The Mathematics Optional Syllabus for UPSC Mains examination requires comprehensive preparation. This is difficult without two factors: understanding the paper pattern and understanding the syllabus. The mathematics optional subject carries a total weightage of 500 marks, which includes 2 papers (250 marks each).
Now, let us walk you through the complete Mathematics Optional Syllabus for UPSC Mains subject for the IAS entrance examination.
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Mathematics Optional Paper 1 Syllabus
The mathematics optional syllabus for the first paper includes the following 6 modules:
A. Linear Algebra
Vector spaces on R and C, linear dependence and independence, subspaces, basis, dimension, linear transformation, rank and nullity, matrix of a linear transformation. Algebra of matrices; Row and column reduction, echelon form, unity, and equality; Rank of the matrix; matrix inverse; Solution of systems of linear equations; Eigenvalues and eigenvectors, characteristic polynomials, Cayley-Hamilton theorem, symmetric, skew-symmetric, Hermitian, skew-Hermitian, orthogonal and unitary matrices, and their eigenvalues.
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B. Calculus
Real numbers, functions of real variables, limits, continuity, variation, mean-value theorem, Taylor’s theorem with remainder, indefinite forms, maximum and minimum, asymptotes; curve tracing; Functions of two or three variables; Limits, continuity, partial derivatives, maxima and minima, Lagrange’s multiplier method, Jacobian. Riemann’s definition of definite integrals; indefinite integrals; infinite and unreasonable integrals; Double and Triple Integrals (Evaluation Techniques Only); Area, surface, and volume.
C. Analytic Geometry
Cartesian and polar coordinates in three dimensions, equations of the second degree in three variables, reduction to canonical forms; Straight lines, minimum distance between two oblique lines, plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.
D. Ordinary Differential Equations
Formulation of differential equations; Equations of first order and first degree, integrating factors. orthogonal trajectory; Equations of first order but not first degree; Clairaut’s equation, singular solution. Second and higher-order linear equations with constant coefficients, complementary functions, special integrals and general solutions. Section order linear equations with variable coefficients, Euler-Cauchy equation. Determination of the complete solution when one solution is known using the method of change of parameters. Laplace and inverse Laplace transforms and their properties, Laplace transform of elementary functions. Application of initial value problems to second-order linear equations with constant coefficients.
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E. Dynamics and Statics
Rectilinear motion, simple harmonic motion, motion in a plane, projectile; hindered movement. Work and energy, conservation of energy; Kepler’s laws, orbits are subject to central forces. Equilibrium of a system of particles; Work and potential energy, friction, general catenary. Theory of Virtual Work; Stability of equilibrium, balance of forces in three dimensions.
F. Vector Analysis
Scalar and vector fields, differentiation of vector fields of scalar variables. Slope, divergence, and curl in Cartesian and cylindrical coordinates; Higher order derivatives; Vector identities and vector equations. Applications in Geometry: Curves, curvature and torsion in space; Serret-Fernet’s formula. Gauss’s and Stokes’ theorems, Green’s identity.
Mathematics Optional Paper 2 Syllabus
The UPSC mathematics optional paper syllabus for part 2 encloses the following 7 topics and many subtopics:
A. Algebra
Groups, subgroups, cyclic groups, cosets, Lagrange’s theorem, normal subgroups, quotient groups, isomorphism of groups, basic isomorphism theorem, permutation groups, Cayley’s theorem. Rings, subrings and ideals, isomorphism of rings; Integral domain, principal ideal domain, Euclidean domain, and unique factorization domain; Fields, quotient fields.
B. Real-Analysis
The real number system is an ordered field with the minimum upper bound property. Sequences, limits of sequences, Cauchy sequences, completeness of the real line. Series and its convergence, complete and conditional convergence of series of real and complex words, rearrangement of series. Continuity of functions and uniform continuity, Properties of continuous functions on compact sets. Riemann integral, improper integral; Fundamental theorem of integral calculus. Uniform convergence, continuity, divergence, and completeness for sequences and series of functions. Partial derivatives of functions of many (two or three) variables, maxima and minima.
C. Complex Analysis
Analytical functions, Cauchy-Riemann equation, Cauchy’s theorem, Cauchy’s integral formula, power series, representation of an analytical function. Taylor’s series; Singularities; Laurent’s series; Cauchy’s residue theorem; contour integration.
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D. Linear Programming
Linear programming problems, basic solutions, basic feasible solutions, and optimal solutions; Graphical method and simplex method of solution; Duality. Transportation and assignment problems.
E. Partial Differential Equations
Representation of families of surfaces and partial differential equations in three dimensions. Solution of first-order quasilinear partial differential equations, Cauchy’s method of characteristics. Linear partial differential equations of the second order with constant coefficients, canonical form. Equation of vibrating string, heat equation, Laplace equation, and their solutions.
F. Numerical Analysis and Computer Programming
Numerical methods: Solution of algebraic and transcendental equations of one variable by bifurcation, Regula-Falcy, and Newton-Raphson methods, solution of systems of linear equations by Gaussian elimination and Gauss-Jordan (direct), Gauss-Seidel (iterative) methods. Newton’s (forward and reverse) and interpolation, Lagrange’s interpolation. Numerical Integration: Trapezoidal rule, Simpson’s law, 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; Conversions to and from the decimal system; Algebra of binary numbers. Elements of computer systems and the concept of memory; Basic logic gates and truth tables, Boolean algebra, general forms. Representations of unsigned integers signed integers, and real, double-precision real, and long integers. Algorithms and flow charts to solve numerical analysis problems.
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G. Mechanics and Fluid Dynamics
Generalised coordinates; D’Alembert’s principle and Lagrange’s equations; Hamilton equation; moment of inertia; Motion of rigid bodies in two dimensions. Continuity equation; Euler’s equation of motion for inviscid flow; Streamline, the path of a particle; potential flow. Two-dimensional and axisymmetric motion; Sources and sinks, vortex motion; Navier–Stokes equations for viscous fluids.
Conclusion
Those were all the topics an aspirant will have to cover if they wish to crack the mathematics optional paper of UPSC. With a comprehension of this, students will have a better shot at scoring well on the exam!
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