**Duration:**

**Number of lectures per week:** 3

**Assessment:** Some assignments, which do not contribute to the final grade

**End-of-year Examination:** One 3-hour end of year examination

**Description: **
Introduction to measures. Definition of Lebesgue integral on the real line.
General integration, monotone and dominated convergence theorems,
Fubini's Theorem.

Derivative as a linear operator for functions between
finite dimensional real vector spaces, partial derivatives in **R**^{n}, chain
rule, equality of mixed partials, criterion for differentiability.
tangent space, differentials, push-forward, pull-back,

inverse function theorem,
C^{n} functions, coordinate systems and partial derivatives, manifold,
implicit function theorem,
Exterior derivative, closed and exact forms, angle form, line integral,
change of variable in multiple integral.
integration of forms,

Functions of a complex variable, differentiability, contour integration. Cauchy's integral formula, Taylor and Laurent series, conformal property, zeros and poles, residues, evaluation of integrals, analytic continuation, maximum modulus principle.

Jun 10, 1998