Preliminaries to calculus : Motivation for calculus. Sets and set operations. Basic propositional logic, quantifiers. Functions and relations. Types of functions. Open and closed sets. Countable and uncountable sets. Axiomatization of real numbers. Well ordering principal. Principle of recursive definition. Principle of mathematical induction. Basic inequalities. Triangle inequality. Some important inequalities. Archimedean property.
Algebra and Basic Combinatorics : Quadratic equations. Polynomials. Polynomial division. Partial fractions. Introduction to complex numbers. Basic counting. Pigeonhole principal. Combinations and permutations. Binomial theorem. Binomial coefficients and their summations, identities. Floor function and ceiling function. Their properties. More complex numbers.
Sequences and Series : Definition of sequences and series. Supremum and infimum. Arithmetic progression. Geometric progression. Arithmetico-geometric progression. Tests of convergence. Radius of convergence. Trigonometrical series.
Limits : Intuitive description of limit. Epsilon delta definition. Limit laws. Limit theorems. Continuity. Mean value theorems.
Integral Calculus : Partitions. Step functions. Area function. Calculating integrals.
Differential Calculus : Rate of change. Derivatives. Formulas of derivatives. Fundamental theorem of calculus. Extremization problems. Introduction to logarithms and exponential functions. Trigonometric functions and their calculus.
Techniques of Integration : All the standard techniques along with some extra techniques such as Feynman’s rule, etc.
Vectors and Vector Calculus : Partial derivatives. Lagrange multipliers. Differential calculus of vector fields. Line integrals. Surface integrals. Multiple integrals. Contour integration (a brief introduction). All the crucial theorems.
Differential Equations : Linear differential equations of first and second order. Some special ODEs and techniques.
Trigonometry : all the basics. Relation to complex numbers. Hyperbolic functions.
Linear algebra : Linear equations. Matrices. Row operations. Reduced row echelon form. Invertibility of matrices. Determinants. Vector spaces. Linear transformations. Properties of determinants and matrices. Eigenvalues and eigenvectors. Eigenspaces. Cayley Hamilton theorem. Special types of matrices.
Probability Theory : Conditional probability. Random variables. Discrete distributions. Continuous distributions. Binomial and Poisson distributions. Normal distribution. Bernoulli trials. Random variables. Law of large numbers.
Genetics : Exploring some applications of probability in biology, with an emphasis on genetics.
Thermodynamics and Statistical Physics : Ideal gas equation. Heat. Temperature. Boltzmann distribution. Maxwell-Boltzmann distribution. Kinetic theory of gases. Pressure. Mean free path. Properties of gases. First law of thermodynamics. Second law of thermodynamics. Carnot’s theorem and Carnot engines. Entropy. Equipartition theorem. Third law.
Rotational Motion and Central Forces : Central force definition. Rutherford scattering. Kepler’s Laws. Gravity. Coulomb’s law. Angular momentum. Torque. Principle axes. Eular’s equations. Motion of a top. Coriolis force. Accelerated frames of reference.
Special Relativity : Postulates. Michelson Morley experiment. Basics of special relativity. Energy mass equivalence.
Waves : Simple harmonic motion. Forced oscillations. Resonance. Normal modes. Waves. Contiuum limit. Elementary Fourier analysis. Longitudinal oscillations. Sound. Travelling waves.
Electromagnetism and Classical Field Theory : Some history. Electric charge. Electric potential. Gauss’s Law. Laplace’s equation. Stokes’s theorem. Electric currents. Conductors. Magnetic field. Electromagnetic Induction. AC. Maxwell’s equations. Electromagnetic waves. Electric and magnetic fields in matter.
Optics and EM waves : Two and three dimensional waves. Spherical waves. Polarization. Interference and diffraction. Double slit experiment. Fermat’s principle. Ray optics. Refraction and reflection laws.