Syllabus of Applied mathematics 1

1. Complex Numbers

Pre‐requisite: Review of Complex Numbers‐Algebra of Complex Number, Different representations of a Complex number and other definitions, DMoivres Theorem.

1.1. Powers and Roots of Exponential and Trigonometric Functions.

1.2. Expansion of sinnθ, cosnθ in terms of sines and cosines of multiples of θ and Expansion of sinnθ, cosnθ in powers of sinθ, cosθ

1.3. Circular functions of complex number and Hyperbolic functions. Inverse Circular and Inverse Hyperbolic functions. Separation of real and imaginary parts of all types of Functions.

2. Logarithm of Complex Numbers , Successive Differentiation

2.1 Logarithmic functions, Separation of real and Imaginary parts of Logarithmic Functions.

2.2 Successive differentiation: nth derivative of standard functions. Leibnitz’s Theorem (without proof) and problems

3. Matrices

Pre‐requisite: Inverse of a matrix, addition, multiplication and transpose of a matrix

Types of Matrices (symmetric, skew‐ symmetric, Hermitian, Skew Hermitian, Unitary, Orthogonal Matrices and properties of Matrices). Rank of a Matrix using Echelon forms, reduction to normal form, PAQ in normal form, system of homogeneous and non – homogeneous equations, their consistency and solutions. Linear dependent and independent vectors. Application of inverse of a matrix to coding theory

4. Partial Differentiation

4.1 Partial Differentiation: Partial derivatives of first and higher order. Total differentials, differentiation of composite and implicit functions.

4.2. Euler’s Theorem on Homogeneous functions with two and three independent variables (with proof). Deductions from Euler’s Theorem

5. Applications of Partial Differentiation , Expansion of Functions

5.1 Maxima and Minima of a function of two independent variables, Jacobian.

5.2 Taylor’s Theorem (Statement only) and Taylor’s series, Maclaurin’s series (Statement only).Expansion of ex, sin(x), cos(x), tan(x), sinh(x), cosh(x), tan(x), log(1+x), $sin^{-1}(x)$, $cos^{-1}(x)$, $tan^{-1}(x)$

Binomial series.

6. Indeterminate forms, Numerical Solutions of Transcendental Equations and System of Linear Equations

6.1 Indeterminate forms, L‐ Hospital Rule, problems involving series.

6.2 Solution of Transcendental Equations: Solution by Newton Raphson method and Regula –Falsi Equation.

6.3 Solution of system of linear algebraic equations, by

(1) Gauss Elimination Method,

(2) Gauss Jacobi Iteration Method,

(3) Gauss Seidal Iteration Method.

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