JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

Course Objectives

To learn

• Types of matrices and their properties.
• Concept of a rank of the matrix and applying this concept to know the consistency and solving the system of linear equations.
• Concept of Eigen values and eigenvectors and to reduce the quadratic form to canonical form.
• Concept of Sequence.
• Concept of nature of the series.
• Geometrical approach to the mean value theorems and their application to the mathematical problems
• Evaluation of surface areas and volumes of revolutions of curves.
• Evaluation of improper integrals using Beta and Gamma functions.
• Partial differentiation, concept of total derivative
• Finding maxima and minima of function of two and three variables.

Course Outcomes

After learning the contents of this paper the student must be able to

• Write the matrix representation of a set of linear equations and to analyse the solution of the system of equations
• Find the Eigen values and Eigen vectors
• Reduce the quadratic form to canonical form using orthogonal transformations.
• Analyse the nature of sequence and series.
• Solve the applications on the mean value theorems.
• Evaluate the improper integrals using Beta and Gamma functions
• Find the extreme values of functions of two variables with/ without constraints.

UNIT - I: Matrices

Matrices: Types of Matrices, Symmetric; Hermitian; Skew-symmetric; Skew-Hermitian; orthogonal matrices; Unitary Matrices; rank of a matrix by Echelon form and Normal form, Inverse of Non-singular matrices by Gauss-Jordan method; System of linear equations; solving system of Homogeneous and Non-Homogeneous equations. Gauss elimination method; Gauss Seidel Iteration Method.

UNIT - II: Eigen values and Eigen vectors

Linear Transformation and Orthogonal Transformation: Eigen values and Eigenvectors and their properties: Diagonalization of a matrix; Cayley-Hamilton Theorem (without proof); finding inverse and power of a matrix by Cayley-Hamilton Theorem; Quadratic forms and Nature of the Quadratic Forms; Reduction of Quadratic form to canonical forms by Orthogonal Transformation

UNIT - III: Sequences & Series

Sequence: Definition of a Sequence, limit; Convergent, Divergent and Oscillatory sequences. Series: Convergent, Divergent and Oscillatory Series; Series of positive terms; Comparison test, p-test, D-Alembert’s ratio test; Raabe’s test; Cauchy’s Integral test; Cauchy’s root test; logarithmic test. Alternating series: Leibnitz test; Alternating Convergent series: Absolute and Conditionally Convergence.

UNIT - IV: Calculus

Mean value theorems: Rolle’s theorem, Lagrange’s Mean value theorem with their Geometrical Interpretation and applications, Cauchy’s Mean value Theorem. Taylor’s Series.

Applications of definite integrals to evaluate surface areas and volumes of revolutions of curves (Only in Cartesian coordinates), Definition of Improper Integral: Beta and Gamma functions and their applications.

UNIT - V: Multivariable calculus (Partial Differentiation and applications)

Definitions of Limit and continuity. Partial Differentiation; Euler’s Theorem; Total derivative; Jacobian; Functional dependence & independence, Maxima and minima of functions of two variables and three variables using method of Lagrange multipliers.

TEXT BOOKS:

• 1. B.S. Grewal, Higher Engineering Mathematics, Khanna Publishers, 36th Edition, 2010
• 2. Erwin kreyszig, Advanced Engineering Mathematics, 9th Edition, John Wiley & Sons, 2006.
• 3. G.B. Thomas and R.L. Finney, Calculus and Analytic geometry, 9th Edition, Pearson, Reprint, 2002.

REFERENCE BOOKS:

• 1. N.P. Bali and Manish Goyal, A text book of Engineering Mathematics, Laxmi Publications, Reprint, 2008.
• 2. Ramana B.V., Higher Engineering Mathematics, Tata McGraw Hill New Delhi, 11th Reprint, 2010.