# Applied Mathematics 1 Semester 1 (FE First Year) BE Production Engineering University of Mumbai Topics and Syllabus

CBCGS [2016 - current]
CBGS [2012 - 2015]
Old [2000 - 2011]

## Syllabus

100 Complex Numbers Old
101 Expansion of Sinn θ,Cosn θ in Terms of Sines and Cosines Of Multiples Of θ And Expansion of Sinnθ, Cosnθ In Powers of Sinθ, Cosθ
102 Separation of Real and Imaginary Parts of All Types of Functions
103 Circular Functions of Complex Number and Hyperbolic Functions.Inverse Circular and Inverse Hyperbolic Functions. Logarithmic Functions.
104 Powers and Roots of Exponential and Trigonometric Functions
200 Matrices and Numerical Methods Old
201 Solution of System Of Linear Algebraic Equations

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)
202 Types of Matrices and Rank 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 forms, system of homogeneous and non –homogeneous equations, their consistency and solutions. Linear dependent and independent vectors.
300 Differential Calculus Old
301 Euler’S Theorem on Homogeneous Functions with Two and Three Independent Variables (With Proof)
• Euler’S Theorem on Homogeneous Functions with Two and Three Independent Variables (With Proof).
• Deductions from Euler’S Theorem
302 Partial Differentiation
• Partial derivatives of first and higher order, total differentials, differentiation of composite and implicit functions.
303 Successive Differentiation
• nth derivative of standard functions.
• Leibnitz’s Thoerem (without proof) and problems.
400 Application of Partial Differentiation, Expansion of Functions , Indeterminate Forms and Curve Fitting Old
401 Fitting of Curves by Least Square Method for Linear, Parabolic, And Exponential
• Regression Analysis (to be introduced for estimation only) (Scilab programming related to fitting of curves is to be taught during lecture hours)
402 Maxima and Minima of a Function of Two Independent Variables
• Lagrange’s method of undetermined multipliers with one constraint.
• Jacobian, Jacobian of implicit function.
• Partial derivative of implicit function using jacobian.
403 Taylor’S Theorem and Taylor’S Series, Maclaurin’S Series
• 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.
500 Complex Numbers

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.

600 Logarithm of Complex Numbers , Successive Differentiation
601 Successive Differentiation
602 Logarithm of Complex Numbers
900 Applications of Partial Differentiation , Expansion of Functions
1000 Indeterminate Forms, Numerical Solutions of Transcendental Equations and System of Linear Equations