#
B.Sc., Mathematics

PROGRAMME OUTCOMES

**P O1:**The Bachelor’s Degree in Mathematics is awarded to the students on the basis of knowledge, understanding, skills, attitudes, values and academic achievements sought to be acquired by learners at the end of this program.

**P O2:**Mathematics is the study of quantity, structure, space and change. The key areas are Calculus, Algebra, Geometry, Analysis, Differential Equations and Mechanics.

**P O3:**Demonstrate the critical thinking mindset and the ability to identify and formulate research problems, research literature, design tools, analyze and interpret data, and synthesize the information to provide valid conclusions and contextual approaches across a variety of subject matter.

**P O4:**This programme will be able to present mathematics clearly and precisely, make vague ideas precise by formulating them in the language of mathematics, describe mathematical ideas from multiple perspectives and explain fundamental concepts or mathematics to non-mathematicians.

**P O5:**Under graduate programme will help the learners to join teaching profession in primary and secondary school. To enhance their employability for government jobs, jobs in banking, insurance and investment sectors, data analyst jobs and jobs in various other public and private enterprises

# PROGRAMME SPECIFIC OUTCOMES

**PS O1:**Formulate and develop mathematical arguments in a logical manner and use suitable mathematical tool for analyzing, identify, locate and evaluate the problem in real life.

**PS O2:**Think in a critical manner, acquire good knowledge, understand, formulate and use quantitative models in advanced area of mathematics, statistics, social science, business and related sciences.

**PS O3:**To analyze, handle issues and encourage the students to develop a range of generic skills helpful for employment, internships and social activities.

# Syllabus

Hours/Week | 5 | Credits | 5 | ||

Semester | I |

Unit-I: | Methods of Successive Differentiation – Leibnitz,s Theorem and its applications- Increasing & Decreasing functions –Maxima and Minima of function of two variables. |

Unit-II: | Curvature – Radius of curvature in Cartesian and in Polar Coordinates – Centre of curvature–Evolutes & Involutes |

Unit-III: | Expansions of sin (nx), cos (nx), tan (nx) – Expansions of sin nx, cos nx –Expansions of sin(x), cos(x), tan(x) in powers of x. |

Unit-IV: | Hyperbolic functions – Relation between hyperbolic & Circular functions- Inverse hyperbolic functions. |

Unit-V: | Logarithm of a complex number –Summation of Trigonometric series – Difference method- Angles in arithmetic progression method –Gregory’s series |

Text and Reference Books (Latest revised edition only) |

S.Narayanan and T.K.Manicavachagom Pillai, Calculus Volume I, S.Viswanathan (Printers&Publishers) Pvt Limited , Chennai -2011. S.Arumugam & others, Trigonometry and Fourier series, New Gamma Publications -1999 |

BOOKS FOR REFERENCE: |

A) S.Arumugam and Isaac, Calculus, Volume1, New Gamma Publishing House, 1991. B) S. Narayanan, T.K. Manichavasagam Pillai, Trigonometry, S. Viswanathan Pvt Limited, and Vijay Nicole Imprints Pvt Ltd, 2004. |

**Course Outcomes (CO) : On completion of the course, students should be able to**

Number | CO Statement |
---|---|

CO01. | Evaluate Maxima and Minima of function of two variables. |

CO02. | Determine the angle of intersection of two curves. |

CO03. | Find radius of curvature and centre of curvature. |

CO04. | Expand sinnθ, cosnθ, and tannθ by using terms Demoivre’s theorems and in terms of θ. |

