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UPSC Civil Services Main Examination 2014 Syllabus: Mathematics Paper-I

(1)  Linear Algebra: Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimension; Linear transformations, rank and nullity, matrix of a linear transformation.
Algebra of Matrices; Row and column reduction, Echelon form, congruence’s and similarity; Rank of a matrix; Inverse of a matrix; Solution of system of linear equations; Eigen values and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem, Symmetric, skew-symmetric, Hermitian, skew-Hermitian, orthogonal and unitary matrices and their Eigen values.


(2) Calculus: Real numbers, functions of a real variable, limits, continuity, differentiability, mean-value theorem, Taylor'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, Lagrange's method of multipliers, Jacobian.

Riemann's definition of definite integrals; Indefinite integrals; Infinite and improper integrals; Double and triple integrals (evaluation techniques only); Areas, surface and volumes.

(3) Analytic Geometry: 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.

(4) 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, Clairaut's equation, singular solution.
Second and higher order linear equations with constant coefficients, complementary function, particular integral and general solution.
Second order linear equations with variable coefficients, Euler-Cauchy equation; Determination of complete solution when one solution is known using method of variation of parameters.
Laplace and Inverse Laplace transforms and their properties; Laplace transforms of elementary functions. Application to initial value problems for 2nd order linear equations with constant coefficients.

(5) Dynamics & Statics: Rectilinear motion, simple harmonic motion, motion in a plane, projectiles; constrained motion; Work and energy, conservation of energy; Kepler'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, equilibrium of forces in three dimensions.

(6) 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-Frenet’s formulae.
Gauss and Stokes’ theorems, Green’s identities.

UPSC Civil Services Examination 2014 Syllabus of more subjects browse from the links below:

