Page 1 / 681Table Of Contents In The Whole BookPart I : The Sword And The ShieldSection 1: Identity, Division, Symmetry, Set Theory, Logic,Numbers and Functions Chapter 1 : Identity, Division & Symmetry Pages 1 to 19Topics : Identity, Division, Symmetry, Difficulty identification 1, Example inSolving Quadratic Equations 3, Cubic Equation 5, Summation Of Series 6,Constants, Principle of equivalence 11, Conservation Laws, Invariants11, Fixed Points, Breaking Symmetry to get the roots etc. 18Chapter 2 : Set Theory, Relations, Functions etc. Pages 1 to 27Topics : Set Theory 1,1Russels Paradox 1, Examples of sets, null set, indexed set, complement, Universal set etc. ,Infinite sets 3, Set Operations 4, Venn Diagrams 7, De Morgans Theorems 8, Power set 12, Relations 12, Mappings 13, Binary relations 13, Binary Operations 14, Operation tables 14, Commutative, associative, distributive properties, Identity and inverse elements, Reflexive, symmetric , transitive Relations, Equivalence Relations15, Partitions 15, Equivalence Classes 15,21, Countability 18, Eulers Circle20, Algebraic structures on sets, such as groups, vector spaces, topological spaces, metric spaces25.Chapter 3 : Symbolic Logic, Mathematical Logic, and Logical gatesPages 1 to 52Topics : Boolean Algebra for making algebra for set theory and logic 1, Symbolic Logic, Statements, Conjunction, Disjunction, negation, contradiction, tautology etc compared to null set, complement, Universal set etc.2, , Duality in Boolean Algebra and symbolic logic 7, Contradictionstatements 6, Compound statements 4, Truth value of logical statements and Truth tables 13, Premises, conclusion, Arguments, validity of argument 22, De Morgans laws in symbolic logic 17, Bi-conditional statements 20, Boolean expressions and functions 26, Arrow Diagram 28, Input- output tables 29, Signals and logical gates 31, AND gate, OR gate, NOT gate, NAND gate, NOR gate, XOR gate, XNOR gate, Equivalent circuits 37, Combinatorial 1The less prominent and unusual topics are marked in bold.Page 2 / 681circuits 41, Rules for making them 42, Use of Boolean algebra in logical gates44.Chapter 4 : The Story Of Numbers Pages 1 to 63Topics : Variable, Constant, Interval 1,Peanos theory of natural numbers 1, Principle Of Mathematical Induction ( in two forms) 3, Recapitulation of permutation, combination and binomial theorem 6, Infinitesimals 9, Natural numbers, integers, rational numbers, irrational numbers, surds, transcendental numbers, Real numbers, Complex numbers 11, Hilberts number 32, Scales of Notation, Decimal, binary etc. 13, Why they call a song or dance as numbers, 17 Division by 0 18, Countability of rational numbers 19, un-countability of real numbers 27, The Napierian number e22, 23, examples of transcendental numbers 28, a set of measure 0 29,Gelfonds theorem 31, Cardinal numbers, transfinite sets 32, equivalent of sets 33, Complex numbers 34, Partial order relation, Total order, chains and lattices 35, Fundamental Theorem Of Algebra 36, Upper bound, lower bound36, Zorns lemma or the axiom of choice 37, Ordered pair representation of numbers, vectors, quaternions 37, Absolute value of a number and triangle inequality 40, Continued fractions representing surds and irrational numbers 43, Conversion into decimal fractions and vice versa, Successive convergents, Conversion of series into continued fractions, better approximation of irrational numbers than decimal representation, Fibonacci numbers and Golden Ratio 59, their natural occurrence and various simple theorems.Chapter 5 : Functions and Graphs Part I Pages 1 to 64Topics : Functions 1, Images and inverse images 2, Single valued Mappings3, Cartesian products 5, sequences as functions of integers 6, Functions as index or labels 6, functions of many variables, independent and dependent variables , Function of continuous variable 7, domain, range Function as single valued correspondence 9 , Onto, Into, one-to-one and one-to-one onto functions, bijective, injective, surjective, inverse function 11, Greatest integer function or step function, or integer function 13, Characteristic function 15, Norm, modulus or distance function 15, fractional part 18, Least integer function or Ceiling function 18, , Linear Function 23, Algebra of functions 18, Composite function, composite operators 20, inverse of composite function22, monotonically increasing ( or monotonically decreasing)functions 23, 26, permutation group as a function 24, Periodic function and period 48, maximum and minimum 27, exponential and logarithmic functions 27, Implicit functions 28, Functions as Expressions 29, Rational algebraic functions, Polynomials 30, transcendental functions 36, Zeroes or roots of a function 30, partial fractions, Rules of decomposition 32 , Function vs operator 39, Linear Operator 37, homomorphism, Graphs of functions 40, algebraic, trigonometric , hyperbolic functions, Inverse trigonometric and inverse Page 3 / 681hyperbolic functions, Simple harmonic function 49, wave function 52 , Some special curves 56 , Even and odd functions 57, Symmetric and alternating functions 59 , parametric equations of curves 60. Chapter 6 : Functions and Graphs Part II Pages 1 to 23Topics : Examples, Problems, and techniques of functions and their graphs1, Simple transformations of graphs 10, Addition, Multiplication etc of functions in graphs of functions 18, points of intersection 22.Chapter 7 : Functions and Graphs Part III Pages 1 to 32Topics : Exercises on domain and range 1 , Identical and equal functions, odd and even functions 17, Odd and even functions 20, Periodic functions and their composites (LCM and GCD) 22, other composite functions 25, Inverse functions and fixed points 28, Section 2: Limit, Continuity, Differentiation and utilities, Theory of Equations Chapter 8 : The Concept of Limit of a Function Part I Pages 1 to 40Topics : Concept of Limit in and definition 1, Left handed and right handed limits 1, concept of limit as division by infinitesimal 2, limit by substitution 3, limit by rationalization 4, limit when x 6, Algebra of Limits7, Squeezing Theorem or Sandwich Theorem10, , Limits of Algebraic functions 11, Limits of Trigonometric Functions 12, exponential series 14, Logarithmic series 17, log table 18, many facets of limits 24, control variables and pre assigned error 24, a limit existing nowhere 25, Sequences and limits26, convergence of sequences along many paths 26, different limits on different paths guarantee non existence of limit26, asymptotes as curves approximating st lines at infinity 30, functions of many variables 32, order of smallness, largeness 33, comparison of infinitesimals and calculation of limits 35, Cauchys necessary and sufficient condition for existence of a limit37.Chapter 9 : The Concept of Limit of a Function Part II Pages 1 to 22Topics : Miscellaneous exercises on calculation of limits, by substitution, by rationalization, by comparison of infinitesimals, by use of exponential forms and many other techniques, triangle of largest area inscribed in any circle is equilateral 21,Chapter 10 : Continuity and Theory Of Equations Pages 1 to 38Topics : The Concept of Continuity in and definition 1, Equivalent definition of continuity in limit being equal to functional value 1, continuity of composite functions 5, Dirichlet function 7, Removable and non-removable discontinuity 9; discontinuity of 2nd kind , piecewise continuous functions, bounded sequences, limit point, neighborhood, Bolzano-Weisstras theorem14, supremum, infimum 15, order completeness property 15, Properties of Page 4 / 681continuous functions 15, Continuous functions must be bounded 17, Continuous function preserves sign in small intervals 18, attains maximum and minimum 17, continuous function takes every value between the end points 21, Theory of Equations and The fundamental theorem of Algebra,Intermediate value theorem and application to theory of equations 19, odd continuous function has a real root, when 0 is a root 21, continuous function taking rational values must be a constant function 22, fixed point mapping22, arithmetic and geometric means of continuous functions, Wavy curve method to analyse an algebraic function (piecewise continuous) 23, Monotonic functions and continuity of inverse functions, Inverse function of polynomials, Monotonic functions 25, Continuous linear operators, Kepplers function 31, Chapter 11 : Furhther theory of Equations and Polynomials Pages 1 to46Topics : Theory of equations and polynomials 4,The division algorithm, remainder theorem and synthetic division 5, Evaluation of a polynomial and other functions 10,synthetic division 11, every polynomial equation of degree n has n roots 12, a polynomial of degree n cannot have more than n roots else it must be identity 13, a polynomial of odd degree with real coefficients has at least one real root and other results 14, Descartes Rule Of Signs and results therefrom16, rational root theorem 19, integer root theorem 20, location theorem 22, evaluation of a polynomial 24, Horners method 25, Relations between roots and coefficients 26, Finding roots when one of them is known 28, upper bound and lower bound theorem 31, transformations of equations 34, reciprocal equation 39, cubic equation 42, symmetric functions of roots 43Chapter 12 : More about theory of equations, roots by numerical methods, Pages 1 to25Topics : Extended synthetic division 1, symmetric functions of roots 3, Multiple roots 5, roots of transcendental functions ( cos sinixe x i x )7, de Moivres theorem 8, Hilberts number ii 10, equations whose roots are multiples of roots of given equation 11, comparison with number system 11, area of a circle 14, Approximate roots, fixed points, Piccards method14,Newtons method , Newton Raphsons method, bisection method 21,secant method 24,Chapter 13 : Differentiation and applications Pages 1 to43Topics : Differentiation 1, results from algebra of limits 4, a differentiable function must be continuous 7, rates of change compared to unity 12, differential coefficient of inverse function 13, differentiation of implicit functions 14, Special result from algebra 14 , differentiation of series 16, Page 5 / 681differentiation of trigonometric functions, hyperbolic functions 18, inverse circular functions 19, logarithmic differentiation 23, inverse hyperbolic functions 25, differentiation of determinants and matrices 26, introduction to LHospitals rule, approximation by linearization 32, successive differentiation 33, differential coefficient as an operator 34, introduction to tangents and normals 35, introduction to maxima and