04-07-2012, 10:59 AM
maths basic
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CALCULUS
Basic properties of R, Absolute value, Intervals on the real line, Functions (Definition and examples), Inverse functions, Graphs of functions, Operations on functions, Composite of functions, Even and odd functions, Monotone functions, Periodic functions.
Definition of limits, Algebra of limits, Limits as (or ), One-sided limits, Continuity (Definitions and Examples, Algebra of continuous functions),
Definition of derivative of a function, Derivatives of some simple functions, Algebra of derivatives, The chain rule, Continuity versus derivability.
Derivatives of the various trigonometric functions, Derivatives of inverse function, The inverse function theorem, Derivatives of inverse trigonometric functions, Use of transformations. Derivative of exponential function, Logarithmic functions, Hyperbolic functions, Inverse hyperbolic functions, Methods of differentiation (Derivative of xr, Logarithmic Differentiation, Derivatives of functions defined in terms of a parameter, Derivatives of implicit functions).
Second and third order derivatives, nth order derivatives, Leibniz theorem, Taylor’s series and Maclaurin’s series Maxima-minima of functions (Definitions and examples, a necessary condition for the existence of extreme points), Mean value theorems (Rolle’s theorem, Lagrange’s mean value theorem), Sufficient conditions for the existence of extreme points (First derivative test, Second derivative test), Concavity/convexity, Points of inflection. Equation of tangents and normals, Angles of intersection of two curves, Tangents at the origin, Classifying singular points, Asymptotes (Parallel to the axes, Oblique asymptotes). Graphing a function, Tracing a curve (given its Cartesian equation, or in parametric form, or Polar equation).
Partitions of a closed interval, Upper and lower product sums, Upper and lower integrals, Definite integral, Fundamental theorem of calculus. Standard integrals, Algebra of integrals, Integration by substitution, Integrals using trigonometric formulas, Trigonometric and Hyperbolic substitutions, Two properties of definite integrals, Integration by parts, Evaluation of . Reduction formulas for and Integrals involving products of trigonometric functions (Integrand of the type ), Integrals involving hyperbolic functions. Integration of some simple rational functions, Partial fraction decomposition, Method of substitution, Integration of rational trigonometric functions, Integration of Irrational functions
Monotonic functions, Inequalities, Approximate value. Area under a curve (Cartesian equation, Polar equations), Area bounded by a closed curve, Numerical integration. (Trapezoidal rule, Simpson’s rule). Length of a plane curve (Cartesian form, Parametric form, Polar form), Volume of a solid of revolution, Area of surface of revolution.
Video Programme: Curves
LINEAR ALGEBRA
Sets, subsets, union and intersection of sets, Venn diagrams, Cartesian product, relations, functions, composition of functions, binary operations, fields. Plane and space vectors, addition and scalar multiplication of vectors, scalar product, orthonormal basis, vector equations of a line, plane and sphere. Definition and basic properties, subspaces, linear combination, algebra of subspaces, quotient spaces. Linear independence and some results about it, basic results about basis and dimension, completion of a linearly independent set to a basis, dimension of subspaces and quotient spaces.
Definitions and examples of linear transformation, kernel, range space, rank and nullity, homomorphism theorems. L (U, V), the dual space, composition of transformations, the minimal polynomial. Definition of a matrix, matrix associated to a linear transformation, the vector space Mmxn(F), transpose, conjugate, diagonal and triangular matrices, matrix multiplication, inverse of a matrix, matrix of a change of basis. Rank of a matrix, elementary operations, row-reduced echelon matrices, applying row reduction to obtain the inverse of a matrix and for solving a system of linear equations.
Definition and properties, product formula, matrix adjoint and its use for obtaining inverses, Cramer’s rule, determinant rank. Definition and how to obtain them, diagonalisation. Cayley-Hamilton theorem, minimal polynomial’s properties.
Definition, norm of a vector, orthogonality.
