze and demonstrate examples of subgroups, Normal Subgroups and quotient groups.

· Obtain the solution of differential equations by the method of separation of variables, homogeneous, Linear and exact differential equations.

· Obtain an integrating factor which may reduce a given differential equation into an exact one and provide its solution.

· Find the complementary function and particular integrals of linear differential equations.

 

CO-4

· Method of Solution of the differential equation of the form dx/P=dy/Q=dz/R.

· Use Lagrange’s method for solving the first order linear partial differential equations, Learn the definition and concept of line integral.

· Evaluations of double integral and triple integrals.

· Find the volume of given surface by using triple integrals.

· Learn the definition of Riemann integral, upper sums and lower sums.

· Criterion for integrability, Fundamental theorem of integral calculus.

· Learn First and Second Mean Value theorems of integral calculus.

CO-5: DSE

· Understand the term convergence.

· Applies this term in to problems.

· Illustrate the convergence properties of infinite series.

· Test the convergence of infinite series by comparision tests. D’ Alembert’s, Raabe’s test, Cauchy’s root test.

· Write precise and accurate Mathematical definitions of ring theory.

· Analyze and Demonstrate examples of ideals and quotient ring.

· Use the concept of isomorphism and homomorphism for rings.

·  Finding the greatest common divisor of polynomials.

CO-5: SEC

· Solve the problems using numerical Differentiation and integration.

· Solve the system of linear equations by using numerical methods.

CO-6: DSE

· Understand the idea about vectors.

· Analyze finite and infinite dimensional vectors space and subspace over a field and properties, including basis structure of vector spaces.

· Use the definition and properties of linear transformation and matrices of linear transformations and change of basis including kernel, range and isomorphism.

· Compute with the characteristic polynomials eigen vectors, eigen spaces.

· Represent complex numbers algebraically and geometrically.

· Apply the concept and consequences of analyticity and Cauchy- Riemann equation and results on harmonic functions.

· Study the concept of Transformation of functions of complex variable from z- plane to w-plane.

· Types of Transformations, Jacobian of a linear transformation and to find Jacobian of some functions.

·  Linear and Bilinear transformations cross ratio of four points with properties, conformal mappings and transformation of some standard functions.

CO-6:SEC

· Evaluate complex contour integrals directly and by the fundamental theorem, apply the Cauchy integral theorem.

· Understand the definition of improper integrals.

· Evaluation of improper integrals using Beta and gamma functions.

COURSE OUTCOME

2020-21

CO-1

· Find the higher order derivative of the product of two functions and maxima, minima, concavity, convexity and point of inflection.

· Solve a system of linear equations using rank of a matrix.

· Familiarize characteristic roots and characters vectors.

· To find inverse of a matrix by Cayley-Hamilton theorem.

· Analyze different form of equations, finding their roots and understand relation between roots and co-efficients.

· Learn about properties of integrals and Reduction formulae for some standard functions.

· Find the angle of intersection of two curves, Find the radius of curvature, circle of curvature and evolutes.

CO-2

· Explain the definition of limits, continuity, and differentiability as related to functions.

· Understand the mean value theorem.

· Expand the functions using Taylor’s and Maclaurin’s theorem.

· Understand the concepts of partial derivatives and functions of several variables.

· Learn the Divisibility, Prime Numbers, Congruences, Wilson’s, Euler’s and Fermat’s theorem and their applications.

CO-3

· Assess properties implied by the definitions of groups.

· Use various canonical types of groups(including cyclic groups and groups of permutation)

· Analyze and demonstrate examples of subgroups, Normal Subgroups and quotient groups.

· Obtain the solution of differential equations by the method of separation of variables, homogeneous, Linear and exact differential equations.

· Obtain an integrating factor which may reduce a given differential equation into an exact one and provide its solution.

· Find the complementary function and particular integrals of linear differential equations.

 

CO-4

· Method of Solution of the differential equation of the form dx/P=dy/Q=dz/R.

· Use Lagrange’s method for solving the first order linear partial differential equations, Learn the definition and concept of line integral.

· Evaluations of double integral and triple integrals.

· Find the volume of given surface by using triple integrals.

· Learn the definition of Riemann integral, upper sums and lower sums.