minima 36, The differential concept, linearization, approximation and sensitivity to change 39, Approximation, error, accuracy, precision and significant figures40, Role of differentials in estimate of absolute and relative error 41, Algebra of errors 42, Minimum error in measuring physical quantities: sum of least squares 43,Chapter 14 : Mean value theorems and Taylors series Pages 1 to43Topics : Turning points and Rolls theorem1, Lagranges mean value theorem 5,A derived function cannot have discontinuity of the first kind 7, Cauchys mean value theorem 7, Darbauxs theorem 8, Taylors theorem or generalized mean value theorem 14, Taylors theorem in Remainder Form 15, The philosophy behind Taylors theorem 16, Lagranges form of Taylors theorem in Remainder theorem 16, Youngs form of Taylors theorem 18, Maxima and minima of a function using Taylors theorem 18, Convexity, concavity and point of inflection 19, Taylors theorem from Remainder theorem in Algebra 22, LHospitals Rule and indeterminate forms 23, its generalization 25, Indeterminate form 26, Indeterminate form- 27, Indeterminate form 00, 0, 1 etc. 28, The indeterminate form - 29,Indeterminate form using Taylors theorem 30, Method of undetermined coefficients 35, McLaurins series 36,Chapter 15 : Miscellaneous problems on differentiation and limitsPages 1 to25Chapter 16 : Partial differentiation and curvilinear co-ordinates Pages 1 to 25Topics : Partial Differentials, linearization and approximation 1, Laplaces equation 5, Poissons equation 7, Equality of xy yxf and f7, Wave equation 7, Eulers theorem on homogeneous functions 11, Converse of Eulers theorem 17, The scope of partial differentiation 18, Choice of independent variables and change of coordinates 19, Orthogonal Curvilinear Coordinate Systems 20, Cylindrical polar coordinates 21, Spherical polar coordinates 23, Orthogonal curvilinear coordinate systems in general 24, Angle element compared to distance element 25,Section 3: Linear Functions, Vectors, St lines and Planes, Determinants and Matrices Chapter 17 : Linear equations and determinants Pages 1 to 38Page 6 / 681Topics : Equations, their degree and order 1, Functions vs Equation 2, Linear equations and their eliminants 2, Determinants 4, Cramers rule 8, Addition of Determinants 16, Additive and multiplicative identities and inverses 18, Cramers rule in Detail 27, Product of two determinants in another form 33, Elimination 36. Chapter 18 : Coordinates and Vectors Pages 1 to 97Topics : Coordinates of a point 1, Coordinate Geometry in three dimensions 2, Distance between two points: In two dimensions2, Distance between two points: In three dimensions 2, Polar Coordinates 3, Change of Coordinates from Cartesian to polar or vice versa 4, Distance between two points in polar coordinates 4, Vector Algebra 6, Addition , subtraction, and resolution of vectors: polygon method 10, The Parallelogram law 11, Vector difference 12, Rectangular Components and Addition and subtraction of vectors 14, Components In two dimensions 16, Explaining Position vectors, norm or magnitude of a vector and coordinates of a point 17, Direction ratios of a line segment or straight line in three dimensions 18, Projections of segments 20, Comparison with imaginary numbers 22 , Product of imaginary numbers24, Meaning of argument 27, Multiplication/ division of a vector by a scalar 28, Multiplication of a vector by a vector. Dot(.) product 29, Commutative, associative and distributive properties 30, Some examples of scalar and vector quantities , Mechanics32, Law of conservation of mass, momentum, energy 36, Free vectors and Localised vectors 37, Angular speed and angular velocity 38, Multiplication of a vector by a vector. Cross (X) product 39, Moment of any vector about a point Moment of any vector about a line 40, Vector addition of rotations 42, Uniform circular motion versus motion along a st line 43, Relation between L and , Moment Of Inertia 43, Newtons laws for rotational motion 44, Conservation of angular momentum 45, Equilibrium 45, Localised force 46, Stability of equilibrium 47, compound angle formulae in Trigonometry 50, Magnitudes of dot and cross products, components along any two directions 51, Direction cosines and dot and cross products, angle between two lines 53, Straight lines perpendicular or parallel to each other 55, In two dimensions 56, Direction cosines and projections 56, Shift of axes of coordinates or frame of reference (linear shift) 58, Rotation of axes and invariants 59, Rotation of axes in 3-dimensions 65, Section formula in vectors and in coordinate geometry of two and three dimensions 72, Collinear points and collinear vectors 73, Internal and external division of a line segment 75, Division formula or section formula in 3 dimensions without vectors 77, Use of complex number 77, The Section formula promises an equation for a st line 81, Parametric equation of a plane 82, More about Linear combinations of vectors in 2-D and 3-D, Components or resolute 83, When linear relations are independent of origin 84, Linear dependence and base 86, Centroid, Moments of mass, centre of mass 87, Centroid of some position vectors is independent Page 7 / 681of choice of origin 88, Centroid of two clusters of points is the centroid of all points taken together 89, Center of mass 92, More explanation on Moments of mass, Moment of Force 93, Vector triple product, scalar triple product 94, Scalar triple product 95, Reciprocal vectors 96 Chapter 19 : Tips, Tricks and Problems on Coordinates and Vectors Pages 1 to 51 Topics : Problems and problem solving techniques , concurrent st lines nd planesChapter 20 : Vectors and Physics Pages 1 to 22Topics : Problems and problem solving techniques especially in PhysicsChapter 21 : Geometry of St lines in 2D and 3D Pages 1 to 52Topics : Equation of a St line from a point in a given direction in vector form1, Intersecting st lines 2, St line through a given point parallel to a given direction in vector form 3, Length of perpendicular and foot of it on this line from an outside point P 4, Parallel or perpendicular St lines; angle between two lines 5, Vector Equation of straight line between two given points in 2-D and 3-D 6, In complex numbers in 2D 7, Length of a perpendicular from a point to a st line joining two points 8, In Cartesian form 9, The Straight Line in two Dimensions; Linear function in one independent variable(Without vectors) 10, Equation of Straight line : y = mx + c form : (slope and intercept form) 10, General equation of 1st degree in x and y is a straight line 11, Results from parametric form of st line 12, Distance of a st line from a fixed point in a given direction in 2D 12, sin and cos 12,Foot of the perpendicular in two dimensions 13, Length of the perpendicular from A(h, k) on 0 Ax By C 14, Angle between two straight lines 14, When the st.lines Ax + By + C = 0 and Px + Qy + R = 0 coincide completely 15, When Ax + By + C = 0 and Px + Qy + R = 0 are parallel or perpendicular 15, Equation of a st line whose intercepts with axes are given :intercepts form 15, Equation of a straight line if perpendicular on it from origin is given: Normal Form 17, Length of the perpendicular from origin 19, Distance between two parallel lines19, Length of perpendicular from a point to a line 19, Points for working out exercises 20, Position of a point with respect to a line 22, Point of intersection of two straight lines in 2D , examples of concurrence of lines 25,26, Image of a point or a line about a st line in 2D 28, Line at infinity 31, A straight line passing through intersection of two straight lines 31, Condition that three st. lines meet at a point 32, Use of complex number 36, Vector formulation of any st line passing through intersection of two st lines 40, Equation of a st. line in polar coordinates 41, One parameter family of st lines in two dimensions 42, Two parameter family of st lines in two dimensions 43, Bisectors of angle between two intersecting Page 8 / 681st lines 43, Bisectors of angle between to st. lines in 2D (without vector concept) 44, Alternative methods 45, 46, Equation of a straight line in three dimensional Coordinate Geometry without vectors 49. Chapter 22 : Tips, Tricks and problems on Geometry of St lines Pages 1 to 24Topics : Problems and problem solving techniques on Geometry of St linesChapter 23 : Geometry of Planes and st lines Pages 1 to 70Topics : A plane contains the whole st line passing through any two points on it 1, Two sides of a plane 2, Vector equation to a plane through three non-collinear points, A,B,C 3, Equation In Cartesian Coordiantes 5, Parametric equation 5, Another form 6, Conversion to Cartesian form 7, More about Triple product and volume 7, Volume of a tetrahedron 9, Volume of a tetrahedron in Cartesian 10, Volume of a tetrahedron whose faces are given planes 11, Volume of a tetrahedron in terms of three concurrent edges and angles between pairs 12, Direct Cartesian derivation equation of a plane through three given points 13, Coplanar points 14, Coplanar points in dot product 15, Comparison of normal form of equation of plane with pedal form 17, Pedal form in Cartesian 17, Intercept form 19, Family of planes passing through a given point 21, Plane perpendicular to a given vector and passing through a point21, Equation of plane passing through a given st line 22, Plane passing though the line 1 1 1x x y y z zl m n and the point 2 2 2, , x y z23, Equation of plane passing through a point and parallel to two given st lines 23, Plane containing a given st line and parallel another st line 23, Angle between two planes 24, Equation to parallel and perpendicular planes25, Angle between a st line and a plane 25, Distance from a point to a plane 26, foot of the perpendicular 27, Distance from a point to a plane in the direction of a given vector 29, Family of planes passing through two intersecting planes( i.e., through one st line) 31, Bisectors of angles between two planes 34, Intersection of two st lines and joint equation of two st lines 34, Intersection of two planes is a st line 35, Converting equation of a st line from Unsymmetrical to Symmetrical form 36, Angle Between the Planes 39, Condition for a line to lie completely in a plane 39, The family of planes passing though a given line 39, The plane though two given points and parallel to a given st line 40, The plane containing a st line and passing through a point 41, Coplanar lines 41, Condition for two st lines to intersect or to be parallel 42, In unsymmetrical form 44, Perpendicular from origin onto a st line 45, Shortest distance between two