Maths

	Chapter	Topics	Time
1.	Real Analysis + Calculus	Sequence	28 March
		Sequence and limit, limit point	28 March
		Cauchy Sequence, Cauchy first and second theorem	28 March
		Monotone bounded sequence, cezaro th	29 March
		Series tests	29 March
		Alternating series	29 March
		Arbitrary term series	29 March
		Uniform convergence	30 March
		Rearrangement of term series	31 March
		Continuity and type of discontinuities	1 April
		Dirichlet , thomae function continuity	2 April
		Uniform continuity	3 April
		Functional equations, maxima minima, increasing decreasing fns of one variable	4 April
		Riemann integral	5 April
		Calculus theorems	6 April
		Improper integral	7 April
		Functions of two variable	8 April
		Differentiability, partial derivatives, mixed derivatives	9 April
		Extreme values of functions of several variables	10 April
		Lagrange’s method of undetermined multipliers, MVTs, Differentiation under integral ISign	11 April
		Leibnitz rule and Multiple integral	12 April
		Asymptotes, Curve tracing, Riemann Re-arrangement	13 April
2.	Linear Algebra	Matrices, Echelon form, reduced EF, Normal form, equivalent matrices	14 April
		Gauss Jordan/Gauss elimination method to get inverse of matrix, System of linear equations	14 April
		Non homogenous linear eq , Eigenvalues /eigenvectors, Similar matrices	15 April
		All type of Diagonalization , similarity	15 April
		Cayley-Hamilton Theorem, Partition of matrices, unitarily similar	16 April
		Vector space and subspace	16 April
		Algebra of subspaces, Intersection , union, basis, Dimension, Span, LI vectors, Linear sum of subpaces.	17 April
		Theorems on Linear dependence of vectors	17 April
		Theorems on Linear dependence ,  Quotient space	18 April
		Linear transformation, Nilpotent transformation, Inverse of transformation, Matrix of Transformation	18 April
		Change of basis matrix, Matrix of LT	18 April
3.	Modern Algebra	Group theory- QSM-Groups, Cayley Table
		Subgroup
		Order of element
		Cyclic Group
		Permutation group
		Coset
		HK, normal subgroup, index
		Quotient group
		Homomorphism
		Class equation
		Cauchy theorem, correspondence  theorem,  direct products
		Introduction to Rings
		Integral domain, division ring, field
		Subrings
		Ideal, Factor Ring
		Prime and maximal ideal
		Homomorphism of rings
		Divisibility of rings
		PID related theorems
		ED,  irred. tests
		Irr tests,Ufd, misc
4.	ODE	Formulation of DE, variable separable , Homogenous DE
		Linear coeff DE, exact DE, integrating factor
		Linear DE
		Normal and tangent ,  orthogonal trajectory
		First order but not first degree
		Solvable for y , Lagranges DE, Solvable for x
		Clairaut DE
		Tac Node cusp, Linear DE with constant coefficient
		CF and PI
		Vexp(ax),Cauchy Euler, Legendre, Variation of parameter first order
		Wronskian , uniqueness theorem, variation of parameter
		Reduction of order Normal method Change of independent variable
		Laplace Transform
		Laplace use to solve ODE, Simultaneous DE, Undetermined coefficient method, Variation of parameter 3rd order
5.	PDE	Formulation of PDE
		Lagrange’s DE, Integral Surface, Orthogonal Surface
		Homogenous Linear PDE with constant coefficient
		Non Homogenous LPDE with constant coefficients -reducible plus irreducible
		Charpit method non linear first order pde
		Cauchy’s characteristics for 1st order and second order PDE
		Canonical form
		Complete integral from one to another
		Wave equation
		2D/3D wave eq Laplace equation
		Heat eqn
		Polar form laplace , Jacobi method
7.	Complex Analysis	Introduction to complex numbers, Limit and continuity of function
		Analytic function, CR equation, Milne thomson, Harmonic Conjugate
		Complex integration, cauchy fundamental theorem
		Cauchy Integral formula, posson’s , gauss MVT, Liouvillen theorem, cauchy inequality , exterior theorem
		Taylor and Laurent series expansion
		Power series
		Expansion + analytic based Qs,
		Singularities
		Identity theorem , Argument Principle, Residue, Cauchy ‘s residue theorem
		Contour integral
		Misc contour, Misc topics, Rouche theorem
8.	Vector Analysis	Vectors, Multiple vector products
		Gradient, Normal vector, level surface
		Vector identities , div, curl
		Serret frenet
		Line integral, Surface integral
		Green Theorem, Gauss Divergence theorem
		Stoke theorem
9.	Analytical Geometry	DCs, DRs etc
		Plane
		Lines and skew lines
		Cone
		Cylinder
		Enveloping Cylinder
		Sphere
		Paraboloid
		Central Conicoid
		Generating lines
		Canonical Form
10.	Mechanics	Moment of Inertia
		D’almbert  Principle
		Lagrangian mechanics
		Charged Particle Lagrangian, Motion abt fixed axis
		Hamiltonian
		Rolling plus sliding
11.	Fluid Mechanics	Streamline pathlines streaklines
		Potential flow, boundary surface, continuity eq
		Curvilinear coordinate system
		Continuity eq in curvilinear coordinates
		Euler eq of motion
		Euler under impulse, energy equation
		Bernoulli method, source in 3D, motion of spheres ,axi-symmetric motion
		Sources ,sinks, doublet, complex potential
		Vortex motion
		Navier stoke eq
		Blasius , image system , transformation
12.	Dynamics	Projectile
		Constrained motion
		Motion in 2D
		Central orbit
		Kepler’s law
		Rectilinear motion (SHM)
13.	Statics	Common Catenary
		Virtual work
		Stability of equilibrium
		MN theorem, Forces in 2D and 3D
14.	LPP
15.	Numerical Analysis