· Criterion for integrability, Fundamental theorem of integral calculus.

· Learn First and Second Mean Value theorems of integral calculus.

CO-5: DSE

· Understand the term convergence.

· Applies this term in to problems.

· Illustrate the convergence properties of infinite series.

· Test the convergence of infinite series by comparision tests. D’ Alembert’s, Raabe’s test, Cauchy’s root test.

· Write precise and accurate Mathematical definitions of ring theory.

· Analyze and Demonstrate examples of ideals and quotient ring.

· Use the concept of isomorphism and homomorphism for rings.

·  Finding the greatest common divisor of polynomials.

CO-5: SEC

· Solve the problems using numerical Differentiation and integration.

· Solve the system of linear equations by using numerical methods.

CO-6: DSE

· Understand the idea about vectors.

· Analyze finite and infinite dimensional vectors space and subspace over a field and properties, including basis structure of vector spaces.

· Use the definition and properties of linear transformation and matrices of linear transformations and change of basis including kernel, range and isomorphism.

· Compute with the characteristic polynomials eigen vectors, eigen spaces.

· Represent complex numbers algebraically and geometrically.

· Apply the concept and consequences of analyticity and Cauchy- Riemann equation and results on harmonic functions.

· Study the concept of Transformation of functions of complex variable from z- plane to w-plane.

· Types of Transformations, Jacobian of a linear transformation and to find Jacobian of some functions.

·  Linear and Bilinear transformations cross ratio of four points with properties, conformal mappings and transformation of some standard functions.

CO-6:SEC

· Evaluate complex contour integrals directly and by the fundamental theorem, apply the Cauchy integral theorem.

· Understand the definition of improper integrals.

· Evaluation of improper integrals using Beta and gamma functions.

COURSE OUTCOME for the academic year 2021-22

 

CO-1(NEP)

· Find the higher order derivative of the product of two functions and maxima, minima, concavity, convexity and point of inflection.

· Solve a system of linear equations using rank of a matrix.

· Familiarize characteristic roots and characters vectors.

· To find inverse of a matrix by Cayley-Hamilton theorem.

· Analyze different form of equations, finding their roots and understand relation between roots and co-efficients.

· Learn about properties of integrals and Reduction formulae for some standard functions.

· Find the angle of intersection of two curves, Find the radius of curvature, circle of curvature.

CO-2(NEP)

· Explain the definition of limits, continuity, and differentiability as related to functions.

· Understand the mean value theorem.

· Expand the functions using Taylor’s and Maclaurin’s theorem.

· Understand the concepts of partial derivatives and functions of several variables.

· Learn the Divisibility, Prime Numbers, Congruences, Wilson’s, Euler’s and Fermat’s theorem and their applications.

· Learn the definition and concept of line integral.

· Evaluations of double integral and triple integrals.

· Find the volume of given surface by using triple integrals.

CO-3

· Assess properties implied by the definitions of groups.

· Use various canonical types of groups(including cyclic groups and groups of permutation)

· Analyze and demonstrate examples of subgroups, Normal Subgroups and quotient groups.

· Obtain the solution of differential equations by the method of separation of variables, homogeneous, Linear and exact differential equations.

· Obtain an integrating factor which may reduce a given differential equation into an exact one and provide its solution.

· Find the complementary function and particular integrals of linear differential equations.

CO-4

· Method of Solution of the differential equation of the form dx/P=dy/Q=dz/R.

· Use Lagrange’s method for solving the first order linear partial differential equations, Learn the definition and concept of line integral.

· Evaluations of double integral and triple integrals.

· Find the volume of given surface by using triple integrals.

· Learn the definition of Riemann integral, upper sums and lower sums.

· Criterion for integrability, Fundamental theorem of integral calculus.

· Learn First and Second Mean Value theorems of integral calculus.

CO-5: DSE

· Understand the term convergence.

· Applies this term in to problems.

· Illustrate the convergence properties of infinite series.

· Test the convergence of infinite series by comparision tests. D’ Alembert’s, Raabe’s test, Cauchy’s root test.

· Write precise and accurate Mathematical definitions of ring theory.

· Analyze and Demonstrate examples of ideals and quotient ring.