skew lines 47, A pitfall: Shortest distance between two parallel lines 49, To find the equation of the common perpendicular to two skew lines 50, The equation of the common perpendicular in vector form 51, The equation of the common perpendicular in a simplified form 51, Three non-parallel planes either meet at a point, orPage 9 / 681on a line or make a prism 54, Point at , line at and plane at 57, One parameter family of planes 57, Point of intersection of a st line with a plane 58, Line and plane to be parallel, perpendicular or to coincide59, Remember the results 62,Chapter 24 : Tips,Tricks and problems on st lines and planes,Vector Equtions Pages1 to 30Topics : Problems and problem solving techniques in st lines and planes1, When a 2nd degree homogeneous equation represents Two Planes 7, Eqn. of a plane passing through two given points 8, Common perpendicular to opposite edges of a tetrahedron and volume 9, Image of a point across a plane10, Image of a point across a st line 12, Image of a line across a plane 20, Surfaces generated by st lines and locus of st line meeting three st lines, intersections 22, Vector equations 26Chapter 25 : Linear Equations and Matrices Pages 1 to 38Topics : Introduction to Matrices as a tool for equation solving1, Different types of matrices and operations on them , inverse matrix up to 9, Crammer Rule 9, Gaussian elimination explained :Alternative treatment for matrices 16, elementary row operations and column operations 17, Rank of a matrix and solution and consistency of a system of equations 19, Normal form, Diagonal form, Triangular forms 23, Of 0 Matrix, Singular and Identity Matrices 26, as characteristic roots or latent roots or simply eigenvalues 27, Matrix as a linear transformation 29, Linear transformations 33, Eigenvectors 34, Reduction to Diagonal form 36, Complex numbers, vectors and matrices compared 37.Part II : Conic Sections And Differential GeometrySection 4: Quadratic Functions, Sections Of Cone , Meanings Of EccentricityChapter 26 : Conic Sections, circle and sphere Pages 1 to 29Topics : The circle and sections of a cone 1, equation of a circle 3, parametric equations 4, circle on a given diameter 7, law of sines 10, extended law of sines 10, law of cosines, extended Pythagoras theorem11, general equation of circle in polar coordinates 12, parametric equations 14, vector equations 14, equation of a chord 14, chords cutting each other 19, a formula for tangent 20, use of complex numbers 23, equation of circle Page 10 / 681through 3 non-collinear points 24, sphere 25, diametric form 26, a plane touches a sphere 26, a plane cuts a sphere 27Chapter 27 : Co-linearity, concurrence and properties of triangle Pages 1 to 66Topics : Concurrence of three st lines 1, Co-linearity of three points 2, Perpendicular Bisectors of Sides of a Triangle are Concurrent 2, Perpendicular Bisectors of Sides of a Triangle are Concurrent ( Brahmaguptas formula or Herons formula)3, coordinates of circum centre 4, Brahmagupta's Theorem 6, The ratios involving half angles 9, Apollonius' theorem 15, centroid 15, Another section formula 16, Cevas theorem 19, Cevas theorem in vector method 20, Menela Desargues theorem us theorem 21, Menelaus theorem in vector method 22, Desargues theorem in vector method 24, Pappus theorem 25, Pappus theorem in vector method 26, Pascals theorem and Brianchons theorem (without proof) 27, Proof of a variant of the Pascals Theorem 27, In-centre 28, ex-center 28, Vector method shall explain the point vividly and just separating the variables, proves the theorem in coordinate geometry 34, a challenging problem 38, Orthocentre of a triangle 40, Orthic triangle, pedal triangles 43, Distance of orthocentre from vertices and from feet of perpendiculars 45, The feet of perpendiculars from a point on sides of a triangle 46, equation of the median 47, equation of the angle bisector 47, Use of complex quantities, some advantage over vectors 49, The three centres are collinear 55, Distance among the three centres 55, Nine point circle 64,Chapter 28 : Concyclic points and quadrilaterals Pages 1 to 16Topics : A circle passing through four corners of a quadrilateral 1, Quadrulatral Inscribed in a Circle 3, A Criteria for four points to be Concyclic 3, Brahmaguptas formula or Herons formula is for area of a quadrilateral 4, cyclic quadrilateral 7, Ptolemys theorem 10, Archimedes' Theorem 13, Newtons problem13, Gauss line 14, Regular polygons 14, Use of complex numbers 16.Chapter 29 : Conic Sections Ellipse Pages 1 to 37Topics : Equation of an ellipse from compression of coordinates1, Change of form and Change of origin 2, Change of Axes 3, Auxiliary Circle 3, Outside and inside of an ellipse 4, Parametric Equations 5, Vector Equations 5, Use of complex numbers 6, A meaning of eccentricity 6, Visualisation of e 7, Second focus and second directrix 9, Distance of focus from the center 9, Sum of distances of any point from two foci 9, Sum of distances of any point from two foci 9, Equation of ellipse from constancy of sum of distances from two foci 10, Another method 12, Construction of ellipse 12, Equation of ellipse from focal distance and directrix distance 12, Determination of focus, Page 11 / 681directrix etc 13, Reduction to standard form 13, Focus and directrix 15, Latus rectum 15, Focal radii 15, Span of the ellipse and sum of the focal radii 16, What does division of major axis by ordinate mean 17, Ellipse as shadow of a circle 18, ellipse in a plane cutting a cylinder 20, ellipse is actually a section of a cone 20, The value of e 24, Another Alternative derivation of ellipse 24, Another alternative derivation of ellipse 25, Another derivation of ellipse 27, Reflection property of ellipse 28, Another alternative 31, Smashing kidney stones 31,Chapter 30 : General Conic Sections and Parabola Pages 1 to 25Topics : Conic sections characterized by their eccentricity, Conic section in polar coordinates: (origin at focus) 1, Eqn to directrix 3, Conic section in Rectangular Coordinates: (origin at directrix) 4, Derivation of Eqn of Ellipse in Cartesian Coordinates 5, Equation of parabola 7, Convex and concave sides , latus rectum or focal width 8, Parabola whose axis is y axis 8, relationship between latus rectum and directrix of ellipses 11, 22 2 0 Ax By gx fy c always represents a parabola 11most general equation of a parabola 12, Any linear and / or angular transformation of axes keeps a parabola a parabola 13, to find its focus, vertex, eccentricity, latus rectum, and distance from focus to vertex 13, 1x ya b represents a parabola 15, Derivation in polar coordinates (vector form)15, Derivation in Cartesian Coordinates 17, Another Derivation 17, Focal distance = x + a if the origin is at the vertex 20, parabola is actually a section of a cone 21, Reflection property of parabola 23, Construction of a parabola 24.Chapter 31 : General Conic Sections and Hyperbola Pages 1 to 31Topics : Equation of rectangular hyperbola 1, An analogy with the circle 1, parametric equations 2, equation of hyperbola 5, Alternative :difference of distances of a point from two fixed points 8, Alternative: center of a circle squeezed through two fixed circles 8, Second focus and second directrix 9, Alternative derivation of a hyperbola. Time difference used by sonar to locate enemy ship 9, Parametric equations 10, Vector equation 11, Use of complex numbers 11, Focal distance of points on hyperbola 11, latus rectum of hyperbola 12, geometric mean of product of focal distances 12, Analogy with ellipse 13, Alternative derivation of a hyperbola 14, Alternative derivation of a Hyperbola 15, hyperbola is actually a section of a cone 15, a meaning of e for the hyperbola 17, reflection property 17, Construction of a hyperbola 18, Alternative construction 19, Alternative derivation of hyperbola 20, another Alternative derivation 20, another derivation 21, another derivation of rectangular hyperbola 21, another derivation of rectangular hyperbola 22, Relation of hyperbola and rectangular hyperbola Page 12 / 68122, Conjugate of hyperbola 23, Asymptotes of a hyperbola 23, Equation of Hyperbola when Asymptotes are Axes 26, Focus, Directrix and latus rectum 28, Conic Sections compared 29, Another comparison 30, Another comparison 31.Chapter 32 : The General Second Degree Equations and Conic Sections Pages 1 to 35Topics : 2ND DEGREE EQUATIONS IN TWO variables represent a conic section 2, condition for circle 4, condition for two st lines 4, Discriminant 5, condition for two parallel lines 7, two st lines through the origin 8, 9,imaginary st lines meeting at a real point 10, homogeneous equation always represents two st. lines passing through the origin 10, Angle between the two st.lines through origin, of ax2 + 2hxy + by2 = 0 11, What is the difference between 2y ax bx c and 20 ax bx c 14, The terms of second degree in 2 2 22 2 2 0 a x haxy aby agx afy ac 15, required condition to represent two st lines 15, area of the equilateral triangle 18, rectangles, triangles enclosed, their orthocenter etc up to 23, Bisectors of angle between the pairs of st lines 2 22 0 ax hxy by 23,in terms of vectors 25, The general conic represents the following curves 27, Compression of coordinates 29, Section 5: Tangents and Normals, Curvature and Meaning Of Parametric EquationsChapter 33 : Secants and Tangents Pages 1 tor 59Topics : what are secants and tangents, tangent to a circle 1, secants to a circle 3, Tangent to a circle, center at origin 4, Tangent to a circle, center not at origin 5, Tangent to a parabola, hyperbola 7, UA golden rule for writing tangents , design an equation for a purpose 8, Important Results 9, Focal chords and harmonic relations 12, The semi latus rectum is harmonic mean of any pair of focal radii 13, U Locus of middle points of focal chords of any conic is a conic of th same type 13, Secant and Tangent to an ellipse if it is in parametric form 14, equation of tangents to conics at points represented by a parameter 15, Secant and Tangent to a general conic in polar coordinates 16, Tangent to a conic section in general 18, Tangent to a conic section at the origin (lowest degree terms 19, Another method to find Tangent to a conic section in general 19, Tangent to a parabola in general, another parametric equation 20, Another Parametric Equation 22, geometric mean Page 13 / 6812. FP FQ FR 28, Tangent to a circle in general 29, Another parametric equation of a circle 30, Tangent to a ellipse in general 30, Another parametric equation of an ellipse 31, A segment of a tangent to an ellipse cut between the directrices 31, Tangent to an ellipse