Topics and Syllabus - Applied Mathematics 1 M.Sc. University of Mumbai

• 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.  

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)

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

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

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

• 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.

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