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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, 21205 downloads) View online
ARW Exercise 1.2 Download PDF (1.27 MiB, 8484 downloads) View online
ARW Exercise 1.3 Download PDF (1.64 MiB, 4251 downloads) View online
ARW Exercise 1.4 Download PDF (653.33 KiB, 21316 downloads) View online
ARW Exercise 1.5 Download PDF (690.22 KiB, 19562 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, 16252 downloads) View online
ARW Exercise 2.2 Download PDF (977.55 KiB, 18080 downloads) View online
ARW Exercise 2.3 Download PDF (410.76 KiB, 12515 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, 9826 downloads) View online
ARW Exercise 3.1 Download PDF (630.02 KiB, 21826 downloads) View online
ARW Exercise 3.2 Download PDF (781.02 KiB, 24726 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, 17761 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, 6065 downloads) View online
ARW Exercise 5.2 Download PDF (395.67 KiB, 21247 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 Download PDF (3.52 MiB, 5385 downloads) View online
ARW Exercise 6.2 Download PDF (1.23 MiB, 4056 downloads) View online
ARW Exercise 6.3 Download PDF (861.66 KiB, 19933 downloads) View online
Geometric series

Infinite series are of great importance in both pure and applied mathematics. They play a significant role in Physics and engineering. In fact many functions can be represented by infinite series. The theory of infinite series is developed through the use of special kind of function called sequence.

  • Sequences
  • Infinite series
  • The basic comparison test
  • The limit comparison test
  • The integral test
  • The ratio test
  • Cauchy's root test
  • Alternating series
  • Absolute and conditional convergence
  • Power series
  • Convergence of power series
  • Differentiation and integration of power series
  • Addition and multiplication of power series
ARW Exercise 8.1 Download PDF (494.55 KiB, 2906 downloads) View online
ARW Exercise 8.2 Download PDF (1.09 MiB, 3939 downloads) View online
ARW Exercise 8.3 Download PDF (658.47 KiB, 3732 downloads) View online
ARW Exercise 8.4 Download PDF (854.59 KiB, 3072 downloads) View online
ARW Exercise 8.5 Download PDF (1.46 MiB, 3449 downloads) View online
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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, 22055 downloads) View online
ARW Exercise 9.2 Download PDF) (457.81 KiB, 3785 downloads) View online
ARW Exercise 9.2 by Umer Asghar Download PDF) (476.62 KiB, 16574 downloads) View online
ARW Exercise 9.3 Download PDF) (992.56 KiB, 7394 downloads) View online
ARW Exercise 9.3 by Umer Asghar Download PDF) (1.06 MiB, 18768 downloads) View online
ARW Exercise 9.4 Download PDF) (450.02 KiB, 4873 downloads) View online
ARW Exercise 9.4 by Umer Asghar Download PDF) (1.21 MiB, 6439 downloads) View online
ARW Exercise 9.5 Download PDF) (1 MiB, 9098 downloads) View online
ARW Exercise 9.6 Download PDF) (508.84 KiB, 27050 downloads) View online
ARW Exercise 9.7 Download PDF) (789.57 KiB, 18427 downloads) View online
ARW Exercise 9.8 Download PDF) (952.79 KiB, 4882 downloads) View online
ARW Exercise 9.9 Download PDF) (975.02 KiB, 23294 downloads) View online
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  • Higher order linear differential equations
  • Exact equations
ARW Exercise 10.1 by Umer Asghar Download PDF (670.29 KiB, 16177 downloads) View online
ARW Exercise 10.1 Download PDF (290.29 KiB, 16200 downloads) View online
ARW Exercise 10.2 by Umer Asghar Download PDF (889.65 KiB, 14717 downloads) View online
ARW Exercise 10.2 Download PDF (1.48 MiB, 8524 downloads) View online
ARW Exercise 10.3 Download PDF (3.9 MiB, 22041 downloads) View online
ARW Exercise 10.4 by Umer Asghar Download PDF (643.86 KiB, 11788 downloads) View online
ARW Exercise 10.4 Download PDF (608.67 KiB, 1299 downloads) View online
ARW Exercise 10.5 Download PDF (1.02 MiB, 684 downloads) View online
ARW Exercise 10.6 Download PDF (749.21 KiB, 16109 downloads) View online
ARW Exercise 10.7 Download PDF (1.54 MiB, 7303 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, 10054 downloads) View online
ARW Exercise 11.1 Download PDF (492.54 KiB, 703 downloads) View online
ARW Exercise 11.2 Download PDF (594.78 KiB, 10175 downloads) View online
ARW Exercise 11.3 Download PDF (1.03 MiB, 8186 downloads) View online
ARW Laplace Transformation Ex 11.1 by Umer Asghar Download PDF (1.39 MiB, 8305 downloads) View online
ARW Inverse Laplace Transformation Ex 11.2 by Umer Asghar Download PDF (1.24 MiB, 10320 downloads) View online

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



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