in general in polar coordinates 32, many of the properties of ellipse may be carried over to hyperbola just replacing 2bby - 2b32, Another parametric equation of hyperbola 32, Tangent to a rectangular hyperbola 33, Condition that a given st.line may be a tangent to a conic 33, Pair of tangents from any point (x1, y1) to the general conic 34, Lengths of tangents to the circle from any point 35, Remember the following results in the exercises 37, When two circles touch each other and two alternative criteria 37, Common tangents to two circles 38, Common tangents to parabolas 40, tangents to other curves in exercises 41, Common tangents to an Ellipse and a Circle, common tangents to ellipses 43, Common Chords 45, Differentiation for getting slope of tangent 47, meaning of derivative 49, Total differential and tangent 51, Tangent at the origin( lowest degree terms equated to 0) 52, Tangents of curves cutting at right angles, director circle & directrix , case of ellipse, case of hyperbola, case of parabola, 54, Angle between two curves at intersection 55, Dip of the horizon 57, Vector equation of tangent, velocity and acceleration58, Chapter 34 : Normal and curvature Pages 1 tor 50Topics : Normal to any curve 1, Normal to any curve 2, Intercepts of normals with axes 2, Reflection property 4, Feet of perpendiculars 4, Auxiliary Circle 6, Feet of perpendiculars joined to foci 6, Normal to a general conic at the point (r, ) in polar coordinates 7, Working rule 8, To find locus of the foot of the perpendicular upon tangent and normals from center of a hyperbola 8, Another form of a normal for parabola; another parametric equation 9, : Theorem of three normals for parabola 9,two of the normal are perpendiculars to each other 10, Feet of the perpendiculars from foci on any tangent to an ellipse 14, Product of distances of any tangent from foci 14, AOF = angle BOG and the angle AFO = angle BFO 14, Any normal to an ellipse 15, Theorem of four normals 15, Geometrical meaning of parametric eqn of parabola 18, Distance of a curve from a given point 19, Pedal equation of the conic25, Subtangent and Subnormal and their lengths 27, focal distance, reflection property, 28, The focus is at middle point of intercepts of tangent and normal with the axis 28, Reflection property and comparison of conic sections 29, Subtangent and subnormal of parabola from simple geometry 30, Angle between radius vector and tangent 30, Angle between two curves at intersection 31, Direct derivation of pedal equation 32, Radius of curvature and curvature 33, A small part of a curve is a small part of a circle 33, Curvature at the origin if one of the axes is a tangent or normal ,Newtons formula 37, Tangents, Normals, Curvature and Torsion 38, binormal 39, Evolutes 40, Evolutes 41, Evolutes may easily be known as its center and radius are known 41, Conic sections as real plane sections of cone, Dandelin spheres 45, Directrix and eccentricity, another meaning of eccentricity 47, Directional derivative and gradient 47, level curves 49, Gradient in different system of coordinates 49,Page 14 / 681Section 6: Parametrization Of Conics and General ProblemsChapter 35 : Parametrization of conics and general problems on circle and parabola Pages 1 tor 59Topics : problems on circles 1, general problems on circles and parabolas 9, Another vector equation of a parabola 50, Another meaning of parameter 51, semicubical parabola 51, Chapter 36 : Parametrization of conics , general problems on Ellipse and hyperbola Pages 1 to 30Topics : General problems on ellipses 1, . . ' PG PH FP F P . 14, 2. OT OG OF 14, General problems on hyperbola 17. Section 7: Further Geometry On ConicsChapter 37 : Parametrization Central Conics Pages 1 tor 8Topics : Central conics 1, eqn to the general conic referred to the new axes through the center 2, Center of a parabola and two st lines as conic sections3, There is only one pair of conjugate diameters common to any two concentric central conics 4, eqn of ellipse , with axes parallel to conjugate diameters 4, Central conics referred to their axes and lengths of the semi-axes 5, Foci of central conics 7 .Chapter 38 : Asymptotes and Envelopes Pages 1 tor 34Topics : An asymptote to a curve is a st line which meets the curve at infinity, but it is not a line at infinity 1, Conjugate hyperbola & Asymptotes 2, Conjugate hyperbola & Asymptotes; eqn in general form of central conics 3, Another definition 4, Asymptotes by expansion and its two generalisations 5, hyperbola for given Asymptotes 7, Theorem of four normals for central conic 7, the feet of the four perpendiculars from any point (h, k) on a central conic lie on a rectangular hyperbola passing through the origin. 7, Asymptotes parallel to y-axis 8, A controversial view of asymptotes 9, Asymptotes of general rational algebraic equation( with exception) 10, two parallel asymptotes 12, Summary 13, Asymptotes in polar coordinates 15, Polar subtangent and polar subnormal 17, An asymptote and the curve at infinity are described by the same equation as it is a tangent at infinity. If there are two asymptotes , the point at infinity is a double point 18, Inverse or reciprocal curves 18, Reciprocal coordinates 18, Reciprocal curves in polar coordinates 19, Pole and polar of circle is a duality relation 22, : Conic sections may be viewed as reciprocals or inverses of a circle 23, Study of curves from their asymptotes 24, Asymptotes as guidelines24, Envelopes 26, Some facts about Envelops29, Pedal curves and pedal point construction, representing conics in yet another way 32, Pedal curve and pedal point construction 33.Page 15 / 681Chapter 39 : Chord Of Contact, Pole and Polar Pages 1 tor 45Topics : Location Of Points Relative To a A Curve 1, Location Of A St Line Relative To Curve 1, Chords Of Contact 2, Working rule and points to remember 4, Pole and Polar 7, Survey Of An Entire Curve From A Point Outside, On, Or Inside It 8, Working rule and points to remember 10, Reciprocity of pole and polar 12, Another relation between them 12, Relation between focus and directrix 13, polar equation of polar of any point (r1, 1) wrt a conic 1 cosler 14, pole of a given line with respect to a given conic 15, Geometrical construction of polar of any given point 15, Joachimshals notation 17, Diameters of conics 22, Locus of middle points of parallel chords is called diameter 23, another parametric equation of the parabola 25, The tangents at end points of any chord of a conic meet on a diameter which bisects the chord. But this never means that the tangents are equal 26, Chords with given middle points 28, equation of a chord bisected at (h, k) 28, Conjugate diameters of ellipse 31, exception 32, Eccentric angles of the ends of a pair of conjugate diameters differ by /2 34, The sum of squares of semi-conjugate diameters is constant and equal to square of radius of the director circle 35, Perpendicular diameters 35, Product of the focal radii 36, Length of a diameter of the ellipse 36, Equi-conjugate diameters 37, Middle point of line joining ends of conjugate diameters 37, Intersection of tangents at the ends of conjugate diameters 38, Locus of foot of perpendicular from focus onto the line joining end points of two conjugate diameters 38, A parallelogram is formed by four tangents at the ends of a pair of conjugate diameters 39, Supplemental chords are parallel to a pair of conjugate diameters 39, Conjugate diameters of hyperbola and parabola 40, Equation of a parabola referred to a diameter and the tangent at its end as axes 40, Equation of a parabola referred to its axes 42, Diameter of a General Conic 43, Conjugate diameters of general conic 44, a pair of st lines through the origin 45, Section 8: Intersection Of ConicsChapter 40 : Superscribed and Inscribed conics, Confocals, orthogonal trajectorie Pages 1 tor15Topics : Confocal conics 1, Touching a given st line 2, Conics Through Intersection Of ST Lines Or Inscribed within 3, Eqn of a conic passing through four given points 3, A unique conic can be made to pass through five given points 5, Eqn of a conic inscribed inside four given st lines 5, Three tangents theorem 6, Pascals theorem 6, Inscribed ellipses 7, Orthogonal Circles and Orthogonal Trajectories7, Radical axes of three circles taken in pairs are concurrent 9, Eqn to family of circles with common radical axis coaxial circles 9, Orthogonal circles to a coaxial system 10, Orthogonal Trajectories 11, Matrix and determinant representation of conics 14.Chapter 41 : Intersection of Conics and Use of Partial Derivatives Pages 1 tor33Topics : Condition that the second degree eqn in two variables represents two st lines 1, Parallel lines meet on the line at infinity 2, Line at infinity 2, Partial differentials met in conic sections 2, Tangent and normal , secant, chord of contact, Polar, the curve itself etc.3, Equation of a chord bisected at any point (s, t) is parallel Page 16 / 681to the polar of the point 5, Centre of the curve 6, Conjugates and Asymptotes 7, Axes of the central conic 7, Foci of the central conic 9, Tracing of parabola, (having nocentre, or centre at infinity) 9, Envelopes 11, four points of intersection 12, conic passing through intersection of two conics 13, Only rectangular hyperbolas can pass through intersection of two rectangular hyperbolas 14, : St lines passing through intersection of two conics 14, : A pair of st lines passing through intersections of two rectangular hyperbolas are perpendicular to each other 14, When two rectangular hyperbolas intersect at four points, each point of intersection is ortho-centre of the triangle formed by the other three points 15, Any conic passing through four points 15, A conic passing through intersection of a conic and a pair of st lines 16, Pair of tangents on the general conic 18, Imaginary tangents from foci of the general conic 19, Foci of the general conic 19, Foci of the central conic 20, conic with a tangent and a normal as axes 21, Chords And Harmonic Relations 23, Symmetric relations like conjugate diameters, pair of tangents etc, 27, Foci of general conic 28, Circle of Curvature 30, Part III: Integral Calculus, Differential Equations and Central ForcesSection 9: Integration, Quadrature and Rectification.Chapter 42 : The Indefinite Integral Pages 1 to 49Topics : Integration as reverse process of differentiation 1, Algebra of integrals 2, Integration by substitution; or change of variable 5, Integration by parts 8, Repeated integration by parts, Order of choice of 1st function 10, Important Results 10, Functions from Algebra and Logarithm 10, Inverse trigonometric