**UPSC Math Syllabus 2019 (UPSC IAS Mains Maths Syllabus and also Hindi Books, Notes and Question Papers) :** Download UPSC Maths Syllabus in PDF for the upcoming year updated as per recent official notifications from the Central UPSC IAS Exams Board. Maths is the study of topics such as quantity (numbers), structure, space, and change. The overall UPSC Maths Syllabus for 2019 has been officially released recently and we have provided below the updated details on that. With Proper preparation, UPSC Maths Syllabus can be easily taken over to the perfection.

**Critical Details on UPSC Maths Syllabus 2019**

Maths Scope | 2 Sections |

UPSC Maths Syllabus | Download |

Maths Subtopics | Fifteen |

Preparation Time | 815 Hours |

**UPSC Maths Syllabus Criticals**

Note First and foremost, you must talk to someone who has taken Maths as an optional earlier and take their advice. There has been a common misconception among the students that UPSC Maths Syllabus is a hard nut to crack and its better to leave it as an optional paper. You are absolutely wrong.

Even through the Maths Syllabus portion in UPSC looks very huge and hard to study, once you start reading through few chapters, you will come to an absolute conclusion that UPSC Maths Syllabus is one of the most interesting and easiest of all the papers in entire UPSC Syllabus.

**Section A : UPSC Maths Syllabus**

**Calculus : UPSC Maths Syllabus**

- Real numbers, functions of a real variable, limits, continuity, differentiability, meanvalue theorem
- Curve tracing; Functions of two or three variables: limits, continuity, partial derivatives, maxima and minima
- Taylor’s theorem with remainders, indeterminate forms, maxima and minima, asymptotes
- Lagrange’s method of multipliers, Jacobian.
- Riemann’s definition of definite integrals
- Indefinite integrals
- Infinite and improper integrals
- Double and triple integrals
- Areas, surface and volumes

**Linear Algebra : UPSC Maths Syllabus**

- Vector spaces over R and C
- Algebra of Matrices
- Row and column reduction
- Rank of a matrix
- Inverse of a matrix
- igenvalues and eigenvectors
- Solution of system of linear equations
- Hamilton theorem, Symmetric, skew-symmetric
- Hermitian, skew-Hermitian, orthogonal and unitary matrices and their eigenvalues

**Analytic Geometry :** ** UPSC Maths Syllabus**

- 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, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties

**Ordinary Differential Equations : UPSC Maths Syllabus**

- Formulation of differential equations; Equations of first order and first degree, integrating factor
- Second order linear equations with variable coefficients, Euler-Cauchy equation
- 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
- Application to initial value problems for 2nd order linear equations with constant coefficients
- Laplace and Inverse Laplace transforms and their properties; Laplace transforms of elementary functions

**Vector Analysis :** ** UPSC Maths Syllabus**

- 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-Frenet’s formulae
- Gauss and Stokes theorems
- Green’s identities

**Section B : UPSC Maths Syllabus**

**Real Analysis : UPSC Maths Syllabus**

- 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 of real and complex terms
- 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

**Algebra : UPSC Maths Syllabus**

- Groups, subgroups, cyclic groups, cosets, Lagrange’s Theorem, normal subgroups
- Cayley’s theorem
- Rings, subrings and ideals, homomorphisms of rings
- Integral domains, principal ideal domains
- Euclidean domains and unique factorization domains; Fields, quotient fields

**Real Analysis : UPSC Maths Syllabus**

- 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 of real and complex terms
- 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

**Linear Programming :** ** UPSC Maths Syllabus**

- Linear programming problems, basic solution, basic feasible solution and optimal solution
- Graphical method and simplex method of solutions
- Duality. Transportation and assignment problems

**Partial differential equations :** ** UPSC Maths Syllabus**

- 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

**Mechanics and Fluid Dynamics :** ** UPSC Maths Syllabus**

- Generalized coordinates
- D’Alembert’s principle and Lagrange’s equations; Hamilton equations
- Moment of inertia; Motion of rigid bodies in two dimensions. Equation of continuity
- Euler’s equation of motion for inviscid flow
- Stream-lines, path of a particle
- Potential flow
- Two-dimensional and axisymmetric motion; Sources and sinks, vortex motion
- Navier-Stokes equation for a viscous fluid

**Mathematics Syllabus Tips from Ranker**

It is highly possible to clear UPSC Mains exam when taking Mathematics as optional but it is possible in single situation if Mathematics is your favorite subject or you are best in subject and you are familiar with the subject

Mathematics could prove to be very high scoring and if you have read it in grad,it will take you less time to cover the syllabus and in turn the revisions as well.

Go through the UPSC Mathematics syllabus and check if you’re well-versed with all the topics mentioned in the syllabus.

Mathematics is a good subject I too had the same subject and was very successful inspite of not joining coaching.

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**Conclusion on UPSC Mains Mathematics Syllabus**

It is always said that the wise men words are often neglected. However, we would like to again stress the fact that the Mathematics Syllabus in UPSC is one of the most important and crucial things that has to be properly taken care of.

Not just for examination point of view, Mathematics Syllabus in UPSC helps you in the overal development to become a good responsible officer who will play a crucial role in the development of the country in the decades to come.

Mathematics can be a good optional if you are extremely brilliant at it but i do have a few problems. Mathematicsis not going to help an ias in any possible way.

There has been a news circulating that, UPSC may remove optionals in coming years.mind you its not a rumor but actual happening in near future.

If readers have any further questions on UPSC Mathematics Syllabus, you can drop us a mail or even post your question below on our comments section.