Essential Topics in Mathematics by


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Acknowledgments I wish to gratefully acknowledge my gratitude for my teachers who gave me the foundations in the methods of scientific inquiry. I also wish to acknowledge the help I have taken from many authors who have written excellent books and internet articles. Mathematics is at the core of all learning and is a very fascinating adventure in itself. The life of an undergraduate student is fast paced and he / she does not have time to refer to many authors and acclimatize to different styles and notations. I have collected some basic material which is at the core of the tools needed to pursue undergraduate studies in physics, mathematics and engineering . I have adopted a conversational style with repetition of the basic ideas in the hope that the reader will remain engaged and those ideas will become the core starting point for further studies. I do not claim any originality and offer this effort in the hope that it will be useful. I am very thankful to Dr. Joydeep Dutta, Professor IIT Kanpur, UP, India, and Dr. S. D. Mahanti, Professor of Physics, Michigan State University, E. Lansing MI, USA for their continued guidance and encouragement . Ved Agrwal Ph.D. To The Reader Of This Book Mathematics is at the core of knowledge. It is a fascinating subject not only for its precision but also for the freedom to exercise imagination. In Engineering and Physics, mathematics is essential. It expands our language and gives us tools for communication not otherwise available. The life of an undergraduate students is busy and fast paced. I have therefore written this guide as a self-study course in the summer months after graduation from High School. A student who has taken AP courses should be able to learn the topics covered in 9-10 weeks. A student who knows basic algebra should be able to cover them in 12 weeks. The material has been presented in conversational style . After reading a few pages, imagine that you are teaching a class. Can you present the material to your students ? If not, then you should read it again until it becomes a part of your thoughts. Congratulations for trying and good luck to you. Ved Agrwal “Knowledge is the best friend and most trustworthy companion.” Copyright © 2016 by Ved P. Agrwal Table of Contents CHAPTER 1 - NUMBER SYSTEMS Properties Of Integers Fractions Irrational Numbers Real Numbers Imaginary Numbers Rules Of Operations On Complex Numbers Graphical Representation Of Complex Numbers CHAPTER 2 - DETERMINANTS Equations in 3 variables Co-Factors CHAPTER 3 - INTRODUCTION TO MATRIX ALGEBRA Equality of Matrices Addition and Subtraction of matrices Multiplication of Matrices Examples of matrix multiplication Commutativity of Matrices Definition of the Inverse (Reciprocal) of a matrix Transpose Of A Matrix Transformation of simultaneous equations Symmetric Matrix CHAPTER 4 - COORDINATE SYSTEMS & COORDINATE TRANSFORMATIONS One Dimensional Coordinate Systems Two Dimensional Coordinate Systems Polar Coordinates Three Dimensional Coordinate Systems Left Handed Coordinate Systems 3-Dimensional Cylindrical Coordinate Systems 3-Dimensional Spherical Coordinates CHAPTER 5 - VECTOR ALGEBRA Addition Of Vectors Vector Addition Is Commutative Multiplication Of A Vector By A Number Projection Of A Vector Multiplication Of A Vector By Another Vector The Distributive Property of the Scalar Product Vector Product Of Two Vectors The Vector Product Is Anti Commutative Distributive Property Of Vector Product Description of Vectors in terms of Coordinate Systems Scalar Product of Two Vectors in 3D coordinates Vector Product in 3D Cartesian Coordinates Vector Triple Products Triple Scalar Product Transformation of vectors in 3-D Cartesian Coordinates Euler's Angles CHAPTER 6 - VECTOR GEOMETRY Representation of a Straight Line in Cartesian Coordinates Equation Of A Straight Line In 3-D Cartesian Coordinate System Parametric Equation of a Vector which passes through two given points in 3-D Cartesian space Equation of a Straight line passing through a given point and parallel to a given vector Intersection Of Two Lines Equation Of A Plane Containing Three Non-Collinear Points Summary Of Parameterization Of Few Curves And Surfaces CHAPTER 7 - FUNCTIONS OF REAL VARIABLES Measurement Of Angles Trigonometric Functions Trigonometric