· Use the concept of isomorphism and homomorphism for rings.

·  Finding the greatest common divisor of polynomials.

CO-5: SEC

· Solve the problems using numerical Differentiation and integration.

· Solve the system of linear equations by using numerical methods.

CO-6: DSE

· Understand the idea about vectors.

· Analyze finite and infinite dimensional vectors space and subspace over a field and properties, including basis structure of vector spaces.

· Use the definition and properties of linear transformation and matrices of linear transformations and change of basis including kernel, range and isomorphism.

· Compute with the characteristic polynomials eigen vectors, eigen spaces.

· Represent complex numbers algebraically and geometrically.

· Apply the concept and consequences of analyticity and Cauchy- Riemann equation and results on harmonic functions.

· Study the concept of Transformation of functions of complex variable from z- plane to w-plane.

· Types of Transformations, Jacobian of a linear transformation and to find Jacobian of some functions.

·  Linear and Bilinear transformations cross ratio of four points with properties, conformal mappings and transformation of some standard functions.

CO-6:SEC

· Evaluate complex contour integrals directly and by the fundamental theorem, apply the Cauchy integral theorem.

· Understand the definition of improper integrals.

· Evaluation of improper integrals using Beta and gamma functions.

COURSE OUTCOME for the academic year 2022-23

CO-1(NEP)

· Find the higher order derivative of the product of two functions and maxima, minima, concavity, convexity and point of inflection.

· Solve a system of linear equations using rank of a matrix.

· Familiarize characteristic roots and characters vectors.

· To find inverse of a matrix by Cay-Hamilton theorem.

· Analysis different form of equations, finding their roots and understand relation between roots and coefficients

· Learn about properties of integrals and Reduction formulae for some standard functions.

· Find the angle of intersection of two curves, Find the radius of curvature, circle of curvature.

CO-2(NEP)

· Explain the definition of limits, continuity, and differentiability as related to functions.

· Understand the mean value theorem.

· Expand the functions using Taylor’s and Maclaurin’s theorem.

· Understand the concepts of partial derivatives and functions of several variables.

· Learn the Divisibility, Prime Numbers, Congruence, Wilson’s, Euler’s and Fermat’s theorem and their applications.

· Learn the definition and concept of line integral.

· Evaluations of double integral and triple integrals.

· Find the volume of given surface by using triple integrals.

 

CO-3(NEP)

· Assess properties implied by the definitions of groups.

· Use various canonical types of groups(including cyclic groups and groups of permutation)

· Analyses and demonstrate examples of subgroups, Normal Subgroups and quotient groups.

· Obtain the solution of differential equations by the method of separation of variables, homogeneous, Linear and exact differential equations.

· Obtain an integrating factor which may reduce a given differential equation into an exact one and provide its solution.

· Find the complementary function and particular integrals of linear differential equations.

CO-4(NEP)

· Understand the term convergence.

· Applies this term in to problems.

· Illustrate the convergence properties of infinite series.

· Test the convergence of infinite series by comparison tests. D’ Alembert’s, Raabe’s test, Cauchy’s root test.

· Method of Solution of the differential equation of the form DX/P=DY/Q=DZ/R.

· Use Lagrange’s method for solving the first order linear partial differential equations, Learn the definition and concept of line integral.

CO-5: DSE (CBCS)

· Understand the term convergence.

· Applies this term in to problems.

· Illustrate the convergence properties of infinite series.

· Test the convergence of infinite series by comparison tests. D’ Alembert’s, Raabe’s test, Cauchy’s root test.

· Write precise and accurate Mathematical definitions of ring theory.

· Analyses and Demonstrate examples of ideals and quotient ring.

· Use the concept of isomorphism and homomorphism for rings.

·  Finding the greatest common divisor of polynomials.

CO-5: SEC (CBCS)

· Solve the problems using numerical Differentiation and integration.

· Solve the system of linear equations by using numerical methods.

CO-6: DSE (CBCS)

· Understand the idea about vectors.

· Analyses finite and infinite dimensional vectors space and subspace over a field and properties, including basis structure of vector spaces.

· Use the definition and properties of linear transformation and matrices of linear transformations and change of basis including kernel, range and isomorphism.

