Study Plan for UPSC Mathematics

22 Weeks Study Plan

Study PLAN to Finish Syllabus in 22 Weeks
Linear Algebra and Matrix:
Week 1, 2:
1Introduction :Algebra of Matrices; Row and column reduction, , congruence’s and similarity; Symmetric, skew-symmetric, Hermitian, skew-Hermitian, orthogonal and unitary matrices.
2Rank of a matrix, Echelon form, Normal Form, Inverse of a matrix
3Solution of system of linear equations
4Eigenvalues and eigenvectors, characteristic polynomial,
5Diagonisable, Cayley Hamilton Theorem. Quadratic Form, Index Signature
6Practice from Krishna Series – Eigen Values, Linear Eqns , Orthogonal
7Practice from Krishna Series – Quadratic, Similarity of Matrices, Diagonalisation
8Vector spaces Definition and problems
9Subspaces, Bases, Dimension
10Linear transformations, rank and nullity,
11LT Isomorphism, Singular, Inveribles
12matrix of a linear transformation, Bases Change
Numerical Analysis:
Week 3:
1Bisection, Newton Raphson, Regula falsi
2Newton Forward , Backward Tabulation, Derivation and Error of Lagrange Interpolation
3Derivation and Error finding of Trapezoid, Simpson
4ODE Euler, RK method
5Linear Equation, Gauss Jordan, Siedel., Errors
6Boolean algebra problems practise
7Algorithm and Flow charts
Vector Analysis
Week 4,
1Gradient, Divergence, Curl in all Cordinate system,
2Vector Identities Problems
3Line Integrals, Green Theorem, Surface Integrals
4Gauss Divergence, Stokes’s thm Problems,
5Curves in spaces, tangent, normal eqns ,
6Curvature , Torsion, Serret Ferret Formula
Week 5,6,
1Formation,Iintegarting factors
2Eqn of 1st Order
4P X Y Solvable
5Claurit, Singular Soln
6Linear Differential eqns with Const Coeff
7Cauchy Euler Eqn
8Method of Variation of parametrs
9Simultaneous Differential Eqn
10Linear Eqn Order 2
11Normal, Variable Transformation
12Laplace Transform
13Inverse Laplace Transform
14Application of Laplace to ODE
Partial Differential Eqns
Week 7,8
1Formation of PDE and examples
2Linear PDE Order 1
3Non Linear PDE ,Charpit method
4Standard form Solns
5Jacobi , Cauchy Strip
6Homogeneous Linear PDE const coeff
7Non Homogeneous PDE
8Euler Cauchy Methods
9Canonical Forms
10Boundary Value Problems, Heat Eqn
11Wave Eqn
13Polar Sysytem,
14Cylindrical, Spherical Syytem
Calculus and Real Analysis:
Week 9,10,11
1Limits , Continuity
2Differentiability and Problems
3Mean Value theorem
4Indeterminate form Problems
5, 6Max min- Lagrange and 2Dimension
8Asymptotes, Partial Derivative
9Riemann Definitions
10Theorem and Problems on Riemann
11,12Improper Integrals
13,14Sequences Basic
15,16Series Problems
17Uniform Convergence
18Uniform Convergence- Continuity, Differentiability, Integrability Problems
19Multiple Integrals
20Double and Triple integrals , Areas
21Surface , Volumes
22Beta Gamma Functions
23Integration under Differentiation sign
Complex Analysis:
Week 12,
1Analytic functions
2,3Complex Integral
4Laurent, Taylor Series
5Singularities, Power Series,Meromorphic functions
6,7Residue Theorem, Contour Integration
Analytic Geometry:
Week 13,14 :
1Intro, Direction Cosines, Projections
3Straight Lines
4Straight Lines
5Shortest Distance, Tetrahedron,
6Skew Lines,
9Cental Conicoid
10Central Conicoid
11Cental Conicoid
13Generating Lines
14Generating Lines
Linear Programming
Week 15
2Graphical Methods
3,4Simplex Methods Practise and Duality
5,6Assignment Problems
7,8Transportaion Problems
Week 16:
1Moment of inertia,
2Lagrange Problems 1
3Lagrange Problems 2
4Hamilton Problems 1
5Hamilton Problems 2
6D Alembert’s principle
7Motion of rigid bodies in two dimensions.(Selected problems only)
8Motion in 2D( Selected Problems Only)
Fluid Dynamics
Week 17,18 :
1Kinematics ,Lagrange , Velocity, Continuity
2Boundary Conditions ,
3Path Line, Velocity Potential
4Euler Eqns
5Impulse Motion, energy Conservation
6Bernaui Eqns
7Stream Fns, Source Sink
8Source Sink, Circle Images
9Irrotational Motion
10Navier Stoke Eqns
11Laminar Flow
Week 19,20
1Intro, Binary Operations, Group
2Subgroups and Problems
3Cyclic Subgroups
5Normal Subgroups and Problems
6Homomorphism and its theorems
7Permutation Group
8Sylow Theorem Problems
9Rings and Sub Rings
10Ideals , Quotient Rings
11Ring Homomorphism, Embedding
12Maximal Ideals, Prime Ideals
14UFD, Eisenstein’s criteria.
Here we will understand the basic concept thoroughly, Do Important Theorems and Important Problems only . This is a Ocean, so we will limit ourself with important things only.
Week 21
1Basic Concept, Rectilinear motion
4Plane Kinematics
5Central Orbits
6Planetary Motion
7Constrained motion in Plane
8Constrained motion in Circle
9Motion in Resisting Medium
Week 22
1,2Principle of Virtual work
3,4Stability , UnStability Problems
6,7Catenary Problems

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