functions 14, Functions from Trigonometry 15, Inverse Circular Functions again, using integration by parts 16, Hyperbolic functions 18, Integrals involving logarithms, ex cos x ,sin x etc 19, Hyperbolic functions and inverse functions 26, Six important integrals using trigonometric transformations 28, Combination of trigonometric and exponential functions and some special types 37, Partial fractions in integration 43, Vector integration 49.Chapter 43 : Transformations of integrals for integration Pages 1 to 41Topics : Integration of rational algebraic functions like 22 nAx Bx C reduction formulae 1, INTEGRATION OF TRIGONOMETRIC FUNCTIONS 3, Recursion formulae and some special types 3, Rational Integral functions of sin x and cos x 15, Integration of hyperbolic functions 23, Integration of some irrational functions 24, various transformations for integration25, Mean value theorem for integral calculus 34, Integrals using differentiation 36, Integrals that cannot be found without numerical integration 40. Page 17 / 681Chapter 44 : The Definite Integral Pages 1 to 52Topics : The definite integral 1, Definite integral as the area bounded by the curve 1, fundamental theorem of Integral Calculus 2, Results To Remember 3, Change in variable 5, another form of fundamental theorem of integral Calculus Definite Integral As The Limit Of A Sum 14, Meanings Of Area , potential difference, work, volume, moment of inertia etc. 15, Summation of series 19, Introduction to Improper integrals 26, Revision of Recursion or Reduction Formula for Definite integrals 33, Transformations for pm nx a bx dx 36, Simple Evaluation of 2 20 0 0sin cos , sin cos sin cosp q p q p qx xdx x xdx and x xdx 47, Simple Evaluation of 2 20 0 0sin cos , sin cos sin cosp q p q p qx xdx x xdx and x xdx 48,Chapter 45 : Limits Of Sums Of series By Integration Pages 1 to 9Topics : Summation of Series 1, Summation of series using Newton Leibnitz rule 5, Chapter 46 : Quadrature and Rectification and Mechanics of Rotation Pages 1 to 36Topics : AREAS OF BOUNDED REGIONS 1, Area bound by a closed curve described by one parameter 8, Area in Polar coordinates 11,Rectification: length of plane curves 13, Simpsons Rule for finding definite integral 14, Centre Of Mass, Centre Of Gravity 16, Algebraic sum of moments of masses about the CM is 0 ;17, Moment Of Inertia And Angular Velocity 19, Angular velocity vector 19, Relation between L and , Moment Of Inertia 20, Newtons laws for rotational motion 20, Conservation of angular momentum21, Calculations of moments of inertia 21, Relation between moment of inertia and centre of mass 22, Theorems of perpendicular axes and parallel axes 22, Misc. examples 24, Surfaces and Volumes of Revolution 31, Calculation of Electric fields, Magnetic inductances etc. 32, Magnetic induction due to a straight current carrying conductor 34,Chapter 47 : Improper Integrals, Beta and Gamma Functions, Differentiation under integral sign Pages 1 to 38Topics : Improper integrals in general 1, An Important result 2 , Comparison Integrals 3, error function 4, TheFunction( Also called Eulers 1st integral) Page 18 / 6815, important results 6, Wallis formula 7, , , m n has many alternative forms 8, The (gamma) function, also called Eulers 2nd integral 12, Graph of Gamma Function 13, Relation between Beta and Gamma Functions 14, Alternative definitions of gamma function 14, Weierstrass Formula for Gamma function 15, Some special cases of gamma function 16, another Relation between beta and gamma function 17, multiplication formula due to Gauss, Euler, complement formula 19, Differentiation under integral sign 20, Leibnitz formula 25, generalization of Fundamental Theorem of Integral Calculus 27, Newton-Leibnitz Formula for evaluating limits 35.Section 10: Differential Equations, Central ForcesChapter 48 : Introduction to Differential Equations Pages 1 to 42Topics :DIFFERENTIAL EQUATIONS: LAWS OF NATURE IN CODES 1, Population growth 2, Radioactive Decay 3, Approximate age of solar system 4, Carbon Dating 5, Newtons empirical law of cooling 6, Discharging of a capacitor 6, Formation Of Differential Equations 8, Wave equation 8, Families of curves : Order and Degree Of Differential Equations 11, General solution and particular integral , singular solution , envelope 11, Boundary values and existence, uniqueness of solutions 13, equations of motion under uniform acceleration 14, DIFFERENTIAL EQUATIONS OF FIRST ORDER AND FIRST DEGREE 15, Variable separation method 15, Exact Differential Equations 16, Integrating Factors 20, Equations of first order and higher degree 22, Equations solvable for y 23, Equations solvable for x 24,Equations solvable or p 25, Reduction of order 25, Independent variable missing 26, dependent variable missing 26, Equations homogeneous in x and y 27, Clairuts Equation, singular solutions and envelops 28, The Lagranges equation 34, Orthogonal Trajectories 34, Linear Equations 38, Homogeneous Equations 42Chapter 49 : 2ndOrder Linear Differential Equations and Central forces Pages 1 to 64Topics : Linear differential equations with constant coefficients 1, Case of Repeated Roots 3, Simple harmonic motion 4, Interpretation 4, phase angle, circle of reference 5, a reverse view of the problem 6, superimposition of two SHMs 7, phenomena of beats 9, Natural frequency 12, Graphs of displacement, velocity and acceleration and energy 13, Damped SHM 13, Critically damped SHM 15, Forced SHM , coupling 15, Particular integral and general solution of non-homogeneous linear differential equations 15, Particular integral of 1 mteF D18, Particular Page 19 / 681Integral of sin mtor cosmt20, Particular integral of 1 mtF D21, Forced oscillations with damping 22 , Particular integral by method of undetermined coefficients 23, travelling time of a tunnel train 25, Elements of electrical circuit 26, self inductance, Lenzs law 27, Examples of Transient current circuits 28, Example of Use of complex quantities in circuit equations, Impedance 33, The power factor 35, Planetary motion: A grand example of 2nd order linear diff. eqn in vectors 36, Keplers second law 38, Projectile motion and vector equation of a parabola 38, Keplers first and third laws 39, Total energy a nd eccentricity: physical meaning of eccentricity 42, Areal velocity and angular momentum and moment of momentum 44, An example Vector integration : Planetary motion (in vectors) 45, Interpretation, finding the orbit and speed 47, expression for e for planetary motion given above and criteria for open / closed orbits 48, hodograph 49, Central force as any function of distance in general 51, Central force directly proportional to distance 52, semi-conjugate diameter 53, Two simple harmonic motions at right angles to each other superimposed on a particle result in an elliptical orbit. 56, Newtons inverse square law from Keplers laws 57, Some more particular integrals 58, Homogeneous linear equation 62, Bertrand s theorem 63.Page 20 / 681FOREWORDThis is an humble endeavour to blend Calculus, Analytical Geometry of Straight Lines, and Conic Sections, Vectors, Matrices and Determinants, Theory of Equations, and Elementary Differential Equations through examples of Uniform Motion, Projectile Motion, Rotational Motion, Central Forces like Uniform Circular Motion, Simple Harmonic Motion, Planetary Motion, Coriolis Force into a single mammoth concept of Identity Division Symmetry, the ultimate key to unravel the mystery of Nature. This is just like every number, say 5, and its additive inverse 5 peacefully coexisting inside the identity of addition 0; or like 5 and its multiplicative inverse 1/5 peacefully coexisting inside the multiplicative identity 1. Eulers celebrated equation the five constants .Identities and constants are treated as same thing in this book, same thing as invariants like the laws of conservation of mass, energy, momentums, etc. and we break open these to get the differential equations of motion or the equation of motion in simple cases. Breaking 8 into 3 and 5 is a division and a series like is a division . The vector is a division in the sense ordered pairs or triplets are. Differential coefficients and integrals are also a remote similarity. Fixed points are some sort of invariants;(Fixed points of x f x are roots of 0 F x f x x ). A locus of a moving point is a static picture, an equation or a conditional identity. A determinant, matrix, inverse circular function or logarithm contains all its properties just in its definition, which is just like a personal identity, just like the name and personality of a real person. As Einstein said, the problem and the answer lie just side by side; approach near and near the problem and the solution is there itself, maybe, just give it a symbol ; as in word problems offered by the village school master or like while taking the limit of 223 03 9 9lim lim3 3 3xxx .A division , 3 and 5 of 8 though not symmetric, a higher level symmetry 8 2 8 25 , 32 2 is there , well behind it. Every topic of the book is an endeavour to link it to identity, division or symmetry, which would give the reader a feeling of rediscovery and confidence of solid fundas which he or she may not forget after the exams. The writer hopes this view would not put an end to whys and hows but definitely help to peep into the world behind them, to enable the average student to see through the concepts taught conventionally. An identity like 0 contains symmetrical pairs like +5 and 5 added together; an identity like 1 contains 5 and 1/5 multiplied together. An identity like 2 2 22 a b a b ab still remains valid if a and b are changed just as someones personal identity which does not change as some one grows up or goes to different places. (not even when someone changes sex!)Such condensation of topics, as mentioned earlier, i.e., integrating many concepts shall not come in the way of rigor; rather it would be a relief in heavy course of curriculum today (some topics of Physics like Mechanics, Circular motion, SHM, Planetary motions, varying currents and 21 0ie 2 31 ..................... x x x 11 x Page 21 / 681transient currents etc. are also covered) and sit would provide an edge for competitive entrance examinations, both in concept building and problems.Conic sections are shown to be really sections of cone just with the help of high school geometry and various meanings of eccentricity are illustrated. They are also derived by simple compression of coordinate axes, as inversion of circles and such alternative treatments of topics abound in the book, which would delight students and teachers alike.Last but