Weekly outline

  • LINEAR ALGEBRA

    Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimension; Linear transformations, rank and nullity, matrix of a linear transformation. Algebra of Matrices; Row and column reduction, Echelon form, congruence’s and similarity; Rank of a matrix; Inverse of a matrix; Solution of system of linear equations; Eigenvalues and eigenvectors, characteristic polynomial, Cayley- Hamilton theorem, Symmetric, skew symmetric, Hermitian, skew-Hermitian, orthogonal and unitary matrices and their eigenvalues.

    • Introduction to Matrices URL
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    • Rank of Matrix URL
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    • Linear Equations URL
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    • Revision of Matrices URL
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    • Determinant URL
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    • Vector Spaces URL
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    • Linear Equations – Vector Space Approach URL
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    • Basis and Dimensions URL
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    • Change of Basis URL
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    • Orthogonality URL
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    • Introduction to Eigenvalues and Eigenvectors URL
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    • Diagonalisation of Matrices and its Applications URL
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    • Properties of Eigenvalues URL
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    • Symmetric Matrices URL
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    • Cayley Hamilton Equation URL
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    • Similar Matrices URL
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    • Introduction to Linear Transformation URL
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    • Matrix of Linear Transformation URL
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  • CALCULUS

    Real numbers, functions of a real variable, limits, continuity, differentiability, mean-value theorem, Taylor’s theorem with remainders, indeterminate forms, maxima and minima, asymptotes; Curve tracing; Functions of two or three variables: limits, continuity, partial derivatives, maxima and minima, Lagrange’s method of multipliers, Jacobian. Riemann’s definition of definite integrals; Indefinite integrals; Infinite and improper integrals; Double and triple integrals (evaluation techniques only); Areas, surface and volumes.

    • Introduction to Double Integrals URL
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    • Change of Order URL
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    • Transformation of Variables URL
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    • Beta and Gamma Function URL
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    • Dirichlet Theorem URL
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    • Introduction and Problems on Volume Calculation URL
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    • Calculating Volumes Using Polar Coordinates URL
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    • centre of mass URL
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    • Moment of Inertia URL
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    • Volume of the Region Bounded by Surface of Revolution URL
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    • Surface Area URL
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    • Introduction to Limits URL
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    • Methods of Finding Limits URL
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    • Continuity URL
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    • Differentiability URL
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    • Monotonicity URL
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    • Critical Points URL
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    • Maxima and Minima URL
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    • Rolle Theorem URL
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    • Lagrange Mean Value Theorem URL
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    • Taylors Theorem URL
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    • Function of Two Variables URL
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    • Maxima and Minima of Function of Several Variables URL
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  • ANALYTIC GEOMETRY

     Cartesian and polar coordinates in three dimensions, second degree equations in three variables, reduction to canonical forms, straight lines, shortest distance between two skew lines; Plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.

  • ORDINARY DIFFERENTIAL EQUATION

    Formulation of differential equations; Equations of first order and first degree, integrating factor; Orthogonal trajectory; Equations of first order but not of first degree, Clairaut’s equation, singular solution. Second and higher order linear equations with constant coefficients, complementary function, particular integral and general solution. Second order linear equations with variable coefficients, Euler-Cauchy equation; Determination of complete solution when one solution is known using method of variation of parameters. Laplace and Inverse Laplace transforms and their properties; Laplace transforms of elementary functions. Application to initial value problems for 2nd order linear equations with constant coefficients.

    • Introduction to Differential Equations URL
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    • Differential Equation of First order and First Degree URL
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    • Orthogonal Trajectory URL
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    • Differential Equation with Constant Coefficient URL
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    • Homogeneous Linear Differential Equation URL
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    • Differential Equation of Second Order URL
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  • DYNAMICS AND STATICS

     Rectilinear motion, simple harmonic motion, motion in a plane, projectiles; constrained motion; Work and energy, conservation of energy; Kepler’s laws, orbits under central forces. Equilibrium of a system of particles; Work and potential energy, friction; common catenary; Principle of virtual work; Stability of equilibrium, equilibrium of forces in three dimensions.

  • VECTOR ANALYSIS

    Scalar and vector fields, differentiation of vector field of a scalar variable; Gradient, divergence and curl in Cartesian and cylindrical coordinates; higher order derivatives; Vector identities and vector equations. Application to geometry: Curves in space, Curvature and torsion; Serret-Frenet’s formulae. Gauss and Stokes’ theorems, Green’s identities.

