Introduction to the idea of a matrix; equality of matrices; special matrices. Algebraic operations of
matrices: commutative property, associative property and distributive property. Transpose of a matrix
(properties (At)t = A, (A+B)t = At + Bt , (cA)t = cAt, (AB)t = BtAt to be stated (without proof) and verified
by simple examples). Symmetric and Skew symmetric matrices.
Properties of determinant (statement only); minor, co-factors and Laplace expansion of determinant;
Cramer's rule and its application in solving system of linear equations of three variables.
Singular and non-singular matrices; adjoint matrix; inverse of a matrix [(AB)-1 = B-1A-1 to be stated and
verified by example. Elementary row and column operations on matrices; definition of rank of a matrix;
determination of rank of a matrix using definition.
System of Linear Equations:
Consistency and Inconsistency. Gauss elimination process for solving a system of linear equations in
three unknowns.
Vector Space:
Basic idea of set, mapping, Binary Composition and Scalar field. Definition of vector space over the field
of real numbers; Examples of vector space; Definition of sub-space of a vector space and a criterion for a
sub-space; Definition of Linear combination, Linear independence and linear dependence of vectors with
examples. Definition of basis and dimension of vector space; Definition of Linear transformation:
Definition of kernel and images of a Linear transformation; Kernel and Images of a Linear
Transformation forming sub-spaces; Nullity and Rank of a Linear Transformation; Dim Ker T + Dim Im
T = Dim V; Definition of Inner product space; Norm of a vector; Orthogonal and Ortho-normal set of
vectors.
Eigenvalues and Eigenvectors of a matrix; Eigenvalues of a Real Symmetric Matrix; Necessary and
Sufficient Condition of diagonalization of matrices (statement only); Diagonalization of a matrix
(problems restricted to 2 x 2 matrix).
Ordinary Differential Equations (ODE):
Definition of order and degree of ODE;
ODE of the first order: Exact equations; Definition and use of integrating factor; Linear equation and
Bernoulli’s equation. ODE of first order and higher degree, simple problems.
General ODE of 2nd order: D-operator method for finding particular integrals. Method of variation of
parameters. Solution of Cauchy-Euler homogeneous linear equations. Solution of simple simultaneous
linear differential equations.
Verification of Legendre function (Pn (x)) and Bessel function (Jn (x)) as the solutions of Legendre and
Bessel equations respectively. Graphical representations of these solutions.
Laplace Transform (LT):
Definition; Existence of LT; LT of elementary functions; First and second shifting properties; Change of
scale property; LT of derivative of functions. LT of (tn f (t)), LT of f (t) / tn; LT of periodic function and
unit step function. Convolution theorem (statement only).
Inverse LT; Solution of ODE's (with constant coefficients) using LT.
Numerical Methods:
Error: Absolute, Percentage, Relative errors. Truncation error, Round off error.
Difference operator (forward, backward, central, shift and average operators); Different table,
Propagation of Error. Definition of Interpolation and Extra-polation. Newton's forward and backward
interpolation formula; Lagrange interpolation formula and corresponding error formulae (statement only).
Numerical Differentiation: Using Newton’s forward and backward interpolation formula.
Numerical Integration: Trapezoidal rule and Simpson's 1/3rd rule and corresponding error terms
(statement only).
Reference Book
1.Kreyszig E.- Advance Engineering Mathematics
2.Krishnamurthy V., Mainra V.P. and Arora J.L. -An Introduction to Linear Algebra
3.Boyce and Diprima- Elementary Differential Equations and Boundary Value Problems
5.Grewal B.S.- Engineering Mathematics
6.S.K.Ratho-r Higher Engineering Mathematics II.EPH
7.Lakshmninarayn- Engg Math,Vikas
8.Jana- UG Engg. Mathematics,Vikas
9.Chakraborty A. -Elements of Ord.Diff. Equations,New Age
10.Bhattacharya P.B. -First Course in Linear Algebra,New Age
11.Rao Sarveswar A. -Engineering Mathematics, Universities Press
12.Gupta S.K. -Numerical Methods for Engineers, New Age
13.Jain M.K. -Numerical Methods for Sc. & Engg Computation, New Age International
14.Jain M.K.- Numerical Solutions of Differential Equations
15.Balachandra Rao- Numerical Methods with Programs in Basic, Fortran Pascal and C++
16.Dutta N.- Computer Programming & Numerical Analysis:An Integral Approach,Universities Press
18.Rao S.B. -Differential Equations with Applications & Programs, Universities Press
19.Murray D.A. -Introductory Course in Differential Equations
20.Bagchi S.C.- First Course on Representation Theory & Linear Lie Groups, Universities Press
Arumugam Engineering mathematics,I,II & III, Scitech