not the least, the author would be very thankful to any one who offers some suggestions for improvement of points out mistakes if any.The Selection of topics includes courses of Calculus, Coordinate Geometry, Vectors, Analytical geometry, Mechanics, determinants, matrices, differential equations of first order and first degree in the +2 curriculums of CBSE, ICSE, and various regional boards, and council s in India and fundamentals for preparation for IITJEE and other entrance exams. The celebrated institutions preparing for these entrance examinations expect this knowledge and they themselves seldom teach the fundamentals. The book is to fill up this gap.Author. Copy right - narayana dash 2011.Page 22 / 681Series : REDISCOVER MATHEMATICS FROM 0 AND 1 PART 1:THE SWORD AND THE SHIELDBook : Calculus And Analytic Geometry Of 2D and 3DSection 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions (Concepts and fundas for IITJEE and other competitive exams) Chapter 1 : Identity, Division ,Symmetry In Math Page 1CHAPTER 1 : IDENTITY, DIVISION ,SYMMETRY1:THE BEAUTY AND POWER OF, IDENTITY, DIVISION AND SYMMETRYYou have a nature and a name, you have your tastes, choices, likes and dislikes, love and hate, relationships, characteristics and personality the sum totality of this is your identity. It does not change, i.e., people still know and recognise you if you grow up in time, change houses or go abroad. You are still the same person with the same identity. Similarly your family has an identity, it does not change , for example, if represented by you or some other member of your family. In the same way the groups or communities or country you belong to, have identities for that matter. In mathematics you have come across the identity 0, celebrated as the identity of addition. You know it contains every number and its negative in pairs, for example 5 and 5 . The identity 0 does not change any number with which it is added or subtracted from . You know 1 is an identity, the identity of multiplication, for example 5 and 1/5 both reside in 1 multiplied with each other. The 1 does not change any number it is multiplied with or divides. Such is the nature of identities. Any number such as 7 is called a constant as it does not vary. You will see this is also an identity in some sense. Note a class of integers which leave a remainder 1 when divided by 7, such as 8, 15, 22 etc. We can denote this class with a symbol [1]. Other such classes of integers are [3], [4] etc. leaving remainder 3 and 4 respectively . Take any two members, one in each class, say 10 from [3] and 25 from [4] and adding them gives 35, which belongs to the class [0] or [7]. This is no fallacy and we can call [7] or [0] each as an identity of addition in these seven classes of integers called congruent modulo 7. An identity does not change ; so also constants. You know (a + b)2= a2+2ab+ b2is an identity. The formula holds no matter how a and b may change. An identity is an equation which holds ( = holds true) for any admissible value of the variable(s). For example, the equationxyy xy1x1 += +holds for any value of x and y except 0. Also ( )( ) 2 2y x y x y x = +is another example. Thus the identity stands on its own and does not depend upon the symbols though it is convenient to express them with the help of these symbols. Instead of identities that are evident at sight, like (a b ) + (b c ) +(c a ) , we can assume or construct identities . If we assume (X x )2+ (Y y )2+(Z z)2= 0 for all values of the variables X, Y, Z, x, y, and z ; or in other words if we assume it to be an identity,( call it a conditional identity) we immediately have the three equations X= x ,Y= y , Z = z. We dwell on this point again and again to prove beautiful results of use like equality of conjugate surds when given surds are equal; equality of complex numbers when given complex numbers are equal; proving that conjugate ofa root of a polynomial is also another root of it; so on and so forth. An identity operation does not change or modify the argument (on which it operates) or the operand. For example, k = (\k)2; an operation like squaring the square root of any numberdoes not change that number. In other words it preserves the identity of the operand and it is Page 23 / 681Series : REDISCOVER MATHEMATICS FROM 0 AND 1 PART 1:THE SWORD AND THE SHIELDBook : Calculus And Analytic Geometry Of 2D and 3DSection 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions (Concepts and fundas for IITJEE and other competitive exams) Chapter 1 : Identity, Division ,Symmetry In Math Page 2therefore only that the operator is called so. Please note that the compound operation of first taking square root and then squaring is not same as first squaring and then taking the square root; simply because, it does not result in identity. Taking square of a number and extracting the square root again are thus seen to be not the exactly reverse operations so as to result in an identity operation. We know reverse operation always takes us farther than we imagined. For example adding two natural numbers results in a natural number ; there is no restriction in adding and we cannot get out of the natural number system in the process of addition. The set natural number is thus said to be closed under the process of addition . We would return to this point a little later. You see the power of identities in the next two examples.RESULT1; equating coefficients of similar powers in either side of an identity.One interesting thing about identities is that , If, t rx qx px d cx bx ax 2 3 2 3+ + + = + + + is taken to be an identity, we must have, a = p, b = q, c = r , and d = t .(coefficients of corresponding powers of a particular variable from both sides shall be equal) This can be easily proved as under :Since the equation is true for any value of x, putting x = 0, we get, d = t and they obviously cancel out from both sides. Then we could assume values of x as 1, -1 and 2 say, and get three equations involving a - p, b - q, and c - r ; and on solving them , we can get, each of a p etc., each = 0. Alternatively, after canceling out d and t from both sides, we can get another identity, or equation which holds for all values of x. Putting x = 0 again in this eqn., we get c = r . Repeating the process, we get the desired result. The utility of this result is indisputable as you are aware when apply this result to knotty problems.RESULT2 ; equating coefficients of similar trigonometric ratios in either side of an identity.If p cos x + q sin x + r = ( a cos x + b sin x + c ) + (b cos x - a sin x) + is taken to be an identity, (in other words, if a given expression in sin x and cos x is changed to another expression in sin x and cos x , for some desired convenience) then we must have the coefficients of sin x to be equal in both sides and so also the coefficients of cos x. We have, p cos x + q sin x + r = ( a cos x + b sin x + c ) + (b cos x - a sin x )+ v for all xOr, p cos x + q sin x + r = (a + b) cos x + (b - a)sin x + (c + v) for all x, (a)Putting cos x = 1 throughout, ( so that sin x = 0), we get,p + r = (a + b) + (c + v).(b)Putting cos x = - 1 throughout, ( so that sin x = 0), we get,Page 24 / 681Series : REDISCOVER MATHEMATICS FROM 0 AND 1 PART 1:THE SWORD AND THE SHIELDBook : Calculus And Analytic Geometry Of 2D and 3DSection 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions (Concepts and fundas for IITJEE and other competitive exams) Chapter 1 : Identity, Division ,Symmetry In Math Page 3- p + r = - (a + b) + (c + v) ...(c)Adding (b) and (c) and dividing throughout by 2 , we get, p = (a + b) and r = (c + v) .....(d)Putting these values in (a) we get, q = b - a .....(e)So we get three independent equations in , , and v from (d) and (e) which may be easily solved to find , , and v in terms of p, q and r. ( the result has many uses in integration chapter)It seems at first sight that identities are trivially true and are of little utility. No. Just like 0 is a trivial nothing , but embodies a world of secrets like a black hole, identities contain a world of secrets inside them. We give below an example how the concept of identity is used to reveal the sum of an infinite series hidden in the symbols we define ,to get rid of a difficulty. The great scientist Albert Einstein perhaps said, if only you could know where exactly the problem lies, the solution lies there itself. In his times, half of the world believed there was the all pervading invisible ether and the other half believed there was no ether. He put an end to the controversy by telling everybody to raise only the questions which probably have answers, and discard all others. Did he take the cue from a small child he taught, who told, there can be another way besides the Earth going around the Sun and the Sun going around the Earth ? If a point of difficulty is pinpointed, and is enclosed in a symbol, the solution lies nearby, just you have to manipulate with the adopted symbol in set procedures and arrive at the removal of difficulty. Do the duo of the difficulty and the solution constitute some sort of identity or invariable ? Maybe. Follow the examples:Example1 ; Quadratic Equations :an example in difficulty identification: First , take the quadratic equation, ax2+ bx + c = 0 with rational coefficients. How easy it would have been, only if the bx term had not been there! So the difficulty seems to be the linear term and let us try to remove it. Now, put bx or x, inside a symbol t, such as t = x - h or x = t + h, proceed mechanically working with the equation so as to get rid of the linear term bx , and discard the t and h when x reveals its value and those extraneous symbols have nothing to do any more. Our equation becomes,a(t +h)2 +b(t +h) + c = 0 at2+ (2ah+b)t + ( ah2+bh +c)=0 If we choose, h = -b/2a, to make the second term 0, the equation reduces to ,at2+ 0+(b2/4a-b2/2a+c)=0 ,(an eqn. in t having no first degree term in t)Page 25 / 681Series : REDISCOVER MATHEMATICS FROM 0 AND 1 PART 1:THE SWORD AND THE SHIELDBook : Calculus And Analytic Geometry Of 2D and 3DSection 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions (Concepts and fundas for IITJEE and other competitive exams) Chapter 1 : Identity, Division ,Symmetry In Math Page 4or, at2= b2/4a c t =a 2 ac 4 b2 x = a 2b a 2 ac 4 b2Now the poor things, h and t have been thrown out mercilessly once the difficulty is removed! The roots of quadratic equation t = f(x) = ax2+ bx + c = 0,are a 2 ac 4 b b - 2 += o and a 2 ac 4 b b - 2 = | . The sum of the two roots is bao+| = and the product of the roots cao| = as may be