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APPLIED MATHEMATICS –III 

DETAILED CONTENTS

 

1. Matrices                              ( Marks Allotted 12 )          

     1.1 Algebra of Matrices, Inverse
           Addition, Multiplication of matrices, Null matrix and a unit matrix, Square matrix, Symmetric, Skew symmetric, Hermitian, Skew hermition, Orthagonal, Unitary, diagonal and Triangular matrix, Determinant of a matrix. Definition and Computation of inverse of a matrix.
     1.2 Elementry Row/Column Transformation 
           Meaning and use in computing inverse and rank of a matrix.
     1.3 Linear Dependence, 
          Rank of a Matrix Linear dependence/independence of vectors, Definition and computation of rank of matrix. Computing rank through determinants, Elementary row transformation and through the concept of a set of independent vectors, Consistency of equations.
     1.4 Eigen Pairs, Cayley-Hamilton Theorem 
           Definition and evaluation of eign values and eign vectors of a matrix of order two and three, Cayley-Hamilton theorem (without Proof)and its verification, Use in finding inverse and powers of a matrix.

2.Differential Calculus             ( Marks Allotted 10 )

     2.1 Function of two variables, identification of surfaces in space, conicoids 

     2.2 Partial Differentiation 

           Directional derivative, Gradient, Use of gradient f, Partial derivatives, Chain rule, Higher order derivatives, Euler’s theorem for homogeneous functions, Jacobians. 

     2.3 Vector Calculus 

           Vector function,Introduction todouble and triple integral,differentiation and integration of vector functions, gradient, divergence and curl, differential derivatives. 

3.Differential Equation            ( Marks Allotted 10 )

     3.1 Formation, Order, Degree, Types, Solution 

           Formation of differential equations through physical, geometrical, mechanical and electrical considerations, Order, Degree of a differential equation, Linear, nonlinear equation. 

     3.2 First Order Equations Variable seperable, equations reducible to seperable forms, Homogeneous equtions, equations reducible to homogeneous forms, Linear and Bernoulli form exact equation and their solutions.

     3.3 Higher Order Linear Equation : 

           Property of solution, Linear differential equation with constant coefficients (PI for X= ax , Sinax, Cosax, Xn , axV, XV

     3.4 Simple Applications 

           LCR circuit, Motion under gravity, Newton's law of cooling, radioactive decay, Population growth, Force vibration of a mass point attached to spring with and without damping effect. Equivalence of electrical and mechanical system.

4.Integral Calculus-II              ( Marks Allotted 09 )

     4.1 Beta and Gamma Functions 

           Definition, Use, Relation between the two, their use in evaluating integrals.

     4.2 Fourier Series

           Fourier series of f(x),-n<x<n, Odd and even function,Half range series. 

     4.3 Laplace Transform 

           Definition, Basic theorem and properties, Unit step and Periodic functions,inverse laplace transform, Solution of ordinary differential equations.

5.Probability and Statistics      ( Marks Allotted 09 )

     5.1 Probability 

           Introduction, Addition and Multiplication theorem and simple problem. 

     5.2 Distribution 

           Discrete and continuous distribution, Bionimal Distribution, Poisson distribution, Normal Distribution. 

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