# Notes of Mathematical Method

Notes of the Mathematical Method written by by S.M. Yusuf, A. Majeed and M. Amin and published by Ilmi Kitab Khana, Lahore.

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### List of chapters

• FEW CHAPTERS ARE GIVEN BELOW

A complex number is an element $(x,y)$ of the set $$\mathbb{R}^2=\{(x,y): x,y \in \mathbb{R}\}$$ obeying the following rules of addition and multiplication.

For $z_1=(x_1,y_1)$, $z_2=(x_2,y_2)$, we put

1. $z_1+z_2= (x_1+x_2, y_1+y_2)$
2. $z_1 z_2 = (x_1 x_2 - y_1 y_2, x_1 y_2+y_1 x_2)$

The set $\mathbb{R}^2$ with operation defined above is denoted by $\mathbb{C}$.

• Complex numbers
• Properties of complex numbers
• The Argand's diagram
• De Moivre's theorem
• Roots of the complex numbers
• Basic elementary functions
• Logarithmic functions
• Inverse hyperbolic functions
• Inverse trigonometric functions
• Complex power
• Summation of series
• Definition (axioms of group)
• Definition ( commutative group )
• Definition (idempotent)
• Properties of Group
• Theorem (The Cancellation Law)
• Theorem (Solution of Linear Equations )
• Subgroups
• Definition ( subgroup )
• Cyclic Groups
• Definition ( cyclic group )
• Cosets-Lagrange’s Theorem
• Permutations
• Cycles
• Transpositions
• Order of a Permutation
• Rings and Fields
• Properties of Rings

The difficulty level of this chapter is very low. Most of the questions involve calculations. This chapter is wide range of applications in Linear Algebra. In many universities teachers include this chapter in the syllabus of Linear Algebra for BS students of mathematics and other subjects.

• Introduction
• Algebra of matrices
• Partitioning of matrices
• Inverse of a matrix
• Elementary row operations
• Elementary column operations

The difficulty level of this chapter is low. Most of the questions involve calculations. This chapter is wide range of applications in Linear Algebra and Operations Research. In many universities teachers include this chapter in the syllabus of Linear Algebra and Operations Research for BS students of mathematics and other subjects.

• Preliminaries
• Equivalent equations
• Gaussian elimination method
• Gauss-Jordan elimination method
• Consistency criterion
• Network flow problems
• Determinant of a square matrix
• Axiomatic definition of a determinant
• Determinant as sum of products of elements
• Determinant of the transpose
• An algorithm to evaluate Det A
• Determinants and inverse of matrices

• Subspaces
• Linear combinations and spanning sets
• Linear dependence and basis
• Row and column spaces
• Rand-Alternative method
• Linear transformations
• Matrix of linear transformation
• D.E and their classification
• Formation of differential equation
• Initial and boundary conditions
• Separable equations
• Homogeneous equations
• D.E. Reducible to homogeneous form
• Higher order linear differential equations
• Exact equations

Let $f$ be a real valued piecewise continuous function defined on $[0,\infty)$. The Laplace transform of $f$, denoted by $\mathcal{L}(f)$, is the function $F$ defined by $$F(s)=\int_0^{\infty} e^{-st} f(t) dt,$$ provided the above improper integral converges. We have $F=\mathcal{L}(f)$. Laplace transformation has lot of applications in engineering and applied mathematics.

The following topics are discussed in this chapter.

• The Laplace transform
• Properties of the Laplace transform
• Inverse Laplace transform
• Convolution
• Solution of initial value problem