CO05. | Define hyperbolic and inverse hyperbolic functions. |

**PO - CO MAPPING MATRIX:**

CO | PO | PSO | ||||||
---|---|---|---|---|---|---|---|---|

PO1 | PO2 | PO3 | PO4 | PO5 | PSO1 | PSO2 | PSO3 | |

CO1 | 1 | 3 | - | - | 2 | - | - | - |

CO2 | - | 3 | - | 2 | 1 | - | - | - |

CO3 | - | 3 | - | - | - | - | - | - |

CO4 | - | 3 | - | 2 | - | - | - | - |

CO5 | - | 3 | - | 2 | - | - | - | - |

Hours/Week | 4 | Credits | 4 | ||

Semester | I |

Unit-I: | Revision of all integral models – simple problems |

Unit-II: | Definite integrals - Integration by parts & reduction formula |

Unit-III: | Geometric Application of Integration-Area under plane curves: Cartesian co- ordinates -Area of a closed curve - Examples - Areas in polar co-ordinates. |

Unit-IV: | Double integrals – changing the order of Integration – Triple Integrals. |

Unit-V: | Beta & Gamma functions and the relation between them – Integration using Beta & Gamma functions |

Text and Reference Books (Latest revised edition only) |

S.Narayanan and T.K.Manicavachagom Pillai, Calculus Volume II, S.Viswanathan (Printers & Publishers) Pvt Limited, Chennai -2011. UNIT I : Chapter 1 section 1 to 10 UNIT II : Chapter 1 section 11, 12 & 13 UNIT III : Chapter 2 section 1.1, 1.2, 1.3 & 1.4 UNIT IV : Chapter 5 section 2.1, 2.2 & 4 UNIT V : Chapter 7 section 2.1 to 2.5 |

BOOKS FOR REFERENCE: |

1. Shanti Narayan, Differential & Integral Calculus. |

**Course Outcomes (CO) : On completion of the course, students should be able to**

Number | CO Statement |
---|---|

CO01. | Solve basic Integral Calculus problems and Explain properties of definite integrals |

CO02. | Prove reduction formulae and solve some problems by using these formulae |

CO03. | Evaluate double and triple integrals and also find the value of double and triple integral by change of variable method |

CO04. | Explain properties of Beta functions and derive relation between Beta and Gamma functions. |

CO05. | Evaluate integrals by using Beta and Gamma functions. |

**PO - CO MAPPING MATRIX:**

CO | PO | PSO | ||||||
---|---|---|---|---|---|---|---|---|

PO1 | PO2 | PO3 | PO4 | PO5 | PSO1 | PSO2 | PSO3 | |

CO1 | 1 | 3 | - | - | - | - | - | 1 |

CO2 | - | 3 | - | - | - | 2 | - | - |

CO3 | 1 | 3 | 1 | 2 | 1 | - | 1 | - |

CO4 | - | 3 | - | 1 | - | - | - | - |

CO5 | - | 3 | - | 1 | - | - | - | - |

Hours/Week | 5 | Credits | 5 | ||

Semester | II |

Unit-I: | First order, higher degree differential equations solvable for x, solvable for y, solvable for dy/dx, Clairauts form – Conditions of integrability of M dx + N dy = 0 – simple problems. |

Unit-II: | Particular integrals of second order differential equations with constant coefficients - Linear equations with variable coefficients – Method of Variation of Parameters ( Omit third & higher order equations). |

Unit-III: | Formation of Partial Differential Equation – General, Particular & Complete integrals – Solution of PDE of the standard forms - Lagrange’s method - Solving of Charpit’s method and a few standard forms. |

Unit-IV: | PDE of second order homogeneous equation with Constant coefficients – Particular integrals of the forms eax+by, Sin(ax+by), Cos(ax+by), xrys and eax+by.f(x,y). |

Unit-V: | Laplace Transforms – Standard formulae – Basic theorems & simple applications – Inverse Laplace Transforms – Use of Laplace Transforms in solving ODE with constant coefficients. |

Text and Reference Books (Latest revised edition only) |

A) T.K.Manicavachagom Pillay & S.Narayanan, Differential Equations, S.Viswanathan Publishers Pvt. Ltd., 1996. B) Arumugam & Isaac, Differential Equations, New Gamma Publishing House, Palayamkottai, 2003. Unit : 1 Chapter IV – Sections 1,2 & 3, Chapter II – Section 6 [1] Unit : 2 Chapter V – Sections 1,2,3,4 & 5, Chapter VIII – Section 4 [1] Unit : 3 Chapter XII – Sections 1 – 6 [1] Unit : 4 Chapter V [2] Unit : 5 Chapter IX – Sections 1 – 8 [1] |