Linear functionals of inner product spaces, adjoint of an operator, self-adjoint and unitary operators, Hermitian and unitary matrices. Definitions, representation as matrix product, transformation under change of basis, rank of a form, orthogonal and normal canonical reductions Definitions, standard equations, description and some geometrical properties of an ellipse, a hyperbola and parabola, the general reduction.
Video Programme: Linear Transformations and Matrices
MATHEMATICAL METHODS
Sets, Equality of Sets, Operations on Sets, Venn diagrams, Functions, Types of Functions, Composite Functions, Operations with Functions. Graphs (Exponential and Logarithmic Functions, Trigonometric functions), Trigonometric ratios. Polynomials and Equations, Sequences and Series, Permutations and Combinations, Binomial Theorem.
Two Dimensional Coordinate System -Distance between Two points, Area of a Triangle, Equation of a Line, , Angle between Two Lines, Distance of a Point from a Line, Circle, Three Dimensional Coordinate System- Equation of a Straight Line in 3-D, The Plane, The Sphere. Vectors as directed line segments, Algebra of Vectors and their applications (addition and subtraction of vectors, resolution of vectors, dot and cross product).
Limit and Continuity, Derivative of a Function at a Point, Its Geometrical Significance, Rules for differentiation, differentiation of Trigonometric, Exponential and Logarithmic Functions, Differentiation of Inverse Algebraic and Inverse Trigonometric Functions, Chain Rule, Differentiation of Implicit Functions and Logarithmic Differentiation, Physical Aspects of Derivatives
Tangents and Normals, Higher Order Derivatives, Maxima and Minima, Asymptotes, Curve-Tracing, Functions of Two Variables, Partial Derivatives of Order Two, Homogeneous Functions, Euler’s Theorem. Antiderivatives, Integration as Inverse of Differentiation, Definite Integral as the Limit of the Sum, Properties of Definite Integrals, Fundamental Theorem of Integral Calculus Standard Integrals, Methods of Integration, Integration by Substitution, Integration by parts, Integration of Trigonometric Functions
Preliminaries, Formation of Differential Equations, Methods of Solving Differential Equations of First Order and First degree (Variables Separable, Homogeneous Equation, Exact Equations and Linear Equations).
Some Basic Definitions in Statistics, Frequency Distribution, Discrete Random Variables, Continuous Random Variables, Measures of Central Tendency and Dispersion (Mean, Mode, Median, Standard Deviation, Mean Deviation). Preliminaries: (Sample Space, Discrete Sample Space, Continuous Sample Space), Rules of Probability, Conditional Probability, Baye’s Theorem. Combination of events, Binomial Distribution, Poisson Distribution (Emphasis Through Illustrations). Continuous Random Variables, Types of Continuous Distributions (Exponential and Normal Distribution – Emphasis Through Illustrations)
Sample Selection, Random Sampling Procedure, Measure of Variation and Accuracy, Standard Error, Unbiased Estimator, Accuracy and Precision of Sample Estimator, Types of Sample Design (Random Sampling, Cluster Sampling).Statistical Hypothesis, Level of Significance, Degrees of Freedom, Chi-square Test, t-test, Analysis of Variance Correlation and scatter diagram, Correlation coefficient, Linear regression, Curve Fitting (Least Square Method).
Video Programme: 1) Sampling a case study
2) Sampling in Life Sciences
ELEMENTARY ALGEBRA
Definition and examples of sets and subsets, Venn diagrams, Complementation, Intersection, Union, Distributive laws, De Morgan’s laws, Cartesian product.
What a complex number is, Geometrical representation, Algebraic operations, De Moivre’s theorem, Trigonometric identities, Roots of a complex number.
Recall of solutions of linear & quadratic equations, Cubic equations (Cardano’s solution, Roots and their relation with coefficients), Biquadratic equations (Ferrari’s solution, Descartes’ solution, Roots and their relation with coefficients)
APPENDIX: Some mathematical symbols (Implication, two-way implication, for all, their exists), Some methods of proof (Direct proof, contrapositive proof, proof by contradiction, proof by counter-example).
