Real Analysis Syllabus (for MS and Preliminary Exams)

Functions of One Variable

  • Axioms for real numbers, sequences, infinite series, compact sets
  • Continuity, continuity and compactness, intermediate value theorem, differentiability, Rolle's theorem, mean value theorem, Taylor's theorem
  • Reimann integral, improper integrals
  • Uniform convergence of sequences and series of functions, interchange of limiting operations
  • Elementary functions
  • Functions of bounded variation

Functions of Several Variables

  • Directional derivatives
  • Differentiability
  • Chain rule
  • Inverse and implicit function theorems
  • Taylor's theorem
  • Change of variables in multiple integrals

Vector Analysis

  • Gradient, divergence and curl
  • Vector identities
  • Line, surface and volume integrals
  • Conservative fields
  • Gauss, Green and Stokes theorem
  • Orthogonal curvilinear coordinates


Ross: Elementary Analysis: The Theory of Calculus
Rudin: Principles of Mathematical Analysis
Buck: Advanced Calculus
Apostol: Mathematical Analysis