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# 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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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
 Exercise 1.1 Download PDF (470.64 KiB, 34090 downloads) View online Exercise 1.2 Download PDF (1.27 MiB, 25685 downloads) View online Exercise 1.3 Download PDF (1.64 MiB, 11634 downloads) View online Exercise 1.4 Download PDF (653.33 KiB, 16046 downloads) View online Exercise 1.5 Download PDF (690.22 KiB, 3481 downloads) View online
• 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: Exercise 2.1 Download PDF (606 KiB, 23673 downloads) View online Exercise 2.2 Download PDF (602.28 KiB, 5762 downloads) View online Exercise 2.3 Download PDF (657.81 KiB, 4706 downloads) View online
 Notes by Shariq Mehtab Syed: Exercise 2.1 Download PDF (774.94 KiB, 1774 downloads) View online Exercise 2.2 Download PDF (977.55 KiB, 688 downloads) View online Exercise 2.3 Download PDF (410.76 KiB, 24113 downloads) View online

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
 Articles of Ex 3.1 Download PDF (327.93 KiB, 31722 downloads) View online Exercise 3.1 Download PDF (630.02 KiB, 873 downloads) View online Exercise 3.2 Download PDF (781.02 KiB, 61096 downloads) View online

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
 Exercise 4 Download PDF) (1 MiB, 1900 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
 Exercise 5.1 Download PDF (1.27 MiB, 9048 downloads) View online Exercise 5.2 Download PDF (395.67 KiB, 15261 downloads) View online

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• Subspaces
• Linear combinations and spanning sets
• Linear dependence and basis
• Row and column spaces
• Rand-Alternative method
• Linear transformations
• Matrix of linear transformation
 Exercise 6.1 Download PDF (3.52 MiB, 2380 downloads) View online Exercise 6.2 Download PDF (1.23 MiB, 11109 downloads) View online Exercise 6.3 Download PDF (861.66 KiB, 3760 downloads) View online

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
 Articles of Exercise 7.1 Download PDF (510.1 KiB, 6024 downloads) View online Solutions of Exercise 7.1 Download PDF (363.36 KiB, 16167 downloads) View online Articles of Exercise 7.2 Download PDF (131.18 KiB, 7258 downloads) View online Solutions of Exercise 7.2 Download PDF (111.31 KiB, 8171 downloads) View online Articles of Exercise 7.3 Download PDF (157.9 KiB, 7526 downloads) View online Solutions of Exercise 7.3 Download PDF (519.81 KiB, 20369 downloads) View online

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
 Exercise 8.1 Download PDF (494.55 KiB, 13290 downloads) View online Exercise 8.2 Download PDF (1.09 MiB, 41215 downloads) View online Exercise 8.3 Download PDF (658.47 KiB, 32278 downloads) View online Exercise 8.4 Download PDF (854.59 KiB, 31770 downloads) View online Exercise 8.5 Download PDF (1.46 MiB, 11091 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.

• D.E and their classification
• Formation of differential equation
• Initial and boundary conditions
• Separable equations
• Homogeneous equations
• D.E. Reducible to homogeneous form
 Exercise 9.1 Download PDF) (829.68 KiB, 24742 downloads) View online Exercise 9.2 Download PDF) (457.81 KiB, 5871 downloads) View online Exercise 9.2 by Umer Asghar Download PDF) (476.62 KiB, 20590 downloads) View online Exercise 9.3 Download PDF) (992.56 KiB, 3161 downloads) View online Exercise 9.4 Download PDF) (450.02 KiB, 4613 downloads) View online Exercise 9.5 Download PDF) (1 MiB, 6771 downloads) View online Exercise 9.6 Download PDF) (508.84 KiB, 7590 downloads) View online Exercise 9.7 Download PDF) (789.57 KiB, 5592 downloads) View online Exercise 9.8 Download PDF) (952.79 KiB, 891 downloads) View online Exercise 9.9 Download PDF) (975.02 KiB, 27539 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.

• Higher order linear differential equations
• Exact equations
 Exercise 10.1 Download PDF (290.29 KiB, 1930 downloads) View online Exercise 10.2 Download PDF (1.48 MiB, 8193 downloads) View online Exercise 10.3 Download PDF (3.9 MiB, 9332 downloads) View online Exercise 10.4 Download PDF (608.67 KiB, 7171 downloads) View online Exercise 10.5 Download PDF (1.02 MiB, 15512 downloads) View online Exercise 10.6 Download PDF (749.21 KiB, 38494 downloads) View online Exercise 10.7 Download PDF (1.54 MiB, 32023 downloads) View online

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 $F=\mathcal{L}(f)$.

Here is the list of other contents given in this chapter.

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

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Here we are posting different suggestion/feedback provided by our valued visitors. You can also give us feedback via About Us page. We will try our best to overcome these deficiencies.

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

We will try our best to find new notes which are clear. If someone else has these notes, they can send us — 2017/05/05 07:25 by Admin

Dear, Can you send me /Uploaded the chapter 11 Excerise.. . Book Mathematicial Method Chapter 11 — 2017/05/24 12:07 by Sami Ullah Zakir

Notes of chapter 11 are available here: http://www.mathcity.org/bsc/notes_of_mathematical_method/ch11_the_laplace_transform 2017/05/24 12:07 by Admin

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