verified direct. We know in high school that this solution has been got by completing square method, either by dividing by a throughout or multiplying by 4a throughout. The difficulty identified in this way is to make ax2+ bx , a part of a complete square (as this one is not a complete square, ) The roots are real if discriminant of the equation , ac 4 b2 is positive, i.e., b2 4ac > 0 and imaginary otherwise( by imaginary we mean, the roots of a negative number, if at all we agree for its existence). They are rational if b2 4ac is a perfect square and irrational otherwise .Actually, the concepts of irrational numbers and imaginary numbers are gifts of the quadratic equations. The roots are equal if the determinant is 0. The expression under the square root is called discriminant because it determines the very nature of roots of the quadratic. If the roots are irrational or imaginary, one is the conjugate of the other, i.e, if one is p+ q or p + iq, the other is p - q or p iq; ( This is evident from the structure of the roots , they are already in p+ q and p - q form. To prove it, assume m+ n is another root, so their product (p+ q)(m+ n) = c/a, a rational number .This is possible only when m+ n is a multiple of conjugate of p+ q, say k(p - q). Now, the sum of the roots is p + q + k(p - q) = p(1 + k) + (1 - k )\q = - b/a. Solving this for \q we see that either 1 k = 0 or \q = {- b/a p(1 + k)}/(1 k), a rational number, which is a contradiction. So 1 k must be 0 or k = 1 or the other root must be p - q . ). In another way, we observe that the sum of the roots is b/a and the product of the roots is c/a; both rational numbers by assumption. Then , if the roots themselves do not occur in conjugate pairs, how can their sum and product be rational ? No matter if the reader does not understand imaginary numbers . We would return to the subjects, quadratic equation and imaginary numbers in detail in later chapters at appropriate places and this is merely one example how to apply the Difficulty Pin-pointing Method.Actually this is a method how second term in a polynomial equation could be removed. (See further the chapters on quadratic equations, theory of equations, Cardans solution of cubic equation etc.) Exercise1 : Remove the second term in the eqn. ax3+ bx2+ cx + d = 0Page 26 / 681Series : REDISCOVER MATHEMATICS FROM 0 AND 1 PART 1:THE SWORD AND THE SHIELDBook : Calculus And Analytic Geometry Of 2D and 3DSection 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions (Concepts and fundas for IITJEE and other competitive exams) Chapter 1 : Identity, Division ,Symmetry In Math Page 5Hint : Put 3bx ta= in the equation and get the cubic equation reduced to 30 t pt q + + = ..(1)where 2233ac bpa= and 3 232 9 2727b abc a dqa += ..(2)Does this removal of square term help in solving the cubic equation ? It is not apparent on first sight. Any playing with (1) sooner or later leads back to the original cubic equation, back to square one. But Cardano decomposed the new variable t into t u v = + which changed equation (2) into ( )( )3 33 0 u v uv p u v q + + + + + = .(3)Could we abruptly put ( )( ) 3uv p u v + + equal to 0. Why not ? Who deters us? ( If we are to put a symbol at the point of difficulty, and there are three unknown roots to search, Cardano might have been motivated to put two symbols instead of one!)Since the number of choices of u and v are infinite as t u v = + and since we are not actually bothered by the relation between u and v, if there be any; we incur no loss assuming a relation between u and v which would make3 03puv p uv + = = (4)with a view just to simplify eqn (3) into 3 30 u v q + + = .(5).And lo, we have got two equations(4) and (5) in u and v and we can immediately solve them.Of course it is a quadratic equation 33 3 3 3,27pu v q and u v + = = in 3u and 3v ,i.e., 3u and 3vbecome the roots of the equation 32027pz qz + = and we get2 332 4 27q q pu = + + and 2 332 4 27q q pv = + (6)And we have not touched the holy cow!Now just retracing our contrivances u v t + = and 3bx ta= , we land ashore quickly,Page 27 / 681Series : REDISCOVER MATHEMATICS FROM 0 AND 1 PART 1:THE SWORD AND THE SHIELDBook : Calculus And Analytic Geometry Of 2D and 3DSection 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions (Concepts and fundas for IITJEE and other competitive exams) Chapter 1 : Identity, Division ,Symmetry In Math Page 62 3 2 33 32 3 2 33 32 4 27 2 4 27 33 2 4 27 2 4 27q q p q q p bu v t xab q q p q q pxa+ = + + + + = = + = + + + + +Only a small assumption 2 304 27q p+ > gives us a real number solution of the cubic equation. We would know later on that any cubic equation has at least a real root. But dont be swayed away with this tempting, small and alluring example which reveals mathematics being so easy. Of course it is, the way you look at it is important.Example2 ; Infinite Sequences and Series :Another example in difficulty identification or use of equating of coefficients : to find out the sum of a series like the following;) A .( .......... .......... terms n ).... 1 n ( n .. upto .......... 4 x 3 3 x 2 2 x 1 tn + + + = Difficulty here is that we do not know the sum of the series. But we think that it must involve n and its powers and some constant, of course, independent of n . So let us assume (like defining symbols as we often do) ,) B ....( .......... .......... .......... .......... .......... Dn Cn Bn A t 3 2n + + + = The tr, the r-th term, or the general term for that matter, may be written as r(r + 1).(Had all of them been equal, we could have simply multiplied n with any term to get the sum. That makes the sum one degree higher in n, than the degree of n in tn. So it is safe to assume that, if n = 6, we dont require more than 7 terms in the series for Etn . So we have taken terms up to n3in (B) when t n has no other power of n larger than 2. Similar must be the things for n+1 terms too; then,2 31 ( 1) ( 1) ( 1) ...............................................( )nt A B n C n D n C+ = + + + + + +1 1 2 2 3 3 4.......... ..( 1)( 2).......................................( )nt x x x upto n n D+ = + + + +Subtracting(B) from (C), and (A) from (D) , we get,) 1 n 3 n 3 ( D ) 1 n 2 ( C B ) 2 n )( 1 n ( t t t 21 n n 1 n + + + + + = + + = = + +) E ( .......... .......... )......... D 3 ( n ) D 3 C 2 ( n ) D C B ( n n 3 2 2 2+ + + + + = + + Equating co-efficients of similar powers of n from both sides, we get,Page 28 / 681Series : REDISCOVER MATHEMATICS FROM 0 AND 1 PART 1:THE SWORD AND THE SHIELDBook : Calculus And Analytic Geometry Of 2D and 3DSection 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions (Concepts and fundas for IITJEE and other competitive exams) Chapter 1 : Identity, Division ,Symmetry In Math Page 73 2 B ... and ,... 1 C ,... 3 1 D 1 D 3 .. and ,... 3 D 3 C 2 , 2 D C B= = = = = + = + +With these values and putting n = 1 (or any integer you like) in (B), which is also an identity, we get, = + + + = = = 0 A .. Or ,.. 3 1 1 3 2 A 2 x 1 t t 1 13 ) 2 n )( 1 n ( n3 n n 3 2n t ..3nn3n 2t 2n32n + +=||.|

\| + += + + = This is not certainly an universal technique for tackling any series whatsoever. But it gives a little bit of that feeling.Exercise 2:Try for expression for sum of these series:1) Sum of 1st n natural numbers, Etn, tn=n, or in other words ( ) 12n nn +=2) Sum of squares of 1st n natural numbers, Etn, tn=n2 or in other words( )( )2 1 2 16n n nn + +=.3) Sum of cubes of 1stn natural numbers, Etn, tn=n3 or ( )223 14n nn +=.Not only we use symbols to avoid difficulties, often we use them to take their advantage. Take the example of Trigonometric ratios; these symbols have been aimed at measurement of heights and distances, but have gone a long way in development of complex numbers, theory of equations and so on. In a similar manner Calculus is developed to deal with infinitesimal numbers (infinitely small numbers) by assigning symbols to them .The process or method is, to identify the difficulty or pinpoint it, adopt some symbol to enclose the difficulty, then proceed with known and standard methods until the symbol reveals itself or until the difficulty vanishes otherwise and toss away the symbol mercilessly. Wait. We can throw the symbol out in a particular problem, or well keep the symbol for future use if it has general importance, like \( - 1) = i or like log28 = 3 for 23= 8 or sin 1 = 300 for sin 300= ., or a symbol for eliminant of equations such as matrices and determinants which find much use elsewhere. We shall return to the subject.Throughout the book series, we have adopted this method to develop a topic, adopting a symbol tacitly carrying its entire properties and characteristics; may it be Matrices and Determinants, Trigonometric functions, Inverse Circular Functions, Logarithms, Limits, Conic Page 29 / 681Series : REDISCOVER MATHEMATICS FROM 0 AND 1 PART 1:THE SWORD AND THE SHIELDBook : Calculus And Analytic Geometry Of 2D and 3DSection 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions (Concepts and fundas for IITJEE and other competitive exams) Chapter 1 : Identity, Division ,Symmetry In Math Page 8sections, Differential Coefficients, Integrals etc. etc. Due to this approach the topics appear in a manner how they were discovered and developed rather than a formal presentation.There are numerous examples throughout the book where we would be using this technique to rediscover and redevelop many topics from identities. Before that the reader may try some identities from high school some of which are given below. The reader can try as many of them as possible and at ease. Use the fact that if a = b put throughout the expression makes it 0, then a b is a factor; similarly if a = - b put throughout the expression makes it 0, then a + b is a factor.4) (a b) + (b c) + (c a) = 0;5) c(a b) + a(b c) + b(c a) = 06) c(a b)3+ a(b c)3+ b(c a)3= (a + b + c)(a b)(b c)(c a)7) c(a b)2+ a(b c)2+ b(c a)2+ 8abc = (a + b)(b + c)(c + a)8) c4(a2 b2) + a4(b2 c2) + b4(c2 a2) = - (a b)(b c)(c a)(a + b)(b + c)(c + a)9) (b - c)3(b + c 2a) + (c - a)3(c + a 2b) + (c - a)3(c + a 2b) = 0 10) (b - c)(b + c 2a)3+ (c - a)(c + a 2b)3+ (c - a)(c + a 2b)3= 0 11) (ab c2)(ac b2) + (bc a2)(ba c2) +(bc a2)(ba c2) = bc(bc a2) + ca(ca b2) + ca(ca b2) 12) bc(b c) + ca(c a) + ab(a b) = - (b c)(c a)(a b) 13) a2(b c) + b2(c a) + c2(a b) = - (b c)(c a)(a b) 14) a(b2c2) + b(c2a2) + c(a2b2) = - (b c)(c a)(a b) 15) a3(b c) + b3(c a) + c3(a b) = - (b c)(c a)(a b)(a + b + c) 16) a3+ b3+ c3 3abc = (a + b + c)(a2+ b2+ c2 bc ca ab)17) a3+ b3+ c3 3abc = (a + b + c)(b2 ca + c2 ab + a2 bc)18) a3+ b3+ c3 3abc = (a + b + c)[(b c)2+ (c a)2+ (a b)2]19) (b