Functions Of Sums And Differences Of Angles The Factorial Function The Binomial Theorem Infinite Series Geometric Series Arithmetic Series Divergent Series Harmonic Series Tests Of Convergence of infinite series Continuity of functions Smoothness Of Functions Differentiation and Integration of Functions Basic Rules about Differentiation of Functions Derivatives of the Trigonometric Functions Integration of Functions The Rules For Integration of Functions Definite and Indefinite Integrals Partial Differentiation & Integration The properties of the exp(x) function The Natural Logarithm The Taylor Series Mclaurin Series Differential Equations CHAPTER 8 - VECTOR CALCULUS Contours In A Scalar Field Directional Derivative Of A Scalar Field Differentiation of vector fields Operations By The The Del Operator Dot Product Interpretation Of ▽.V (div V), The 'Divergence Of Vector Field V The Equation of Continuity Convention for assigning “Direction” to an Area Vector Integration Green's Theorem Stoke's Equation Surface Integrals Volume Integrals Divergence Theorem CHAPTER 9 - FUNCTIONS Of COMPLEX VARIABLES Operations on Complex Numbers Graphical Representation Of Addition In Complex Plane Representation Of A Complex Number Using Trigonometric Function De Moivre Theorem Roots Of A Complex Number Functions Of Complex Variables Roots of a Complex Number Derivative Of A Function Of A Complex Variable Cauchy – Riemann Equations Complex Integration Cauchy Integral Theorem For An Analytic Function Cauchy Integral Formula Greens Theorem CHAPTER 10 - n-DIMENSIONAL VECTOR SPACES Algebra Of N-Dimensional (n-D) Vector Spaces Vectors In N-D Space Definition Of Euclidean Vector Space (E) Transformation Of Coordinates Transformation of coordinates in n_D space Contra-Variant & Co-Variant Vectors The Basis Of 3-D Non-Rectilinear Space Contra-variant & Co-variant Components in 2-D space 2-D Skew Axes Contra-Variant Components In Skew Axes Combining Vector Spaces to get higher dimensional spaces Tensor Product Other Types of Vector Spaces Functionals Dual Vector Space Definition of a Linear Form CHAPTER 11 - TENSORS Notations Used In Tensor Algebra Einstein's Summation Convention Symmetric and Skew Symmetric Systems (& Tensors) Properties Of Tensor As Products Of Vectors Transformation of Coordinates in n-D spaces Transformation among Orthogonal Coordinates in 3-D The Metric Orthogonal Curvilinear Coordinates Cylindrical Coordinates (r, ϕ, z): Invariance of Physical Laws Tensor Calculus Appendix To Chapter 11 - Conductivity Tensor For Anisotropic Crystals Inertia Tensor CHAPTER-1 NUMBER SYSTEMS The history of number systems is very fascinating and you may be interested to know that the simple math you learned in the first grade took a very long time to evolve. If you like history, please read the book “The Nothing That is 0 , A Natural History of Zero” by Robert Kaplan. The concept of counting became necessary to keep track of the wealth of the rich, and the food available to the poor to survive the winter. Scribes (secretaries to the Ruler of a nation) adopted different methods in different of the world. They counted the coins in heaps of, say 5, 10 or 60 coins or bushels etc. and then counted the number of heaps. In the early stages there was no “Zero”. Even the mighty Romans did not have a zero, and in their counting system once a certain quantity was reached, a new symbol was created. Thus the first four items were marked by scratch marks like “ ││││” and at the next point, they used the symbol 'V' for 5. They used symbol X for 10, L for 50, C for 100, D for 500 and M for 1000. In absence of a zero, arithmetic in the Roman system was quite an adventure and luckily, they did not have to deal with interstellar distances and the U.S. budgets. The present number system with the idea of Zero and the place system was developed in 5th century in India and later in Egypt. This was a great discovery because the 10 has one zero, 100 has 2 zeros and a million has 6 zeros and a billion has 9 zeros after 1 and you do not have to memorize an endless series of symbols. You only need 9 symbols for 1 to 9 and a zero. It is pretty compact and easy to work with and counting the money in the king's treasury or talking about budget deficits or excesses became easy. These numbers 1, 2, , 1000, ...were called “Integers”. Just as wealth was represented by positive