Differential geometry and complex analysis 1984/85, 1985/86. For 3rd year mathematics students. Official name: Istituzioni di Geometria superiore.

DIFFERENTIAL GEOMETRY Complex numbers n-th roots The complex numbers as matrices Differentiable functions of several real variables Curve integrals Surfaces and hypersurfaces Classical tensor calculus Tensor product of modules Tensor algebra Exterior algebra Determinants from exterior algebra Alternating differential forms Rotation and divergence General transformation formulas for integrals Differentiable manifolds Differentiable mappings between manifolds Directional derivative in R^d Tangent space The tangent bundle as a differentiable variety Tangent vectors as derivations Tangent vectors as elements of (Ann/Ann^2)' Tangent vectors as tangent vectors of curves Vector fields and sections Integral curves of a vector field Differential forms on a variety Mapping and derivation of differential forms The simplicial Stokes theorem Stokes' theorem for chains HOLOMORPHIC FUNCTIONS Cauchy's integral theorem Cauchy's integral formula Power series in the complex field Every holomorphic function is analytic First applications The identity theorem Isolated singular points and Laurent series Theorem of residues Applications of the theorem of residues Harmonic functions