Green-Riemann Formula: Curvilinear Integrals

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Explores curvilinear integration in the complex plane, including regular curves, properties, examples, antiderivatives, Cauchy theorem, and integrability criteria.

Vector Calculus: Line Integrals

Covers the concept of line integrals and their application in vector fields.

Curves with Poritsky Property and Liouville Nets

Explores curves with Poritsky property, Birkhoff integrability, and Liouville nets in billiards.

Topology: Classification of Surfaces and Fundamental Groups

Discusses the classification of surfaces and their fundamental groups using the Seifert-van Kampen theorem and polygonal presentations.

Residue Theorem: Calculating Integrals on Closed Curves

Covers the application of the residue theorem in calculating integrals on closed curves in complex analysis.

Model Categories: Properties and Structures

Covers the properties and structures of model categories, focusing on factorizations, model structures, and homotopy of continuous maps.

Homotopy Theory in Care Complexes

Explores the construction of cylinder objects in chain complexes over a field, focusing on left homotopy and interval chain complexes.

Cellular Approximation: Homotopy and CW Complexes

Explores the cellular approximation theorem for CW complexes and its implications on homotopy groups.

Geometric Areas: Integrals and Regions

Covers the calculation of areas using integrals for geometric regions defined by curves and parametric equations.

The Whitehead Lemma: Homotopy Equivalence in Model Categories

Explores the Whitehead Lemma, showing when a morphism is a weak equivalence.