## JNTU B Tech 1st Sem Syllabus for Mathematics I

**University:** Jawaharlal Nehru Technological University Hyderabad

MATHEMATICS- I

B.Tech. I Year I Sem. L T/P/D C

Course Code: MA101BS 3 1/0/0 3

**UNIT – I Sequences – Series**

Basic definitions of Sequences and series – Convergences and divergence – Ratio test – Comparison test– Integral test – Cauchy’s root test – Raabe’s test – Absolute and conditional convergence

**UNIT – II Functions of Single Variable**

Rolle’s Theorem – Lagrange’s Mean Value Theorem – Cauchy’s mean value Theorem – Generalized Mean, Value theorem (all theorems without proof) Functions of several variables – Functional dependence-Jacobian- Maxima and Minima of functions of two variables with constraints and without constraints

**UNIT – III Application of Single variables**

Radius, Centre, and Circle of Curvature – Evolutes and Envelopes Curve tracing – Cartesian, polar and parametric curves.

**UNIT – IV Integration & its applications**

Riemann Sums , Integral Representation for lengths, Areas, Volumes and Surface areas in Cartesian and polar coordinates multiple integrals – double and triple integrals – change of order of integration- change

of variable

**UNIT – V Differential equations of first order and their applications**

Overview of differential equations- exact, linear and Bernoulli. Applications to Newton’s Law of cooling, Law of natural growth and decay, orthogonal trajectories and geometrical applications.

**UNIT – VI Higher Order Linear differential equations and their applications**

Linear differential equations of second and higher order with constant coefficients, RHS term of the type f(X)= e ax , Sin ax, Cos ax, and xn, e ax V(x), x n V(x), method of variation of parameters. Applications bending of beams, Electrical circuits, simple harmonic motion.

**UNIT – VII Laplace transform and its applications to Ordinary differential equations **

Laplace transform of standard functions – Inverse transform – first shifting Theorem, Transforms of derivatives and integrals – Unit step function – second shifting theorem – Dirac’s delta function – Convolution theorem – Periodic function – Differentiation and integration of transforms-Application of Laplace transforms to ordinary differential equations.

**UNIT – VIII Vector Calculus**

Vector Calculus: Gradient- Divergence- Curl and their related properties Potential function – Laplacian and second-order operators. Line integral – work is done ––- Surface integrals – Flux of a vector-valued function.

Vector integrals theorems: Green’s -Stoke’s and Gauss’s Divergence Theorems (Statement & their Verification).

### TEXTBOOKS:

1. Engineering Mathematics – I by P.B. Bhaskara Rao, S.K.V.S. Rama Chary, M. Bhujanga Rao.

2. Engineering Mathematics – I by C. Shankaraiah, VGS Booklinks.

### REFERENCES:

1. Engineering Mathematics – I by T.K. V. Iyengar, B. Krishna Gandhi & Others, S. Chand.

2. Engineering Mathematics – I by D. S. Chandrasekhar, Prison Books Pvt. Ltd.

3. Engineering Mathematics – I by G. Shanker Rao & Others I.K. International Publications.

4. Higher Engineering Mathematics – B.S. Grewal, Khanna Publications.

5. Advanced Engineering Mathematics by Jain and S.R.K. Iyengar, Narosa Publications.

6. A text Book of KREYSZIG’S Engineering Mathematics, Vol-1 Dr .A. Ramakrishna Prasad. WILEY

publications

2009-2010

## Linear Algebra and Differential Equations JNTU B.Tech 1st Semester Mathematics

**UNIT – I : Solution for linear systems**

Matrices and Linear systems of equations: Elementary row transformations-Rank-Echelon form, Normal form – Solution of Linear Systems – Direct Methods- LU Decomposition- LU Decomposition from Gauss Elimination –Solution of Tridiagonal Systems-Solution of Linear Systems

**UNIT – II: Eigen Values & Eigen Vectors**

Eigenvalues, eigenvectors – properties – Condition number of rank, Cayley-Hamilton Theorem (without Proof) – Inverse and powers of a matrix by Cayley-Hamilton theorem – Diagonalization of the matrix. Calculation

of powers of matrix – Modal and spectral matrices.

**UNIT – III: Linear Transformations**

Real matrices – Symmetric, skew-symmetric, orthogonal, Linear Transformation – Orthogonal Transformation. Complex matrices: Hermitian, Skew-Hermitian, and Unitary – Eigenvalues and eigenvectors of complex matrices and their properties. Quadratic forms- Reduction of quadratic form to canonical form – Rank – Positive, negative definite – semi definite – index – signature – Sylvester law, Singular value

decomposition.

**UNIT – IV: Solution of Non- linear Systems**

The solution of Algebraic and Transcendental Equations: Introduction – The Bisection Method – The Method of False Position – The Iteration Method – Newton-Raphson Method. Interpolation: Introduction- Errors in Polynomial Interpolation – Finite differences- Forward Differences- Backward differences –Central differences – Symbolic relations and separation of symbols- Difference Equations – Differences of a polynomial-Newton’s formulae for interpolation – Central difference interpolation

Formulae – Gauss Central Difference Formulae –Interpolation with unevenly spaced points-Lagrange’s Interpolation formula. B. Spline interpolation – Cubic spline.

**UNIT – V: Curve fitting & Numerical Integration**

Curve fitting: Fitting a straight line –Second-degree curve-exponential curve-power curve by the method of least squares. Numerical Differentiation – Simpson’s 3/8 Rule, Gaussian Integration, Evaluation of principal

value integrals, Generalized Quadrature.

**UNIT – VI: Numerical solution of IVP’s in ODE**

Numerical solution of Ordinary Differential equations: Solution by Taylor’s series-Picard’s Method of successive Approximations-Euler’s Method-Runge-Kutta Methods –Predictor-Corrector Methods- Adams-

Bashforth Method.

**UNIT – VII Fourier Series**

Fourier Series: Determination of Fourier coefficients – Fourier series – even and odd functions – Fourier series in an arbitrary interval – even and odd periodic continuation – Half-range Fourier sine and cosine expansions.

**UNIT – VIII Partial differential equations**

Introduction and Formation of partial differential equation by the elimination of arbitrary constants and arbitrary functions, solutions of first-order linear (Lagrange) equation and nonlinear (Standard type) equations, Method of separation of variables for second-order equations -Two-dimensional wave equation.

### TEXTBOOKS:

1. Mathematical Methods by P.B.Bhaskara Rao, S.K.V.S. Rama Chary, M.Bhujanga Rao,

B.S.Publications.

2. Mathematical Methods by K.V.Suryanarayana Rao by Scitech Publications.

### REFERENCES:

1. Mathematical Methods by T.K.V. Iyengar, B.Krishna Gandhi & Others, S. Chand.

2. Introductory Methods by Numerical Analysis by S.S. Sastry, PHI Learning Pvt. Ltd.

3. Mathematical Methods by G.Shankar Rao, I.K. International Publications, N.Delhi

4. Higher Engineering Mathematics by B.S. Grewal, Khanna Publications.

5. Mathematical Methods by V. Ravindranath, Etl, Himalaya Publications.

2009-2010

6. A text Book of KREYSZIG’S Mathematical Methods, Dr .A. Ramakrishna Prasad. WILEY

publications.