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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.

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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, 21277 downloads) View online
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ARW Exercise 1.3 Download PDF (1.64 MiB, 1748 downloads) View online
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ARW Exercise 1.5 Download PDF (690.22 KiB, 9499 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
Notes by Shahzad Ahmed Khan: NEW
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ARW Exercise 2.2 Download PDF (602.28 KiB, 100 downloads) View online
ARW Exercise 2.3 Download PDF (657.81 KiB, 14503 downloads) View online
Notes by Shariq Mehtab Syed:
ARW Exercise 2.1 Download PDF (774.94 KiB, 28204 downloads) View online
ARW Exercise 2.2 Download PDF (977.55 KiB, 387 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, 24397 downloads) View online
ARW Exercise 3.1 Download PDF (630.02 KiB, 46742 downloads) View online
ARW Exercise 3.2 Download PDF (781.02 KiB, 51937 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, 16118 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, 10290 downloads) View online
ARW Exercise 5.2 Download PDF (395.67 KiB, 7192 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
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ARW Exercise 6.2 Download PDF (1.23 MiB, 1538 downloads) View online
ARW Exercise 6.3 Download PDF (861.66 KiB, 39835 downloads) View online
Spaces

Inner product spaces form and important topic of Functional Analysis. These are simply vector space over the field of real or complex numbers and with an inner product defined on them.

  • Definition and examples
  • Orthogonality
  • Orthogonal matrices
  • Eigenvalues and Eigenvectors
  • Similar matrices
  • Symmetric matrices
  • Diagonalization of matrices
ARW Articles of Exercise 7.1 Download PDF (510.1 KiB, 453 downloads) View online
ARW Solutions of Exercise 7.1 Download PDF (363.36 KiB, 11571 downloads) View online
ARW Articles of Exercise 7.2 Download PDF (131.18 KiB, 4953 downloads) View online
ARW Solutions of Exercise 7.2 Download PDF (111.31 KiB, 5738 downloads) View online
ARW Articles of Exercise 7.3 Download PDF (157.9 KiB, 5022 downloads) View online
ARW Solutions of Exercise 7.3 Download PDF (519.81 KiB, 13432 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
ARW Exercise 8.1 Download PDF (494.55 KiB, 7078 downloads) View online
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ARW Exercise 8.4 Download PDF (854.59 KiB, 24362 downloads) View online
ARW Exercise 8.5 Download PDF (1.46 MiB, 3782 downloads) View online

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

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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, 11922 downloads) View online
ARW Exercise 9.2 Download PDF) (457.81 KiB, 35648 downloads) View online
ARW Exercise 9.2 by Umer Asghar Download PDF) (476.62 KiB, 10273 downloads) View online
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ARW Exercise 9.4 Download PDF) (450.02 KiB, 20695 downloads) View online
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Notes of the book Mathematical Method written by S.M. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN.

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  • Higher order linear differential equations
  • Exact equations
ARW Exercise 10.1 Download PDF (290.29 KiB, 3204 downloads) View online
ARW Exercise 10.2 Download PDF (1.48 MiB, 10860 downloads) View online
ARW Exercise 10.3 Download PDF (3.9 MiB, 27389 downloads) View online
ARW Exercise 10.4 Download PDF (608.67 KiB, 16767 downloads) View online
ARW Exercise 10.5 Download PDF (1.02 MiB, 3785 downloads) View online
ARW Exercise 10.6 Download PDF (749.21 KiB, 29179 downloads) View online
ARW Exercise 10.7 Download PDF (1.54 MiB, 21373 downloads) View online

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  • Notes of Exercise 10.3 is not cleared — 2017/03/20 06:39 by Talat Mehmood

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