integers, the loans (indebtedness) were represented by a negative sign before the number. Thus if you had 10 dollars, your wealth was +10 dollars. But if you splurged and spent $14 to entertain your friend, you became a debtor and your wealth became -4 dollars. All these considerations led to the concepts of addition, subtraction, multiplication and division . These integers extend from - ∞ to + ∞ . ( ∞ means a very large number, larger than any number you can imagine. If you can imagine any large number, ∞ is bigger than that. Thus, infinity “∞” does not have a defined specific value like a trillion. It is just bigger than anything you can come up with. To this extent infinity is really not defined as a specific value. You should note it because if in solving a problem, you get infinity as an answer, then you have not found a well defined value. ) The entire range of infinite set of “Integers” is represented on what is called the Number Line. (See Fig 1_1) Positive numbers are placed on the right of “zero” and negative numbers are placed on the left of “Zero”. Properties Of Integers Four operations namely Addition, Subtraction, Multiplication, and Division are defined,, However multiplication is extended addition and division is extended subtraction. If symbols 'a', 'b', 'c' etc are integers and '0' represents Zero, then: (In the following, ( * means multiplication.) (1) a + 0 = a. '0' is called “Additive Identity” (2) a * 1 = a. The number '1' is called multiplicative identity. (3) a + b = b + a. Addition is commutative (order of numbers does not matterand can be reversed) (4) a * b = b * a. Multiplication is Commutative. (5) If the parentheses are computed first, then; (a*b)*c = a*(b*c). Multiplication is associative. (6) a*(b + c) = a*b + a*c. The multiplication is distributive. a*(b + c) = (b+c)*a = b*a +c*a. (7) We write a*a as a2 , a shorthand notation. Similarly a*a*a* (m times) as am . Now, a2*a3 = (a*a)*(a*a*a) = a5.= a(2+3). Similarly am*an = a(m+n). Fractions It must have become obvious quite soon that “Integers” were not adequate. What if a baker has to split a loaf of bread in 2 equal parts in order to sell them. You can describe each part as one part out of two. What if the baker splices the bread in 10 equal parts and manages to sell 5 slices. He finds that remaining 5 slices out of ten have same weight as one out of two. So 5/10 = ½. Such considerations lead to arithmetic of fractions. It is obvious that if you represent the fractions on a Number Line, there will be an infinite number of fractions between any two integers. Note that a fraction is defined as an integer divided by another integer. The set of Integers and Fractions has an infinite number of members. These numbers are called “Rational” numbers. The name somehow creates an idea of “Rationality” which has a meaning in terms of human behavior. Well, there is nothing rational or irrational about numbers. In number theory, 'Rational' is just a name. Irrational Numbers Consider the equation 3*3 = 9. You now ask the question: “what number multiplied by itself will give us 9 “. Answer is 3. You know that 7*7 = 49. If I ask you what number multiplied by itself will give 49, the answer would be 7. In these examples, 3 is called the 'Square Root' of 9, and, 7 is called 'Square Root' of 49. Now we ask the question: “What is the square root of 2 ? “ We write the square root of 2 symbolically as √2. We ask “What number multiplied by itself will give us 2 ?” When you calculate square root of 2 (i.e. √2 ), you find that the calculation gives you a number in which the fractional part never ends. Thus √2 = 1.414213... (non stop). This number is quite different from the number 0.333333333333...(for ever) because 0.333333... can be proved to be equal to 1/3, a ratio of two integers and thus a proper fraction. Thus 1/3 or its decimal equivalent are members of the set of “Rational Numbers”. On the other hand, √2 simply cannot be expressed as a ratio of two integers. It is certainly a number that does not fit in the category of 'Rational' numbers, and yet, it is a legitimate number. Such numbers occur often. (Look at a right triangle with equal base and height of length 1, and try to find the length of hypotenuse of the right triangle. It is √2.) This is not a unique example. In fact there are an infinite number of such numbers which cannot be expressed as ratio