Linear systems, Solving by substitution, Solving by elimination. Definition of a matrix, Determinants, Cramer’s rule. Inequalities known to the ancients (Inequality of the means, Triangle inequality), Less ancient inequalities (Cauchy-Schwarz inequality, Weierstrass’ inequalities, Tchebyshev’s inequalities)
ANALYTICAL GEOMETRY
Equations of a line, Symmetry, Change of axes (Translating the axes, rotating the axes), Polar coordinates. Focus-directrix property, Description of standard form of parabola, ellipse and hyperbola; Tangents and normals of parabola, ellipse, hyperbola; Polar equation of conics. General second degree equation, Central and non-central conics, tracing a conic (Central conics, Parabola), Tangents, Intersection of conics.
Points, Lines (Direction cosines, Equations of a straight line, Angle between two lines), Planes (Equations of a plane, Intersecting planes and lines). Equations of a sphere, Tangent lines and planes, Two intersecting spheres, Spheres through a given circle.
Cones, Tangent plane to a cone, Cylinders.
Definition of a conicoid, Change of axes (Translation of axes, projection, Rotation of Axes), Reduction to standard form. A conicoid’s centre, Classification of central conicoids, Ellipsoid, Hyperboloid of one sheet, Hyperboloid of two sheets, Intersection with a line or a plane.Standard equation, Tracing the paraboloids, Intersection with a line or a plane.
ABSTRACT ALGEBRA
Sets, Cartesian Product, Relations, Functions, Some number theory – Principle of induction and divisibility in Z. Binary operations, Definition of a group, Properties of a Group, Some details of Zn, Sn, C, and appendix on some properties of complex numbers. Subgroups and their properties, Cyclic groups. Cosets; Statement, proof and applications of Lagrange’s theorem.
Definition and standard properties of normal subgroups, Quotient groups. Definition and examples, Isomorphisms, Isomorphism theorems, Automorphisms. Definition, Examples, Cayley’s theorem. Direct product, Sylow theorems (without proof), Classifying groups of order 1 to 10.
Elementary properties, Examples of commutative and non-commutative rings and rings with and without identity. Definitions, Examples, Standard properties, Quotient Rings (in the context of commutative rings).
Definition and properties of integral domains, Fields, Prime and maximal ideals, Fields of quotients. Examples, Division Algorithm and Roots of Polynomials. Euclidean domain, PID, UFD. Eisenstein’s criterion, Prime fields, Finite fields
Video Programme: Groups of Symmetries
ADVANCED CALCULUS
The Extended Real Number System R (Arithmetic Operations in R , Bounds in R . Extension of Exponential and Logarithmic Functions to R ). The concept of infinite limits (infinite limits as the independent variable , One-sided Infinite Limits, Limits as the independent variable tends to or - , Algebra of limits).
Indeterminate Forms, L’Hopital’s rule for form (Simplest form of L’Hopital’s Rule, Another form of L’Hopital’s rule for form), L’Hoptal’s rule for form, other types of Indeterminate Forms (indeterminate forms of the type - , indeterminate forms of the type 0. , indeterminate forms of the type 00, 0,1 ). The Space Rn (Cartesian products, algebraic structure of Rn, Distance in Rn), Functions from Rn to Rm.
Limits of Real-Valued Functions, Continuity of Real-Valued Functions, Limit and Continuity of Functions from Rn Rm, Repeated limits. First Order Partial Derivatives (Definition and Examples, Geometric interpretation. Continuity and Partial Derivatives), Differentiability of Functions from R2 to R, Differentiability of functions from Rn R, n > 2. Higher Order Partial Derivatives, Equality of Mixed Partial Derivatives. Chain Rule, Homogeneous Functions, Directional Derivatives
Taylor’s Theorem (Taylor’s theorem for functions of one variables, Taylor’s theorem for functions of two variables), Maxima and Minima (local extrema, Second derivative test for local extrema), Lagrange’s multipliers; Jacobians (Definition and examples, Partial derivatives of Implicit Functions), Chain rule, Functional Dependence (Domains in Rn, Dependence). Implicit Function Theorem (Implicit Function Theorem for two variables, implicit Function Theorem for three variables), inverse Function Theorem.