# Best Books for UPSC Optional Maths π

The UPSC Exam, officially known as the Civil Services Examination (CSE), is a highly competitive nationwide test conducted by the Union Public Service Commission (UPSC) in India. This examination recruits officers for various civil services of the Government of India, including the Indian Administrative Service (IAS), Indian Foreign Service (IFS), and Indian Police Service (IPS). The exam is known for its high level of competition and rigorous selection process, consisting of three stages: UPSC Prelims, UPSC Mains and UPSC Interview.

## Best Books for UPSC Optional Maths βοΈπ

We have cherry-picked some of the best books for UPSC Optional Maths. If you are looking for the same, check out the table below-

## Tips to Prepare for UPSC Optional Maths for UPSC βοΈ

Preparing for UPSC Optional Maths demands a strategic and dedicated approach. Here are concise tips to guide your preparation:

1. Know the Syllabus
2. Have a Structured Study Plan
3. Choose Good Reference Books
4. Work on Foundational Knowledge
5. Regular Practice
6. Seek Guidance
7. Effective Revision
8. Time Management

## What are the Pros and Cons of Having UPSC Optional Maths?

Some pros and cons of having UPSC Optional Maths-

β Pros of UPSC Optional Maths

• High Scoring Potential.
• Less Competition.
• Logical and Analytical Approach.
• Distinct Profile.
• Transferable Skills.

β Cons of UPSC Optional Maths

• Time-Consuming Preparation.
• Limited Overlap with General Studies.
• No Marks for Attempting (Objective Questions).
• Requires Strong Mathematical Background.
• Limited Guidance and Resources.

## UPSC Optional Maths Syllabusπ

Hereβs the syllabus of UPSC Optional Maths-

Paper I:

1. Linear Algebra
• Vector spaces over R and C, linear dependence and independence.
• Subspaces, bases, dimensions, linear transformations.
• Rank and nullity, matrix of a linear transformation.
• Algebra of Matrices; Row and column reduction, Echelon form.
• Congruences and similarity; Rank of a matrix; Inverse of a matrix.
• Solution of a system of linear equations.
• Eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem.
• Symmetric, skew-symmetric, Hermitian, skew-Hermitian, orthogonal, and unitary matrices.
1. Calculus
• Real numbers, functions of a real variable, limits, continuity, differentiability.
• Riemannβs definition of definite integrals; Indefinite integrals.Riemannβs definition of definite integrals; Indefinite integrals.Riemannβs definition of definite integrals; Indefinite integrals.Riemannβs definition of definite integrals; Indefinite integrals.Mean-value theorem, Taylorβ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, Lagrangeβs method of multipliers, Jacobian.
• Riemannβs definition of definite integrals; Indefinite integrals.
• Areas, surface, and volumes.
1. Analytic Geometry
• Cartesian and polar coordinates in three dimensions.
• Second-degree equations in three variables, reduction to canonical forms.
• Straight lines are the shortest distance between two skew lines.
• Plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets.
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, Clairautβs equation, singular solution.
• Second and higher-order linear equations with constant coefficients.
• Complementary function, particular integral, and general solution.
• Section order linear equations with variable coefficients, Euler-Cauchy equation.
• Determination of a complete solution when one solution is known.
• Laplace and Inverse Laplace transforms, application to initial value problems.
1. Dynamics and Statics
• Rectilinear motion, simple harmonic motion, motion in a plane, projectiles.
• Constrained motion; Work and energy, conservation of energy; Keplerβ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 in three dimensions.
1. 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-Furenet’s formulae.

Also Read: How to Prepare for UPSC 2024 at Home?

Paper II:

1. Algebra
• Groups, subgroups, cyclic groups, cosets, Lagrangeβs Theorem, normal subgroups.
• Quotient groups, homomorphism of groups, basic isomorphism theorems.
• Permutation groups, Cayleyβs theorem. Rings, subrings and ideals.
• Homomorphisms of rings; Integral domains, principal ideal domains.
• Euclidean domains and unique factorization domains; Fields, quotient fields.
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.
• Series and its convergence, absolute and conditional convergence 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 variables, maxima and minima.
1. Complex Analysis
• Analytic function, Cauchy-Riemann equations, Cauchy’s theorem.
• Cauchy’s integral formula, power series, representation of an analytic function.
• Taylorβs series; Singularities; Laurentβs series; Cauchyβs residue theorem.
• Contour integration.
1. Linear Programming
• Linear programming problems, basic solutions, basic feasible solutions, and optimal solutions.
• Graphical method and simplex method of solutions; Duality.
• Transportation and assignment problems.
1. 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, Cauchyβ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.
1. Numerical Analysis and Computer Programming
• Numerical methods: Solution of algebraic and transcendental equations.
• Newtonβs and interpolation, Lagrangeβs interpolation.
• Numerical integration: Trapezoidal rule, Simpsonβs rule, Gaussian quadrature formula.
• Numerical solution of ordinary differential equations: Euler and Runge-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.

## FAQs

What are the recommended books for UPSC Optional Maths?

Some recommended books for UPSC Optional Maths include “Linear Algebra” by Kenneth Hoffman and Ray Kunze, “Real Analysis” by H.L. Royden, and “Introduction to the Theory of Numbers” by Ivan Niven.

Are there specific books for each topic within UPSC Optional Maths?

Yes, for Linear Algebra, “Linear Algebra” by Kenneth Hoffman and Ray Kunze is comprehensive. For Real Analysis, “Real Analysis” by H.L. Royden is widely used. Additional books like “Higher Algebra” by Hall and Knight can be beneficial for specific topics.

Should I rely on multiple books or stick to one for UPSC Optional Maths preparation?

It’s advisable to have a primary reference book for each topic. While multiple perspectives can be helpful, sticking to a few well-regarded books ensures in-depth understanding. Balance depth with coverage to manage preparation effectively.

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