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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