c)3+ (c a)3+ (a b) 3 3(b c)(c a)(a b) = 020) (a + b + c)3= Ea3+ 3Ea2b + 6abc21) (a + b + c + d)3= Ea3+ 3Ea2b + 6Eabc22) bc(b +c) + ca(c +a) + ab(a +b) + 2abc = (b + c)(c + a)(a + b) 23) a2(b +c) + b2(c +a) + c2(a +b) + 2abc = (b + c)(c + a)(a + b) 24) (b + c)(c + a)(a + b) + abc = (a + b + c)( bc + ca + ab)25) (a + b + c) (-a + b + c) (a - b + c) (a + b - c) = 2b2c2+ 2c2a2+ 2a2b2 a4 b4 c426) c (a4 b4) + a (b4 c4) + b (c4 a4) = (b c)(c a)(a b)(a2+ b2+ c2 bc ca ab)27) (a + b)5= a5+ b5+ 5ab(a + b)( a2+ ab + b2)28) (a + b + c)5= a5+ b5+ c5+ 5(a + b) (b + c)(c + a)( a2+ b2+ c2+ bc + ca + ab)29) c2(a3 b3) + a2(b3 c3) + b2(c3 a3) = (ab + bc + ca)(a b)(b c)(c a)30) bc(c2 b2) + ca(a2 c2) + ab(b2 a2) = (b c)(c a)(a b)(a + b + c)31) (b +c){(r + p)(x + y) (p + q)(z + x)}+ (c +a){(p + q)(y + z) (q + r)(x + y)}+ (c +a){(p + q)(y + z) (q + r)(x + y)} = 2[a(qz ry) + b(rx pz) + c(py qx)]32) (a x)2{(b y)2(c z)2- (b z)2(c y)2}+(a x)2{(b y)2(c z)2- (b z)2(c y)2} +(a x)2{(b y)2(c z)2- (b z)2(c y)2} = 2(b c) (c a) (a b) (y z) (z x) (x y)and so on and so forth.Page 30 / 681Series : REDISCOVER MATHEMATICS FROM 0 AND 1 PART 1:THE SWORD AND THE SHIELDBook : Calculus And Analytic Geometry Of 2D and 3DSection 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions (Concepts and fundas for IITJEE and other competitive exams) Chapter 1 : Identity, Division ,Symmetry In Math Page 9All Mathematical formulae read in high school are identities or conditional identities and we know their utility. As examples of some conditional identities put a + b + c = 0 in expressions above where a + b + c appears: the result may be taken as a conditional identity.If a + b + c = 0, prove the following : 33) 2bc = a2 b2 c234) 8a2b2c2= (a2 b2 c2)( b2 c2 a2)( c2 a2b2)35) a3+ b3+ c3= 3abc36) 2(a4+ b4+ c4) = (a2+ b2+ c2)237) 3a2b2c2 2(bc + ca + ab)3= a6+ b6+ c638) a5+ b5+ c5+ 5abc(bc + ca + ab)39)7 c b a5 c b a2 c b a 7 7 7 5 5 5 2 2 2+ +=+ +=+ +40) |.|

\| ++= |.|

\|++ c b aba ca c b/ 3 3 /b a ca c bc ba2:Symmetry , Anti-symmetry And Asymmetry Are some Aspects Of Beauty.A look at the previous examples of identities makes us think about symmetry anti-symmetry and cyclic symmetry of the expressions. Discoverer of electrical generator must have observed the change in electric field due to motion of chares, i.e., electric current causes a magnetic field; and it must have occurred to him that a change in magnetic field may generate electric current. Take the previous example of equalizing coefficients of similar terms from both sides of an identity equation. This reveals the beautiful feature of symmetry. Take the algebrical identities in the previous exercises. The hint is to put a = b in the expression. If the expression reduces to 0, then (a b) is a factor. This is cyclic symmetry. Analyse any identity and you find some symmetry. Symmetry reveals the things that are not explicit. It also helps us to write expressions in brief. For example, Ea stands for a + b if two elements are taken; it stands for a + b + c or a + b + c + d if 3 or 4 elements are taken. Similarly Ea2stands for a2+ b2or a2+ b2+ c2or a2+ b2+ c2+ d2if 2 or 3 or 4 elements are taken. The expression a2+ b2is symmetrical with respect to a and b in a sense that a and b can be replaced with each other without affecting the value of the expression. The feature may be termed bilateral symmetry. The expressions such as a2+ b2+ c2or bc + ca + ab are bilaterally symmetrical, as any two of them can be interchanged without changing the value of the expression. In addition, the latter expressions are of cyclic symmetry; i.e., if a is replaced by b, b is replaced by c and c is replaced by a simultaneously, the expression is unchanged.To illustrate the method of applying symmetry concept in working out problems, consider factorizing the expression(a + b + c)5- a5- b5- c5. Page 31 / 681Series : REDISCOVER MATHEMATICS FROM 0 AND 1 PART 1:THE SWORD AND THE SHIELDBook : Calculus And Analytic Geometry Of 2D and 3DSection 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions (Concepts and fundas for IITJEE and other competitive exams) Chapter 1 : Identity, Division ,Symmetry In Math Page 10The value of the expression is unchanged if we put b in place of a , c in place of b and a in place of c. But the expression becomes 0 if b = - a throughout. Hence b + a or a + b must be a factor of it. Remember this is due to factor theorem read in high school. Similarly b + c and c + a must be factors of it. As such the expression contains (b + c)(c + a)(a + b) as a factor. The latter factor is of third degree whereas the expression to be factorised is of 5thdegree. So it contains another factor of 2nddegree. As the expression is symmetric in a, b and c; so also all its factors must be symmetric cyclically. A general expression in three elements and in 2nddegree would be A(a2+ b2+ c2) + B(bc + ca + ab) where A and B have to be determined. So we have complete factorization as (a + b + c)5- a5- b5- c5= (b + c)(c + a)(a + b)[A (a2+ b2+ c2)+ B(bc + ca + ab)]Now since this is an identity, the expression holds for any value of a, b and c. Putting each equal to 1 and each equal to 2 in turn we get, A + B = 10 and 5A + 2B = 35.Solving the equations, we get, the values of A and B , both equal to 5 and the complete factorization becomes,(a + b + c)5- a5- b5- c5=5 (b + c)(c + a)(a + b)(a2+ b2+ c2+ bc + ca + ab)An expression such as a2(b c) + b2(c a) + c2(a b) is changed to[ a2(b c) + b2(c a) + c2(a b)], i.e., its own negative when any two of its variables are interchanged with each other. Such an expression is said to be alternating or anti-symmetric.More about symmetry and its uses shall be discussed from topic to topic later on; especially in transformation of graphs. Puzzle: In finding out factors of 3 3 33 a b c abc + + one observes that the expression reduces to 0 if a b c = = is put into it. But neither of the expressions , , a b b c c a is a factor of it. Explain how.ans. assuming stand alone expression like a b = etc, does not make the expression 0, it becomes 0 only when all of a b c = = is assumed. So it indicates, neither of , , a b b c c a is a factor of the expression 3 3 33 a b c abc + + , but a factor of the expression involves some combination of all of , , a b b c c a at the same time! observe that ( ) ( ) ( ) ( ){ }2 2 23 3 3 132a b c abc a b c a b b c c a + + = + + + + Page 32 / 681Series : REDISCOVER MATHEMATICS FROM 0 AND 1 PART 1:THE SWORD AND THE SHIELDBook : Calculus And Analytic Geometry Of 2D and 3DSection 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions (Concepts and fundas for IITJEE and other competitive exams) Chapter 1 : Identity, Division ,Symmetry In Math Page 11Some other aspects of beauty are continuity, completeness, compactness, connectedness, convergence and uniformity. Examples shall follow throughout the book. The concept of continuity of functions shall be discussed in Calculus in a later chapter. Convergence concept shall be discussed in the chapters for sequences a, series and limits. While watching a movie or drama we note a touching sequence of events and keep guessing what should happen at last. If the end comes of our expectation we feel continuity in the story line. If the end of the story keeps us guessing still , we must feel something lacking in the story, e.g., there may not be an end to the drama and it may not be said to be complete. What happens in this case is a sequence of chosen events leads to a limiting event, a point which is not included in the story; As such the sequence of events is not continuous and the story is not complete. The concepts as such, are better illustrated in Topology, which is a set of some subsets called open subsets with a structure closed under arbitrary unions and finite intersections. The topic of topology is an attempt to provide a common platform to Algebra, Analysis, Differential equations, etc. etc.The principle of equivalence in mechanics , as propounded by Newton, states that the laws of mechanics are same with respect to all inertial frames; i.e. , they do not change if we change frames of reference with a new one moving at a constant velocity with the old one. Anillustration would be given at the appropriate place +. It would be shown that acceleration of somebody measured in one frame of reference will just be same in be same as measured in a different frame of reference moving at a constant velocity from the initial frame of reference taken. It would be just childs play and the reader even might have done the derivation is high school. Einstein derived the epoch making theory of relativity only from two assumptions; one : the principle of equivalence with only one word changed ; he wrote Physics in place of Mechanics. The second assumption is that the velocity of light in empty space does not depend on ( not added to nor subtracted from) the velocity of its source. The latter assumption is nothing but wise acceptance of failure to observe the expected result in the famous Michelson Morley experiment to measure absolute velocity of earth . ( Doesnt it seem that the theory of relativity was derived from the very antithesis of relativity ?)The conservation laws in Physics like conservation of mass, conservation of energy, conservation of momentum, conservation of angular momentum, conservation of spin etc. tell us +for those who refuse to wait until then. Let us measure acceleration a of some object while we stand still on the ground. Acceleration is t v va 0 1 = , where v1, v0are its final and initial velocities and t is time taken for this change of velocity. If we observe the same from object from a train with velocity u, do not observe the initial and final velocities, but observe the initial and final relative velocities instead , v1u and v0 u . Now acceleration observed from the train is at v vt ) u v ( ) u v ( 0 1 0 1== , again, which proves the proposition. One can change all these symbols except for time to vectors and prove the proposition in case of vector velocities too. Page 33 / 681Series : REDISCOVER MATHEMATICS FROM 0 AND 1 PART 1:THE SWORD AND THE SHIELDBook : Calculus And Analytic Geometry Of 2D and 3DSection 1: Identity, Division ,Symmetry Set Theory, Numbers And Functions (Concepts and fundas for IITJEE and other competitive exams) Chapter 1 : Identity, Division ,Symmetry In Math Page 12that the totalities of quantities like mass, momentum, energy, spin etc before an event like collision or explosion remains unchanged after the collision or explosion etc., as the event may be. So total mass, momentum, energy etc. are invariants .The concept is s