Course topics

  • Basic concepts of topology of metric spaces: open and closed sets, connectedness, compactness, completeness.
  • Normed spaces and inner product spaces. All norms on $\mathbb{R}^n$ are equivalent.
  • Theorem on existence of a unique fixed point for a contraction mapping on a complete metric space.
  • Differentiability of a map between Euclidean spaces. Partial derivatives. Gradient. Chain rule. Multivariable Taylor expansion.
  • Open mapping theorem and implicit function theorem. Lagrange multipliers. Maxima and minima problems.
  • Riemann integral. Subsets of zero measure and the Lebesgue integrability criterion. Jordan content.
  • Fubini theorem. Jacobian and the change of variables formula.
  • Path integrals. Closed and exact forms. Green’s theorem.
  • Time permitting, surface integrals, Stokes’s theorem, Gauss’ theorem

Course Information

University course catalogue:
Advanced Undergraduate
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