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## SEMESTER I

• Paper Title: Calculus
• Paper Code: MATHS101
• Pre Requisite: 10+2
• Max (Univ. Exam) Marks: 50 Time of examination: 3hrs.
• Internal Assessment: 50
• Course Duration: 45 lectures of one hour each.

Note for the examiner: The semester question paper will be of 50 Marks having 7 questions of equal marks. Students are required to attempt 5 questions in all. First question, covering the whole syllabus and having questions of conceptual nature, will be compulsory. Rest of the paper will be divided into two parts having three questions each and the candidate is required to attempt two questions from each section.

###### Objectives
• To understand the behaviour of infinite series and its use.
• To learn the concepts of functions of two and more than two variables and their applications.
• To learn the methods to evaluate multiple integrals and their applications to various problems.
• To understand the concepts of Vector calculus and their use in engineering problems.
###### PART A
1. FUNCTIONS OF ONE VARIABLE

Sequences and Series: Sequences, Limits of sequences, Infinite series, series of positive terms, Integral test, Comparison test, Ratio test, Root test. Alternating series, Absolute and Conditional Convergence, Leibnitz test. Power series: radius of convergence of power series, Taylor's and Maclaurin's Series, Formulae for remainder term in Taylor and Maclaurin series, Error estimates. (Scope as in Chapter 8, Sections 8.1 – 8.10 of Reference 1).

Integral Calculus: Areas of curves, Length of curves, Volume and surface areas of revolution (Scope as in Chapter 5, Sections 5.1, 5.3, 5.5, 5.6 of Reference 1).

2. DIFFERENTIAL CALCULUS OF FUNCTIONS OF TWO AND THREE VARIABLES

Concept of limit and continuity of a function of two and three variables, Partial derivatives, total derivative, Euler's theorem for homogeneous functions, composite function, differentiation of an implicit function, chain rule, change of variables, Jacobian, Taylor's theorem, Errors and increments, Maxima and minima of a function of two and three variables, Lagrange's method of multipliers (Scope as in Chapter 12, Sections 12.1 – 12.6, 12.8 – 12.9 of Reference 1).

3. SOLID GEOMETRY

Cylinder, Cone, Quadric surfaces, Surfaces of revolution.
(Scope as in: 10.6, 10.7 of Reference 1).

###### PART B
4. INTEGRAL CALCULUS OF FUNCTIONS OF TWO AND THREE VARIABLES

Double and triple integrals, Change of order of integration, Change of Variables, Applications to area, volume and surface area.(Scope as in Chapter 13 of Reference 1).

5. VECTOR DIFFERENTIAL CALCULUS

Vector-valued functions and space curves, arc lengths, unit tangent vector, Curvature and torsion of a curve, Gradient of a Scalar field, Directional Derivative (Scope as in Chapter 11, Sections 11.1, 11.3, 11.4, Chapter 12, Section 12.7 of Reference 1).

6. VECTOR INTEGRAL CALCULUS

Line integrals, Vector fields, Work, Circulation and Flux, Path Independence, Potential functions and Conservative fields, Green's theorem in the plane, Surface Areas and Surface Integrals, Stoke's Theorem, Gauss Divergence Theorem (Statements only) (Scope as in Chapter 14 of Reference 1).

###### Outcomes
• The students are able to test the behavior of infinite series.
• Ability to analyze functions of more than two variables and their applications.
• Ability to evaluate multiple integrals and apply them to practical problems.
• Ability to apply vector calculus to engineering problems
###### References:
• 1. G. B. Thomas, R. L. Finney. Calculus and Analytic Geometry, Ninth Edition, Pearson Education.
• 2. E. Kreyszig. Advanced Engineering Mathematics, Eighth Edition, John Wiley.
• 3. Michael D. Greenberg. Advanced Engineering Mathematics, Second Edition, Pearson Education.
• 4. Advanced Engineering Mathematics, Wylie and Barrett, McGraw Hill
• 5. B. V. Ramana. Higher Engineering Mathematics, Tata McGraw Hill.
• 6. R. K. Jain, S. R. K. Iyenger. Advanced Engineering Mathematics, Narosa

## SEMESTER II

• Paper Title: Differential Equations and Transforms
• Paper Code: MATHS201
• Pre Requisite: Calculus (MATHS101)
• Course Duration: 45 lectures of one hour each.
• Max (Univ. Exam) Marks: 50 Time of examination: 3hrs.
• Internal Assessment: 50

Note for the examiner: The semester question paper will be of 50 Marks having 7 questions of equal marks. Students are required to attempt 5 questions in all. First question, covering the whole syllabus and having questions of conceptual nature, will be compulsory. Rest of the paper will be divided into two parts having three questions each and the candidate is required to attempt two questions from each section.

###### Objectives
• To learn the methods to formulate and solve linear differential equations and their applications to engineering problems
• To learn the concepts of Laplace transforms and to evaluate Laplace transforms and inverse Laplace transform
• To apply Laplace transforms to solve ordinary differential equations
• To learn the concept of Fourier series, integrals and transforms.
• To learn how to solve heat, wave and Laplace equations.
###### PART A
1. ORDINARY DIFFERENTIAL EQUATIONS

Review of geometrical meaning of the differential equation y'=f(x,y), directional fields, Exact differential equations (Scope as in Chapter 8, Section 8.7 of Reference 5), Integrating factors (Scope as in Chapter 8, Section 8.8 of Reference 5), Solution of differential equations with constant coefficients: method of differential operators.

Non – homogeneous equations of second order with constant coefficients: Solution by method of variation of parameters (Scope as in Chapter 9, Section 9.7 of Reference 5). Power series method of solution (Scope as in Chapter 10, Section 10.2 of Reference 2)

2. Laplace Transforms

Laplace transform, Inverse transforms, shifting, transform of derivatives and integrals. Unit step function, second shifting theorem, Dirac's Delta function. Differentiation and integration of transforms.

Convolution Theorem on Laplace Transforms. Application of Laplace transforms to solve ordinary differential equations with initial conditions. (Scope as in Chapter 5, Sections 5.1 – 5.5 of Reference 1).

###### PART B
3. Fourier Series and Transforms:

Periodic functions, Fourier series, Even and odd series, half range expansions, Complex Fourier Series, Approximation by trigonometric polynomials. Fourier integrals, Fourier Cosine and Sine transform, Fourier Transforms.
(Scope as in Chapter 10, Sections 10.1 – 10.5, 10.7 – 10.10 of Reference 1).

4. Partial Differential Equations:

Partial differential equations of first order, origin, solution of linear partial differential equations of first order, Integral surfaces passing through a given curve
(Scope as in Chapter 2, Sections 1, 2, 4, 5 of Reference 4).

5. Boundary Value Problems:

"D'Alembert's solution of wave equation, separation of variables: one dimension and two dimensions heat and wave equation, Laplace equation in Cartesian and Polar coordinates
(Scope as in Chapter 11, Sections 11.1, 11.3 – 11.5, 11.8 – 11.9 of Reference 1).

###### Outcomes
• 1. The student will learn to solve Ordinary Differential equations.
• 2. The students will be able to apply the tools of Laplace Transforms to model engineering problems and solve the resulting differential equations.
• 3. Students will understand the nature and behavior of trigonometric (Fourier) series and apply it to solve boundary value problems.
###### References:
• 1. E. Kreyszig. Advanced Engineering Mathematics, Eighth Edition, John Wiley.
• 2. B. V. Ramana. Higher Engineering Mathematics, Tata McGraw Hill.