Real Analysis III

Syllabus

• Sequences and series of functions: pointwise vs uniform convergence; when limits preserve continuity, integrability, and differentiability.
• Uniform convergence criteria; Weierstrass M-test; term-by-term integration and differentiation for series of functions (conditions and examples).
• Power series: radius of convergence; Abel-type results; term-by-term differentiation and integration.
• Taylor series and real-analytic functions; binomial series; Taylor formula with Lagrange and integral remainder; applications (error bounds, numerical integration, limits, ODEs via power series).
• Multivariable calculus foundations: Euclidean spaces, coordinate systems (polar/cylindrical/spherical), inner products and Cauchy–Schwarz.
• Topology in ℝ^n: open/closed sets, limit/accumulation points, compactness; continuity and transfer principles; continuous functions on compact sets.
• Partial derivatives and total differentiability; tangent plane; directional derivatives; gradient; Jacobian matrix; chain rule.
• Higher derivatives, Taylor formula in several variables; quadratic forms and Hessian; local extrema and Hessian tests; constrained extrema and Lagrange multipliers.
• Implicit differentiation and the implicit function theorem (statement); extremum problems on compact domains; principal axes theorem for symmetric matrices (use in optimization).
• Vector fields: gradient fields and potentials; divergence and curl; Laplacian; interpretation in mechanics/fluids (overview).
• Line integrals and work; parameterization invariance; Newton–Leibniz for line integrals; characterizations of conservative fields; finding potentials.
• Multiple integrals: definition and properties; Fubini theorem; integration over normal domains; applications (areas/volumes, iterated integrals).
• Change of variables in 2D/3D (statements); polar/cylindrical/spherical substitutions; mass/centroid applications; Green’s theorem (statement and vector form).