BOOKS FOR REFERENCE: |

M.D.Raisinghania , Ordinary and Partial Differential Equations, S.Chand & Co M.K. Venkatraman, Engineering Mathematics, S.V. Publications, 1985 Revised Edition |

**Course Outcomes (CO) : On completion of the course, students should be able to**

Number | CO Statement |
---|---|

CO01. | Extract the solution of differential equations of the first order and of the first degree by variables separable, Homogeneous and Non-Homogeneous methods. |

CO02. | find a solution of differential equations of the first order and of a degree higher than the first by using methods of solvable for p.x and y. |

CO03. | Compute all the solutions of second and higher order linear differential equations with constant coefficients, linear equations with variable coefficients. |

CO04. | Solve simultaneous linear equations with constant coefficients, total differential equations, Form partial differential equations and find the solution of First order partial differential equations for some standard types. |

CO05. | Use inverse Laplace transform to return familiar functions and Apply Laplace Transform to solve second order linear differential equation and simultaneous linear differential equations. |

**PO - CO MAPPING MATRIX:**

CO | PO | PSO | ||||||
---|---|---|---|---|---|---|---|---|

PO1 | PO2 | PO3 | PO4 | PO5 | PSO1 | PSO2 | PSO3 | |

CO1 | 1 | 3 | - | 1 | - | 2 | - | 1 |

CO2 | - | 3 | - | 1 | 1 | - | - | - |

CO3 | - | 3 | - | 1 | 2 | - | - | - |

CO4 | - | 3 | - | 1 | 2 | - | - | - |

CO5 | - | 3 | - | 1 | - | - | - | - |

Hours/Week | 4 | Credits | 3 | ||

Semester | II |

Unit-I: | Coordinates in space-Direction consines of a line in space-angle between lines in space – equation of a plane in normal form. Angle between planes – Distance of a plane from a point. |

Unit-II: | Straight lines in space – line of intersection of planes – plane containing a line. Coplanar lines – skew lines and shortest distance between skew lines- length of the perpendicular from point to line. |

Unit-III: | General equation of a sphere-Section of sphere by plane-tangent planes –condition of tangency-system of spheres generated by two spheres - System of spheres generated by a sphere and plane. |

Unit-IV: | The equation of surface – cone – intersection of straight line and quadric cone – tangent plane and normal |

Unit-V: | Condition for plane to touch the quadric cone - angle between the lines in which the plane cuts the cone. Condition that the cone has three mutually perpendicular generators- Central quadrics – intersection of a line and quadric – tangents and tangent planes – condition for the plane to touch the coincoid |

Text and Reference Books (Latest revised edition only) |

Shanthi Narayanan and Mittal P.K:Analytical Solid Geometry 16th Edition S.Chand & Co., New Delhi. Narayanan and Manickavasagam Pillay, T.K. Treatment as Analytical Gementry S.Viswanathan (Printers & Publishers ) Pvt. Ltd., Unit I : Chapter I, Sec 1.5 to 1.9, Chapter II Sec 2.1 to 2.3, Pages : 10-31, Chapter II Sec 2.4 to 2.8 pages : 32-47 of [1] Unit II : chapter III section 3.1-3.7, pages 55-89 of [1] Unit III : Chapter VI Sec. 6.1 to 6.6 pages : 121-143 of [1] Unit IV : Chapter V Sec.43 to 47 pages : 103-113 of [2] Unit V : Chapter V Sec.49 to 53, Pages:115-125 of [2] |

BOOKS FOR REFERENCE: |

1. P.Duraipandian & others- Analytical Geometry 3 Dimensional – Edition. |

**Course Outcomes (CO) : On completion of the course, students should be able to**

Number | CO Statement |
---|---|

CO01. | Describe the various forms of equation of a plane, straight line, Sphere, Cone and Cylinder. |