· Compute with the characteristic polynomials Eigen vectors, Eigen spaces.

· Represent complex numbers algebraically and geometrically.

· Apply the concept and consequences of analyticity and Cauchy- Riemann equation and results on harmonic functions.

· Study the concept of Transformation of functions of complex variable from z- plane to w-plane.

· Types of Transformations, jacobian of a linear transformation and to find jacobian of some functions.

·  Linear and Bilinear transformations cross ratio of four points with properties, conformal mappings and transformation of some standard functions.

CO-6:SEC (CBCS)

· Evaluate complex contour integrals directly and by the fundamental theorem, apply the Cauchy integral theorem.

· Understand the definition of improper integrals.

· Evaluation of improper integrals using Beta and gamma functions.

AVK COLLEGE FOR WOMEN, HASSAN

DEPARTMENT OF MATHEMATICS

RESULT ANALYSIS

2018-19 Batch:

SEMESTER	I	II	III	IV	V	VI
Total Appeared	65	64	63	63	48	47
High (>60=3)	26	42	24	28	34	23
Medium(50-59=2)	23	10	17	9	13	16
Low(40-49=1)	11	10	8	12	0	5
Below(<40=0)	5	2	14	14	1	3

 

 

SEMESTER	I	II	III	IV	V	VI
PASS PERCENTAGE	92%	97%	78%	78%	98%	94%

 

AVK COLLEGE FOR WOMEN, HASSAN

DEPARTMENT OF MATHEMATICS

RESULT ANALYSIS

2019-20 Batch:

SEMESTER	I	II	III	IV	V	VI
Total Appeared	54	52	62	62	60	60
High (>60=3)	10	48	32	53	36	26
Medium(50-59=2)	22	0	13	0	18	19
Low(40-49=1)	19	2	12	4	1	12
Below(<40=0)	3	2	5	5	5	3

 

 

 

SEMESTER	I	II	III	IV	V	VI
PASS PERCENTAGE	94%	96%	92%	92%	92%	95%

 

 

AVK COLLEGE FOR WOMEN, HASSAN

DEPARTMENT OF MATHEMATICS

RESULT ANALYSIS

2020-21 Batch:

SEMESTER	I	II	III	IV	V
Total Appeared	38	30	49	47	61
High (>60=3)	24	30	38	44	46
Medium(50-59=2)	3	0	6	0	6
Low(40-49=1)	5	0	2	0	6
Below(<40=0)	6	0	3	3	3

 

 

SEMESTER	I	II	III	IV	V
PASS PERCENTAGE	84%	100%	94%	94%	95%

 

 

AVK COLLEGE FOR WOMEN, HASSAN

DEPARTMENT OF MATHEMATICS

RESULT ANALYSIS

2021-22 Batch:

SEMESTER	I	II	III	IV	V	VI
Total Appeared	16	16	36	35	41	41
High (>60=3)	7	4	22	25	40	39
Medium(50-59=2)	5	8	7	4	1	2
Low(40-49=1)	2	1	3	1	0	0
Below(<40=0)	2	3	4	5	0	0

 

 

	SEMESTER	I	II	III	IV	V	VI
	PASS PERCENTAGE	88%	81%	89%	86%	100%	100%

 

AVK COLLEGE FOR WOMEN, HASSAN

DEPARTMENT OF MATHEMATICS

RESULT ANALYSIS

2022-23 Batch:

SEMESTER	I	II	III	IV	V	VI
Total Appeared	3	3	16	16	31	31
High (>60=3)	1	2	11	15	27	29
Medium(50-59=2)	2	1	3	0	2	2
Low(40-49=1)	0	0	0	0	2	0
Below(<40=0)	0	0	2	1	0	0

 

 

	SEMESTER	I	II	III	IV	V	VI
	PASS PERCENTAGE	100%	100%	88%	94%	100%	100%

 

Name	Designation	Department	Qualification	Experience
Jaishree M R	Lecturer	Mathematics	M.Sc	1.2 Years
Meghana A K	Lecturer	Mathematics	M.Sc,B.ed	3 Months
Sushmitha K C	H.O.D	Mathematics	M.Sc	10 Years