    • Gradient URL
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    • Tangent Plane and Normal URL
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    • Problems on Gradient URL
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    • Divergence and Curl
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    • Line Integral URL
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    • Greens Theorem URL
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    • Surface Integral URL
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    • Gauss Divergence Theorem URL
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    • Stokes Theorem URL
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    • Conservative Vector Field URL
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  • ALGEBRA

    Groups, subgroups, cyclic groups, cosets, Lagrange’s Theorem, normal subgroups, quotient groups, homomorphism of groups, basic isomorphism theorems, permutation groups, Cayley’s theorem. Rings, subrings and ideals, homomorphisms of rings; Integral domains, principal ideal domains, Euclidean domains and unique factorization domains; Fields, quotient fields.

    • Defining Groups URL
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    • Modulo Arithmetic URL
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    • Cayley Table URL
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    • Unit Group URL
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    • Group of Matrices URL
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    • Dihedral Groups URL
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    • Subgroup URL
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    • Problems on Groups URL
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    • Cyclic Groups URL
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    • Permutation Group URL
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    • Isomorphism of Groups URL
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    • External Direct Product URL
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    • Homomorphism of Groups URL
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  • REAL ANALYSIS

    Real number system as an ordered field with least upper bound property; Sequences, limit of a sequence, Cauchy sequence, completeness of real line; Series and its convergence, absolute and conditional convergence of series of real and complex terms, rearrangement of series. Continuity and uniform continuity of functions, properties of continuous functions on compact sets. Riemann integral, improper integrals; Fundamental theorems of integral calculus. Uniform convergence, continuity, differentiability and integrability for sequences and series of functions; Partial derivatives of functions of several (two or three) variables, maxima and minima.

    • Set Theory URL
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    • Relation and Functions URL
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    • Countability of Sets URL
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    • Real Numbers URL
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    • Metric Space URL
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    • Open Sets URL
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    • Closed Sets URL
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    • Limit Points URL
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    • Compactness and Connectedness URL
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    • Introduction to Real Sequences URL
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    • Monotonic Sequence URL
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    • Cauchy Theorems on Limits URL
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    • Limit Points URL
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    • Subsequence URL
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    • Cauchy Sequence URL
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    • Introduction to Infinite Series URL
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    • Test of Convergence of Infinite Series URL
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    • Alternating Series URL
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    • Power Series URL
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  • COMPLEX ANALYSIS

     Analytic functions, Cauchy-Riemann equations, Cauchy’s theorem, Cauchy’s integral formula, power series representation of an analytic function, Taylor’s series; Singularities; Laurent’s series; Cauchy’s residue theorem; Contour integration.

  • LINEAR PROGRAMMING

     Linear programming problems, basic solution, basic feasible solution and optimal solution; Graphical method and simplex method of solutions; Duality. Transportation and assignment problems.

  • PARTIAL DIFFERENTIAL EQUATIONS

     Family of surfaces in three dimensions and formulation of partial differential equations; Solution of quasilinear partial differential equations of the first order, Cauchy’s method of characteristics; Linear partial differential equations of the second order with constant coefficients, canonical form; Equation of a vibrating string, heat equation, Laplace equation and their solutions.

  • NUMERICAL ANALYSIS AND COMPUTER PROGRAMMING

     Numerical methods: Solution of algebraic and transcendental equations of one variable by bisection, Regula-Falsi and Newton-Raphson methods; solution of system of linear equations by Gaussian elimination and Gauss-Jordan (direct), Gauss-Seidel(iterative) methods.
    Newton’s (forward and backward) interpolation, Lagrange’s interpolation.
    Numerical integration: Trapezoidal rule, Simpson’s rules, Gaussian quadrature formula.
    Numerical solution of ordinary differential equations: Euler and Runga Kutta-methods. Computer Programming: Binary system; Arithmetic and logical operations on numbers; Octal and Hexadecimal systems; Conversion to and from decimal systems; Algebra of binary numbers.
    Elements of computer systems and concept of memory; Basic logic gates and truth tables, Boolean algebra, normal forms.
    Representation of unsigned integers, signed integers and reals, double precision reals and long integers. Algorithms and flow charts for solving numerical analysis problems.

  • MECHANICS AND FLUID DYNAMICS

     Generalized coordinates; D’ Alembert’s principle and Lagrange’s equations; Hamilton equations; Moment of inertia; Motion of rigid bodies in two dimensions. Equation of continuity; Euler’s equation of motion for inviscid flow; Stream-lines, path of a particle; Potential flow; Two-dimensional and axisymmetric motion; Sources and sinks, vortex motion; Navier-Stokes equation for a viscous fluid.