of two integers. These numbers could not be ignored, and were given the name ”Irrational Numbers” because the name Rational was already used up. Again note that the feelings associated with irrationality should not interfere with acceptance and use of these numbers. “Irrational numbers” is just a name. Real Numbers The combined set of Rational and Irrational numbers is called the “Set of Real Numbers” and we should remember that these are simply numbers which are useful. Imaginary Numbers There is nothing imaginary about imaginary numbers. They occur naturally, and without them our number system is incomplete. We know that in the real number system a negative number multiplied by itself gives a positive number. Thus (-a * -a) = (a * a) = +a2, which is positive. We ask the question: “ Is there some number, say 'i', whose square is -1 (i * i = -1). The answer is that such a number actually appears in solving quadratic equations, and in absence of this number we will lose out on a very rich area of mathematics. Consider the following quadratic equation in variable 'x'. a*x2 + b*x + c = 0 1_1 We can solve this equation by, say, “completion of squares”. The solution is: x = {-b + √(b2 – 4ac)}/2a 1_2 If (b2 > 4ac) the quantity under the square root sign is positive and we get two solutions both of which are real numbers. An example is: a=1, b=3, c=1. The two solutions are: x = {-3 + √ (9 – 4)}/2 = {-3 + √ 5}/2 Both solutions are real numbers. Now try the following values: a=1, b=1, c=1. The solutions are: x = {-1 + √ (1 – 4)}/2 = {-1 + √ (-3)}/2 Since -3 = (3)*(-1) we get x = {-1 + √ (3*(–1)}/2 = {-1 + (√ 3* √(–1)}/2 1_3 The number √(–1) appears quite naturally in solving a valid quadratic equation. Now we have two options: (a) Declare that we do not want to deal with √(–1). In that case we have to give up a fertile and powerful part of mathematics by arbitrarily excluding certain types of mathematical expressions. We do not want that. (b) Accept that our “Real Number” system needs to be expanded by accepting the number √(– 1). This opens up a new horizon and enriches the number system. We accept this option and see how it helps. After all nobody proved that the Real Number system was the end of discoveries in this area. So, let us define a number 'i' by the equation: 'i' = √(-1) 1_4 This number was assigned the name “Imaginary Number”. The unfortunate choice of the name was probably prompted by the fact that this number was different from real numbers. However, the barrier in acceptance of this number will be reduced if we remember that despite its name, it is a perfectly good number which exists in its own right. If b2 < 4ac, we can write the two solutions of Eq.1_2 as: x = {-b + i*√ (4ac - b2 )}/2a 1_5 If X and Y are two Real numbers, then the equation 1_5 gives the solution in the general form: Z = X + i*Y 1_6 The number Z in 1_6 is called a “Complex Number”. The complex number Z has two components: The real component X and the imaginary component (i*Y). The complex number Z was introduced as a necessity in solving quadratic equation. Let us generalize the idea and write: z = x + i*y 1_7 Here x and y are real numbers, “i“= √(-1) so that i2 = (-1) Do not get deterred by the word “Complex”. It is just a name. Also, look at the idea and not the symbol. Often, we use 'x' or 'a' for the real and 'y' or 'b' for the imaginary component. Rules Of Operations On Complex Numbers These rules are similar to the rules applicable to Integers and Real Numbers. The “additive identity” for complex numbers is (0 + i*0) = 0. The multiplicative identity is '1'. The commutative, distributive, and associative properties etc. apply. Graphical Representation Of Complex Numbers Let us start with a real positive number b. On the number line, it will be represented by a line on the positive side of zero. Now we multiply it once by 'i' to get i*b, a purely imaginary number. If we multiply it again with 'i', we get i2b = -b. Thus we can consider 'i' to be an “Operator”, which, in two successive operations, rotates the line representing a real positive number in counterclockwise direction through 180o to change the number from positive to negative. Thus, we can consider that the first application of 'i' rotated the line counter-clockwise through 90o. Therefore, using a two dimensional coordinate system, we can represent real numbers on the x-axis, and