CO02. | Find the angle between planes, Bisector planes, Perpendicular distance from a point to a plane, Image of a line on a plane, Intersection of two lines. |

CO03. | Define coplanar lines and illustrate. |

CO04. | Compute the angle between a line and a plane, length of perpendicular from a point to a line. |

CO05. | Define skew lines and calculate the shortest distance between two skew lines. |

**PO - CO MAPPING MATRIX:**

CO | PO | PSO | ||||||
---|---|---|---|---|---|---|---|---|

PO1 | PO2 | PO3 | PO4 | PO5 | PSO1 | PSO2 | PSO3 | |

CO1 | - | 3 | - | 2 | 1 | - | - | - |

CO2 | 1 | 2 | - | 1 | 1 | 2 | - | - |

CO3 | - | 3 | - | 1 | - | - | - | - |

CO4 | 1 | 2 | - | 1 | 1 | 2 | - | - |

CO5 | - | 2 | - | 1 | 1 | - | - | - |

Hours/Week | 5 | Credits | 4 | ||

Semester | III |

Unit-I: | Sequences – Bounded Sequences – Monotonic Sequences – Convergent Sequence – Divergent Sequences – Oscillating sequences |

Unit-II: | Algebra of Limits – Behavior of Monotonic functions |

Unit-III: | Some theorems on limits – subsequences – limit points : Cauchy sequences |

Unit-IV: | Series – infinite series – Cauchy’s general principal of convergence – Comparison – test theorem and test of convergence using comparison test (comparison test statement only, no proof) |

Unit-V: | Test of convergence using D Alembert’s ratio test – Cauchy’s root test – Alternating Series – Absolute Convergence (Statement only for all tests) |

Text and Reference Books (Latest revised edition only) |

Dr. S.Arumugam & Mr.A.Thangapandi Isaac Sequences and Series – New Gamma Publishing House – 2002 Edition. Unit I: Chapter 3 : Sec. 3.0 – 3.5 Page No : 39-55 Unit II : Chapter 3 : Sec. 3.6, 3.7 Page No:56 – 82 Unit III : Chapter 3 : Sec. 3.8-3.11, Page No:82-102 Unit IV : Chapter 4 : Sec. (4.1 & 4.2) Page No : 112-128 Unit V : Relevant part of Chapter 4 and Chapter 5: Sec. 5.1 & 5.2 Page No:157-167 |

BOOKS FOR REFERENCE: |

1. Algebra – Prof. S.Surya Narayan Iyer 2. Algebra – Prof. M.I.Francis Raj |

**Course Outcomes (CO) : On completion of the course, students should be able to**

Number | CO Statement |
---|---|

CO01. | Define different types of sequences and discuss the behaviour of the geometric sequence. Prove properties of convergent and divergent sequence and verify the given sequence in convergent and divergent by using behaviour of Monotonic sequence. |

CO02. | Prove Cauchy’s first limit theorem, Cesaro’s theorem, and Cauchy’s Second limit theorem. |

CO03. | Explain subsequence, upper and lower limits of a sequence. |

CO04. | Classify convergence, divergence and oscillating series. |

CO05. | Prove theorems on different test of convergence and divergence of a series of positive terms and Verify the given series is convergent or divergent by using different test. |

**PO - CO MAPPING MATRIX:**

CO | PO | PSO | ||||||
---|---|---|---|---|---|---|---|---|

PO1 | PO2 | PO3 | PO4 | PO5 | PSO1 | PSO2 | PSO3 | |

CO1 | 1 | 2 | - | 1 | 2 | - | 2 | - |

CO2 | - | 2 | - | 1 | - | - | - | - |

CO3 | - | 2 | - | 1 | - | - | - | - |

CO4 | - | 2 | - | 1 | 1 | - | - | - |

CO5 | 1 | 2 | - | 1 | 1 | - | - | - |