## University of Mumbai Semester 1 (FE First Year) Applied Mathematics 1 Revised Syllabus

University of Mumbai Semester 1 (FE First Year) Applied Mathematics 1 and their Unit wise marks distribution

### Units and Topics

## Syllabus

Solution of system of linear algebraic equations, by

- Gauss Elimination Method (Review)
- Guass Jordan Method
- Crouts Method (LU)
- Gauss Seidal Method and
- Jacobi iteration (Scilab programming for above methods is to be taught during lecture hours)

- 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 forms, system of homogeneous and non –homogeneous equations, their consistency and solutions. Linear dependent and independent vectors.

- Euler’S Theorem on Homogeneous Functions with Two and Three Independent Variables (With Proof).
- Deductions from Euler’S Theorem

- Partial derivatives of first and higher order, total differentials, differentiation of composite and implicit functions.

- nth derivative of standard functions.
- Leibnitz’s Thoerem (without proof) and problems.

- Regression Analysis (to be introduced for estimation only) (Scilab programming related to fitting of curves is to be taught during lecture hours)

- Lagrange’s method of undetermined multipliers with one constraint.
- Jacobian, Jacobian of implicit function.
- Partial derivative of implicit function using jacobian.

- Taylor’s Theorem (Statement only) and Taylor’s series, Maclaurin’s series (Statement only).
- Expansion of ex , sinx, cosx, tanx, sinhx, coshx, tanhx, log(1+x), sin‐1 x, cos1 x, Binomial series.
- Indeterminate forms, L‐ Hospital Rule, problems involving series also.

- Review of Complex Numbers‐Algebra of Complex Number
- Different Representations of a Complex Number and Other Definitions
- D’Moivre’S Theorem
- Powers and Roots of Exponential Function
- Powers and Roots of Trigonometric Functions
- Expansion of sinn θ, cosn θ in terms of sines and cosines of multiples of θ
- Expansion of sinnθ, cosnθ in powers of sinθ, cosθ
- .Circular Functions of Complex Number
- Hyperbolic functions of complex number
- Inverse Circular Functions
- Inverse Hyperbolic Functions
- Separation of Real and Imaginary Parts of All Types of Functions

Pre‐requisite: Review of Complex Numbers‐Algebra of Complex Number, Different

representations of a Complex number and other definitions, D’Moivre’s Theorem.

1.1.Powers and Roots of Exponential and Trigonometric Functions.

1.2. Expansion of sin nθ, cos nθ 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 a complex number and Hyperbolic functions. Inverse Circular and

Inverse Hyperbolic functions. Separation of real and imaginary parts of all types

of Functions.

- Inverse of a Matrix
- Addition of a Matrix
- Multiplication of a Matrix
- 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
- consistency and solutions of homogeneous and non – homogeneous equations
- Linear Dependent and Independent Vectors
- Application of Inverse of a Matrix to Coding Theory

- Indeterminate Forms
- L‐ Hospital Rule
- Problems Involving Series
- Solution of Transcendental Equations
- Solution by Newton Raphson Method
- Regula – Falsi Equation
- Solution of System of Linear Algebraic Equations by Gauss Elimination Method
- Gauss Jacobi Iteration Method
- Gauss Seidal Iteration Method
(Scilab programming for above methods is to be taught during lecture hours)