MATHEMATICS â?? I

UNIT - I: Matrix Theory
Introduction to matrices- Elementary row and column operations on a matrix- Finding rank of a matrix by reducing to Echelon and Normal forms-Consistency of system of linear equations (homogeneous and non- homogeneous) using the rank of a matrix-Cayley - Hamilton Theorem (without proof) â?? Verification- finding inverse of a matrix and powers of a matrix by Cayley-Hamilton theorem- Linear dependence and Independence of Vectors- Eigen values and eigen vectors of a matrix-Properties of eigen values and eigen vectors of real and complex matrices,Diagonalisation of matrix.

UNIT â?? II: Differential Calculus
Mean Value Theorms: Rolleâ??s Theorem â?? Lagrangeâ??s Mean Value Theorem â?? Cauchyâ??s mean value Theorems with geometrical interpretations (all theorems without proof)- verification of the Theorems and testing the applicability of these theorems to the given function- Taylorâ??s series -Maclaurinâ??s series.
Functions of Several Variables: Jacobian-Functional dependence- Maxima and Minima of functions of two variables without constraints and with constraints - Method of Lagrange multipliers.

UNIT â?? III: Ordinary Differential Equations of First Order and Applications
Formation of differential equation- Solution of D.E - Variable Separable form- Homogeneous-Non homogeneous- Exact-Non Exact-Linear and Bernoulliâ??s equations-Applications of first order differential equations â?? Newtonâ??s Law of cooling- Law of natural growth and decay- Orthogonal trajectories.

 

UNIT - IV: Linear Differential Equations of Higher Order and Applications
Linear differential equations of second and higher order with constant coefficients- Non-homogeneous term of the type f(x) = eax , sinax, cosax, xn , eax V and xn V- Method of variation of parameters-Applications to bending of beams, Electrical circuits and simple harmonic motion.

UNIT â?? V Laplace Transforms and Applications

Definition of Laplace transform- Domain of the function and Kernel for the Laplace transforms- Existence of Laplace transform- Laplace transform of standard functions- first shifting Theorem,-Laplace transform of functions when they are multiplied or divided by â??tâ??- Laplace transforms of derivatives and integrals of functions â?? Unit step function â?? second shifting theorem â?? Diracâ??s delta function- Periodic function â?? Inverse Laplace transform by Partial fractions-Inverse Laplace transforms of functions when they are multiplied or divided by â??sâ??, Inverse Laplace Transforms of derivatives and integrals of functions- Convolution theorem â??Solving ordinary differential equations by Laplace transforms.

TEXT BOOKS:
1. Engineering Mathematics â?? I by T.K. V. Iyengar, B. Krishna Gandhi & Others, S. Chand.
2. Higher Engineering Mathematics by B.S. Grewal, Khanna Publishers.

REFERENCES:
1. Advanced Engineering Mathematics by R.K. Jain & S.R.K. Iyengar, 3rd edition, Narosa
Publishing House, Delhi.
2. Advanced engineering Mathematics by Kreyszig, John Wiley & Sons Publishers.
3. Engineering Mathematics â?? I by D. S. Chandrasekhar, Prison Books Pvt. Ltd.