# HPSC Maths Optional Syllabus

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### HPSC Maths Optional Syllabus

#### PART I HPSC Maths Optional Syllabus

1. Linear Algebra:

Vector space, linear dependence and independence, subspaces, bases, dimensions. Finite dimensional vector spaces. Matrices, Cayley Hamilton theorem, Eigen values and Eigenvectors. Matrix of linear transformation, row and column reduction, Echelon form, equivalence, congruence and similarity, reduction to canonical form, rank, orthogonal, symmetrical, skew symmetrical, unitary, Hermitian, skew Hermitian forms their Eigen values.  Orthogonal and unitary reduction of quadratic and Hermitian forms, positive definite quadratic form.

2. Calculus:

Real numbers, limits, continuity, differentiability, mean value theorems, Taylor

#### PART II HPSC Maths Optional Syllabus

1. Vector Analysis:

Scalar and vector fields, triple products. Differentiation of vector function of a scalar variable.   Gradient, Divergence and curl in cartesian, cylindrical and spherical coordinates and their physical interpretations.  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.

2. Real Analysis:

Real number system, ordered sets, Bounds, ordered field, real number system as an ordered field with least upper bound.  Cauchy sequence. Completeness. Continuous Functions. Uniform continuity. Properties of continuous functions on compact sets.  Riemann integral, improper integrals. Differentiation of functions of several variables, maxima and minima. absolute and conditional convergence of series of real and complex terms, rearrangement of series. Uniform convergence. Infinite Products. Continuity, differentiability and integrability for series.  Multiple integrals. Infinite and alternating series.

3. Numerical Analysis:

Numerical methods: Solution of algebraic and transcendental equations of one variable by bisection, Regula? Falsi and Newton?Raphson methods, solution of system of linear equations by Gaussian elimination and Gauss?Jordan (direct) methods, Gauss?Seidel(iterative) method.

Interpolations: Newton

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