UPSC Civil Services Main Examination 2014 Syllabus: Zoology Paper-II
UPSC Civil Services Examination 2014 Syllabus: Zoology Paper-I
UPSC Civil Services Main Examination 2014 Syllabus: Statistics Paper-II
UPSC Civil Services Main Examination 2014 Syllabus: Statistics Paper-I
UPSC Civil Services Main Examination 2014 Syllabus: Sociology-II
UPSC Civil Services Main Examination 2014 Syllabus: Sociology Paper-I
UPSC Civil Services Main Examination 2014 Syllabus: Public Administration Paper-II
UPSC Civil Services Main Examination 2014 Syllabus: Public Administration Paper-I
UPSC Civil Services Main Examination 2014 Syllabus Psychology-II
UPSC Civil Services Main Examination 2014 Syllabus: Psychology Paper-I
UPSC Civil Services Main Examination 2014 Syllabus Political Science and International Relations Paper-I
UPSC Civil Services Examination 2014 Syllabus: Political Science and International Relations (PS&IR) Paper-II
UPSC Civil Services Main Examination 2014 Syllabus: Physics Paper-II
UPSC Civil Services Main Examination 2014 Syllabus: Physics Paper-I
UPSC Civil Services Main Examination 2014 Syllabus: Philosophy Paper-II
UPSC Civil Services Main Examination 2014 Syllabus: Philosophy Paper-I
UPSC Civil Services Main Examination 2014 Syllabus: Medical Science-II
UPSC Civil Services Main Examination 2014 Syllabus: Medical Science Paper-I
UPSC Civil Services Main Examination 2014 Syllabus: Mechanical Engineering Paper-II
UPSC Civil Services Main Examination 2014 Syllabus: Mechanical Engineering Paper-I
UPSC Civil Services Main Examination 2014 Syllabus: Mathematics Paper-II
UPSC Civil Services Main Examination 2014 Syllabus: Mathematics Paper-I
UPSC Civil Services Main Examination 2014 Syllabus: Management Paper-II
UPSC Civil Services Examination 2014 Syllabus: Management Paper-I
UPSC Civil Services Main Examination 2014 Syllabus: Urdu Paper-II
UPSC Civil Services Main Examination 2014 Syllabus: Urdu Paper-I
UPSC Civil Services Main Examination 2014 Syllabus: Telugu Paper-II
UPSC Civil Services Main Examination 2014 Syllabus: Telugu Paper-I
UPSC Civil Services Main Examination 2014 Syllabus: Tamil Paper-II
UPSC Civil Services Examination 2014 Syllabus: Tamil Paper-I
UPSC Civil Services Main Examination 2014 Syllabus: Sindhi Paper-II
UPSC Civil Services Main Examination 2014 Syllabus: Sindhi Paper-I
UPSC Civil Services Main Examination 2014 Syllabus: Sindhi Paper-I
UPSC Civil Services Main Examination 2014 Syllabus: Santhali Paper-II
UPSC Civil Services Examination 2014 Syllabus: Santhali Paper-I
UPSC Civil Services Examination 2014 Syllabus: Sanskrit Paper-II
UPSC Civil Services Main Examination 2014 Syllabus: Sanskrit Paper-I
UPSC Civil Services Main Examination 2014 Syllabus: Punjabi Paper-II
UPSC Civil Services Main Examination 2014 Syllabus: Punjabi Paper-I
UPSC Civil Services Main Examination Syllabus: Oriya Paper-II
UPSC Civil Services Main Examination 2014 Syllabus: Oriya Paper-I
UPSC Civil Services Main Examination 2014 Syllabus: Nepali Paper-II
UPSC Civil Services Main Examination 2014: Nepali Paper-I
UPSC Civil Services Main Examination 2014 Syllabus Marathi Paper-II
UPSC Civil Services Main Examination 2014 Syllabus Marathi Paper-I
UPSC Civil Services Main Examination 2014 Syllabus Manipuri Paper-II
UPSC Civil Services Examination 2014 Syllabus Manipuri Paper-I
UPSC Civil Services Examination 2014 Syllabus: Malyalam Paper-II
UPSC Civil Services Main Examination 2014 Syllabus Malayalam Paper-I
UPSC Civil Services Main Examination 2014 Syllabus Maithili-II
UPSC Civil Services Main Examination 2014 Syllabus Maithili-I
UPSC Civil Services Main Examination 2014 Syllabus Konkani-II
UPSC Civil Services Main Examination 2014 Syllabus Konkani-I
UPSC Civil Services Main Examination 2014 Syllabus Kashmiri-II
UPSC Civil Service Main Examination 2014 Syllabus Kashmiri-I
UPSC Civil Services Main Examination 2014 Syllabus Kannada-II
UPSC Civil Services Main Examination 2014 Syllabus Kannada-I
UPSC Civil Services Main Examination 2014 Syllabus Hindi-II
UPSC Civil Services Main Examination 2014 Syllabus Hindi-I
UPSC Civil Services Main Examination 2014 Syllabus Gugarati-II
UPSC Civil Services Main Examination 2014 Syllabus Gujarati Paper I
UPSC Civil Services Main Examination 2014 Syllabus English-II
UPSC Civil Services Examination 2014 Syllabus English - I
UPSC Civil Services Main Examination 2014 Syllabus: Dogri-II
UPSC Civil Services Main Examination 2014 Syllabus Dogri Paper-I
UPSC Civil Services Main Examination 2014 Syllabus: Bodo-II
UPSC Civil Services Main Examination 2014 Syllabus: Bodo-I
UPSC Civil Services Main Examination 2014 Syllabus: Bengali Paper-II
UPSC Civil Services Main Examination 2014 Syllabus: Bengali Paper I
UPSC Civil Services Main Exam 2014 Syllabus: Assamese-II
UPSC Civil Services Main Exam 2014 Syllabus: Assamese-I
UPSC Civil Services Main Examination 2014 Syllabus: Law-II
UPSC Civil Services Main Examination 2014 Syllabus: Law-I
UPSC Civil Services Main Examination 2014 Syllabus: History-II
UPSC Civil Services Main Examination 2014 Syllabus: History-I
UPSC Civil Services Main Examination 2014 Syllabus: Geology-II
UPSC Civil Services Main Examination 2014 Syllabus: Geology-I
UPSC Civil Services Main Examination 2014 Syllabus: Geography-II
UPSC Civil Service Main Exam Syllabus 2014: Geography-I
UPSC Civil Services Main Examination 2014 Syllabus Electrical Engineering-II
UPSC Civil Services Main Examination 2014 Syllabus Electrical Engineering-I
UPSC Civil Services Main Exam 2014 Syllabus Paper VII: Economics-II
UPSC Civil Services Main Exam 2014 Syllabus: Paper VI: Economics-I
UPSC Civil Services Main Examination 2014 Syllabus Paper VII: Commerce and Accountancy-II
UPSC Civil Services Main Examination 2014 Syllabus Paper VI: Commerce and Accountancy-I
UPSC Civil Services Main Examination 2014 Syllabus Paper VII: Civil Engineering-II
UPSC Civil Services Main Examination 2014 Syllabus Paper VI: Civil Engineering-I
UPSC Civil Services Main Examination 2014 Syllabus Paper VII: Chemistry-II
UPSC Civil Services Main Examination Syllabus Paper VI: Chemistry-I
UPSC Civil Services Main Examination 2014 Syllabus Paper VII: Botany-II
UPSC Civil Services Main Examination 2014 Syllabus Paper VI: Botany-I
UPSC Civil Services Main Examination 2014 Syllabus Paper VII: Anthropology-II
UPSC Civil Services Main Examination 2014: Syllabus Paper VI: Anthropology-I
UPSC Civil Services Main Examination 2014 Syllabus Paper VII
UPSC Civil Services Main Examination 2014: Syllabus Paper VI
UPSC Civil Services Main Examination 2014: Syllabus Paper VII: Agriculture-II
UPSC Civil Services Main Examination 2014: Syllabus Paper VI: Agriculture-I
UPSC Civil Services Main Examination 2014 Syllabus: Paper-V: General Studies-IV
UPSC Civil Services Main Examination 2014 Syllabus: Paper-IV: General Studies-III
UPSC Civil Services Main Examination 2014 Syllabus: Paper-III: General Studies-II
UPSC Civil Services Main Examination 2014 Syllabus: Paper-II: General Studies-I
UPSC Civil Services Main Examination 2014 Syllabus: Paper-I: Essay
UPSC Civil Services Main Examination 2014 Syllabus: Indian Language and English

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