Mathematical Science Syllabus Paper 2 Unit 1 To 7

Mathematical Science Syllabus Paper 2 Unit 1 To 7

1. Real Analysis: Riemann integrable functions; improper integrals, their convergence and uniform convergence. Eulidean space R", Bolzano-Weierstrass theorem, compact Subsets of R", Heine-Borel theorem, Fourier series.

Continuity of functions on R", Differentiability of F: R"-> R^m, Properties of differential, partial and directional derivatives, continuously differentiable functions. Taylor's series. Inverse function theorem, Implicit function theorem.

Integral functions, line and surface integrals, Green's theorem, Stoke's theorem.

2. Complex Analysis: Cauchy's theorem for convex regions, Power series representation of Analytic functions. Liouville's theorem, Fundamental theorem of algebra, Riemann's theorem on removable singularities, maximum modulus principle, Schwarz lemma, Open Mapping theorem, Casoratti-Weierstrass-theorem, Weierstrass's theorem on uniform convergence on compact sets, Bilinear transformations, Multivalued Analytic Functions, Riemann Surfaces.

3. Algebra: Symmetric groups, alternating groups, Simple groups, Rings, Maximal Ideals, Prime Ideals, Integral domains, Euclidean domains, principal Ideal domains, Unique Factorisation domains, quotient fields, Finite fields, Algebra of Linear Transformations, Reduction of matrices to Canonical Forms, Inner Product Spaces, Orthogonality, quadratic Forms, Reduction of quadratic forms.

4. Advanced Analysis: Elements of Metric Spaces, Convergence, continuity, compactness, Connectedness, Weierstrass's approximation Theorem, Completeness, Bare category theorem, Labesgue measure, Labesgue Integral, Differentiation and Integration.

5. Advanced Algebra: Conjugate elements and class equations of finite groups, Sylow theorems, solvable groups, Jordan Holder Theorem, Direct Products, Structure Theorem for finite abelian groups, Chain conditions on Rings; Characteristic of Field, Field extensions, Elements of Galois theory, solvability by Radicals, Ruler and compass construction.

6. Functional Analysis: Banach Spaces, Hahn-Banach Theorem, Open mapping and closed Graph Theorems. Principle of Uniform boundedness, Boundedness and continuity of Linear Transformations, Dual Space, Embedding in the second dual, Hilbert Spaces, Projections. Orthonormal Basis, Riesz-representation theorem, Bessel's Inequality, parsaval's identity, self-adjoined operators, Normal Operators.

7. Topology: Elements of Topological Spaces, Continuity, Convergence, Homeomorphism, Compactness, Connectedness, Separation Axioms, First and Second Countability, Separability, Subspaces, Product Spaces, quotient spaces. Tychonoff's Theorem, Urysohn's Metrization theorem, Homotopy and Fundamental Group.