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Book–1 : (Vol. – 1) Matrices and Linear Algebra:
  • 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.
  • Book–1 : (Vol. – 2) Matrices and 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.
    Book–2 : (Vol – 1) 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
    Book–2 : (Vol – 2) Advance Calculus
    • Functions of two or three variables
    • Limits, continuity
    • Partial derivatives
    • Maxima and minima
    • Lagrange's method of multipliers
    • Jacobian
    Book–2 : (Vol – 3) Advance Calculus
    • Riemann's definition of definite integrals
    • Indefinite integrals
    • Infinite and improper integrals
    • Double and triple integrals (evaluation techniques only)
    • Areas, surface and volumes.
    Book–3 : (Vol – 1) 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
    Book–3 : (Vol – 2) Analytic Geometry
    • Paraboloid
    • Ellipsoid
    • Hyperboloid of one and two sheets and their properties.
    Book–4 : (Vol.–1) Ordinary Differential Equations
    • 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.
    Book–4 : (Vol.–2) Ordinary Differential Equations
    • 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.
    Book–5 : (Vol.-1) Dynamics
    • 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
    Book–5 : (Vol.-2) Statics
    • 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.
    Book–6 : 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.
    • Gauss and Stokes' theorems, Green's identities.
    Book–7 : Differential Geometry
    • Application to geometry
    • Curves in space, Curvature and torsion; Serret-Frenet's formulae.
    Book–8 : (Vol.-1) Modern Algebra (Group Theory)
    • Groups
    • Subgroups
    • Cyclic groups
    • Cosets, Lagrange's Theorem
    • Normal subgroups, quotient groups
    • Homomorphism of groups
    • Basic isomorphism theorems
    • Permutation groups, Cayley's theorem.
    Book–8 : (Vol.-2) Modern Algebra (Ring and Field Theory)
    • Rings
    • Subrings and ideals
    • Homomorphisms of rings
    • Integral domains
    • Principal ideal domains, Euclidean domains and unique factorization domains
    • Fields, quotient fields.
    Book–9 : (Vol.-1) 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
    Book–9 : (Vol.-2) Real Analysis
    • Series and its convergence
    • Absolute and conditional convergence of series of real and complex terms
    • Rearrangement of series.
    Book–9 : (Vol.-3) Real Analysis
    • 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.
    Book–10 : 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.
    Book–11 : Linear Programming:
    • Linear programming problems
    • Basic solution, basic feasible solution and optimal solution
    • Graphical method and simplex method of solutions
    • Duality
    • Transportation
    • Assignment problems.
    Book–12 : 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.
    Book–13 : Numerical Analysis
    • 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.