purely imaginary numbers on the y-axis. Thus, a complex number z = (x + iy) stands for a point in Complex Plane and can be represented by the line OA as shown in Fig.1_2. Fig.1_2 From Fig.1_2, Length of OA = r = √(a2 + b2) 1_8 This is called “Absolute Value” or “Modulus” of 'z'. The Real and Imaginary parts of z in FIG_1_2 are also denoted by Re(z) = a ; Im(z) = i b We also define a complex quantity called “Complex-conjugate” of “z” as z = a – ib The product of z and its complex conjugate z is: z * z = (a+ib)*(a–ib) = a2 -iba + iba – (ib)*(ib) = a2 – (i*i)*b*b = a2 -(-1)*b2 1_9 1_10 or z * z = (a2 + b2) 1_11 Thus the product (z * z ) = the square of the modulus (Absolute Value) of 'z'. From Fig.1_2, we see that a = r*Cos(θ) ; b = r*Sin(θ) 1_12 From the above description, we see that a Complex number has a real component and an Imaginary components. These components can vary independently and the complex number z traces a path in a two-dimensional complex plane (just like a crawler crawling on a piece of cardboard). We will learn more about complex numbers when we deal with functions of a complex variable. In this chapter, we described the evolution of number systems from integers to fractions, rational and irrational numbers, and real and complex numbers. We showed that to pursue science, all the above numbers including complex numbers are essential CHAPTER-2 DETERMINANTS Determinants are used in solving simultaneous sequins. A determinant is a single number calculated from a square array of numbers. Determinants can be introduced formally, as is often done in mathematics classes. That approach is elegant, however we want to show how the determinants arise naturally in solving simultaneous equations. Once you are comfortable with the basic idea, then the subject can be covered formally. We will start with two simultaneous equations in two variables x and y. Ax + By + C = 0 Dx + Ey + F = 0 2_1 If you multiply first equation by E and the second eq. by B, you get: (Note: Here AE = EA = A*E) AEx + BEy + CE = 0 BDx + BEy + BF = 0 Subtracting, we eliminate term in 'y ' and get (AE – BD)*x + (CE - BF) = 0 Therefore X = (BF – CE)/(AE – BD) Similarly y = (CD – AF)/(AE - BD) 2_2 You can verify this solution by writing the two simultaneous equations with numeric coefficients. It will be a good exercise. [ Make sure that coefficients A and B are not simple multiples of coefficients D and E because, if they are simple multiples, (e.g. A= k*D and B = k*E where k is any number), then (AE - BD) = 0 and there is no solution. In such a case, we say that equations are linearly dependent.] (Remember this result.) What is important to note is that the denominators in the values of x and y are the same, viz. (AEBD). Clearly, if (AE-BD is zero, the equations do not have a solution because division by zero is not allowed. If (AE – BD) is not zero, a solution will exist. Thus the expression (AE – BD) determines if a solution exists. It is called the “Determinant”. Before you even solve the equations, you can calculate the determinant and determine if the solution exists. Also note that the determinant depends only on the coefficients of x and y in the two equations. As we solve equations in three variables later, we will find a similar, though more complicated, expression for the determinant. Also note that in the value of x, only the coefficients of y and the constant terms appear. Similarly, the value of y depends only on the coefficients of x and the constant terms. A similar pattern will be observed in case of solving three simultaneous equations,

Author Isbn File size 11.84MB Year 2017 Pages 312 Language English File format PDF Category Mathematics Book Description: FacebookTwitterGoogle+TumblrDiggMySpaceShare This study guide is for aspiring undergraduate mathematicians, engineers, or physicists. The style of instruction in this book is a refreshing change from typical teaching methods. It is a boot camp for the mind, starting from the basics and advancing systematically through more complex topics, focusing on the practical, not dwelling on theory.     Download (11.84MB) General Theory of Functions and Integration Dynamics In One Complex Variable. Intermediate Algebra An Introduction to Mathematical Taxonomy An Introduction to Continuous-Time Stochastic Processes Load more posts

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