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Notes of Mathematical Method

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

We always try our best to add new solutions of more chapters as we are able to manage. If you have notes which you thing are worth to share with other then please contact us from here or email at Admin@MathCity.org.

List of chapters

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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
ARW Exercise 1.1 Download PDF (470.64 KiB, 2638 downloads) View online
ARW Exercise 1.2 Download PDF (1.27 MiB, 3787 downloads) View online
ARW Exercise 1.3 Download PDF (1.64 MiB, 2914 downloads) View online
ARW Exercise 1.4 Download PDF (653.33 KiB, 1997 downloads) View online
ARW Exercise 1.5 Download PDF (690.22 KiB, 1814 downloads) View online
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  • 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
ARW Exercise 2.1 Download PDF (774.94 KiB, 2014 downloads) View online
ARW Exercise 2.2 Download PDF (977.55 KiB, 1818 downloads) View online
ARW Exercise 2.3 Download PDF (410.76 KiB, 1435 downloads) View online
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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
ARW Articles of Ex 3.1 Download PDF (327.93 KiB, 1600 downloads) View online
ARW Exercise 3.1 Download PDF (630.02 KiB, 2454 downloads) View online
ARW Exercise 3.2 Download PDF (781.02 KiB, 2415 downloads) View online
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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
ARW Exercise 4 Download PDF) (1 MiB, 7182 downloads) View online
  • 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
ARW Exercise 5.1 Download PDF (1.27 MiB, 7938 downloads) View online
ARW Exercise 5.2 Download PDF (395.67 KiB, 5738 downloads) View online

Vetor Space

  • Subspaces
  • Linear combinations and spanning sets
  • Linear dependence and basis
  • Row and column spaces
  • Rand-Alternative method
  • Linear transformations
  • Matrix of linear transformation
ARW Exercise 6.1 NEW Download PDF (3.52 MiB, 9490 downloads) View online
ARW Exercise 6.2 NEW Download PDF (1.23 MiB, 1337 downloads) View online
ARW Exercise 6.3 NEW Download PDF (861.66 KiB, 5532 downloads) View online

Chapter 09: First Order Differential Equations

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  • D.E and their classification
  • Formation of differential equation
  • Initial and boundary conditions
  • Separable equations
  • Homogeneous equations
  • D.E. Reducible to homogeneous form
ARW Exercise 9.1 Download PDF) (829.68 KiB, 4810 downloads) View online
ARW Exercise 9.2 Download PDF) (457.81 KiB, 6116 downloads) View online
ARW Exercise 9.2 by Umer Asghar Download PDF) (476.62 KiB, 1263 downloads) View online
ARW Exercise 9.3 Download PDF) (992.56 KiB, 5161 downloads) View online
ARW Exercise 9.4 Download PDF) (450.02 KiB, 5776 downloads) View online
ARW Exercise 9.5 Download PDF) (1 MiB, 10054 downloads) View online
ARW Exercise 9.6 Download PDF) (508.84 KiB, 3997 downloads) View online
ARW Exercise 9.7 Download PDF) (789.57 KiB, 1973 downloads) View online
ARW Exercise 9.8 Download PDF) (952.79 KiB, 5219 downloads) View online
ARW Exercise 9.9 Download PDF) (975.02 KiB, 5116 downloads) View online
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  • Higher order linear differential equations
  • Exact equations
ARW Exercise 10.1 Download PDF (290.29 KiB, 5187 downloads) View online
ARW Exercise 10.2 Download PDF (1.48 MiB, 3972 downloads) View online
ARW Exercise 10.3 Download PDF (3.9 MiB, 3364 downloads) View online
ARW Exercise 10.4 Download PDF (608.67 KiB, 5608 downloads) View online
ARW Exercise 10.5 Download PDF (1.02 MiB, 4781 downloads) View online
ARW Exercise 10.6 Download PDF (749.21 KiB, 4396 downloads) View online
ARW Exercise 10.7 Download PDF (1.54 MiB, 1830 downloads) View online
The Laplace Transform

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
ARW Articles of Ex 11.1 Download PDF (486.17 KiB, 3931 downloads) View online
ARW Exercise 11.1 Download PDF (492.54 KiB, 1683 downloads) View online
ARW Exercise 11.2 Download PDF (594.78 KiB, 3730 downloads) View online
ARW Exercise 11.3 Download PDF (1.03 MiB, 480 downloads) View online
ARW Laplace Transformation Ex 11.1 by Umer Asghar Download PDF (1.39 MiB, 1560 downloads) View online
ARW Inverse Laplace Transformation Ex 11.2 by Umer Asghar Download PDF (1.24 MiB, 1218 downloads) View online

We are very thankful to Mr. Umer Asghar for providing the notes.



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