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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, 682 downloads) View online
ARW Exercise 1.2 Download PDF (1.27 MiB, 11562 downloads) View online
ARW Exercise 1.3 Download PDF (1.64 MiB, 9676 downloads) View online
ARW Exercise 1.4 Download PDF (653.33 KiB, 6859 downloads) View online
ARW Exercise 1.5 Download PDF (690.22 KiB, 6390 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, 3156 downloads) View online
ARW Exercise 2.2 Download PDF (977.55 KiB, 5964 downloads) View online
ARW Exercise 2.3 Download PDF (410.76 KiB, 4528 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, 430 downloads) View online
ARW Exercise 3.1 Download PDF (630.02 KiB, 7408 downloads) View online
ARW Exercise 3.2 Download PDF (781.02 KiB, 7585 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, 12756 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, 2265 downloads) View online
ARW Exercise 5.2 Download PDF (395.67 KiB, 9011 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, 5776 downloads) View online
ARW Exercise 6.2 Download PDF (1.23 MiB, 4320 downloads) View online
ARW Exercise 6.3 Download PDF (861.66 KiB, 9348 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, 847 downloads) View online
ARW Exercise 9.2 Download PDF) (457.81 KiB, 11495 downloads) View online
ARW Exercise 9.2 by Umer Asghar Download PDF) (476.62 KiB, 563 downloads) View online
ARW Exercise 9.3 Download PDF) (992.56 KiB, 11673 downloads) View online
ARW Exercise 9.3 by Umer Asghar Download PDF) (1.06 MiB, 4259 downloads) View online
ARW Exercise 9.4 Download PDF) (450.02 KiB, 10423 downloads) View online
ARW Exercise 9.4 by Umer Asghar Download PDF) (1.21 MiB, 3204 downloads) View online
ARW Exercise 9.5 Download PDF) (1 MiB, 907 downloads) View online
ARW Exercise 9.6 Download PDF) (508.84 KiB, 8685 downloads) View online
ARW Exercise 9.7 Download PDF) (789.57 KiB, 755 downloads) View online
ARW Exercise 9.8 Download PDF) (952.79 KiB, 230 downloads) View online
ARW Exercise 9.9 Download PDF) (975.02 KiB, 8987 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, 3162 downloads) View online
ARW Exercise 10.1 Download PDF (290.29 KiB, 7685 downloads) View online
ARW Exercise 10.2 by Umer Asghar Download PDF (889.65 KiB, 1711 downloads) View online
ARW Exercise 10.2 Download PDF (1.48 MiB, 2768 downloads) View online
ARW Exercise 10.3 Download PDF (3.9 MiB, 2455 downloads) View online
ARW Exercise 10.4 by Umer Asghar Download PDF (643.86 KiB, 1459 downloads) View online
ARW Exercise 10.4 Download PDF (608.67 KiB, 8445 downloads) View online
ARW Exercise 10.5 Download PDF (1.02 MiB, 7765 downloads) View online
ARW Exercise 10.6 Download PDF (749.21 KiB, 6880 downloads) View online
ARW Exercise 10.7 Download PDF (1.54 MiB, 4732 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, 5362 downloads) View online
ARW Exercise 11.1 Download PDF (492.54 KiB, 3197 downloads) View online
ARW Exercise 11.2 Download PDF (594.78 KiB, 5250 downloads) View online
ARW Exercise 11.3 Download PDF (1.03 MiB, 2321 downloads) View online
ARW Laplace Transformation Ex 11.1 by Umer Asghar Download PDF (1.39 MiB, 1296 downloads) View online
ARW Inverse Laplace Transformation Ex 11.2 by Umer Asghar Download PDF (1.24 MiB, 3635 downloads) View online

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



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