Advanced Analysis II: Demonstration of Functions and Tangent Planes

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Explores the geometry of surfaces in R², including implicit descriptions, critical points, and regular surfaces.

Explores surfaces in space, including paraboloids, spheres, and hyperboloids, and their equations and intersections.

Introduces the basics of differential geometry for parametric curves and surfaces, covering curvature, tangent vectors, and surface optimization.

Explains the parametric equations of surfaces of revolution generated by curves in space.

Explores parametric curves, focusing on finding envelopes and caustics.

Explores the differential geometry of parametric surfaces, covering tangent space, normal curvature, principal curvatures, and asymptotic curves.

Discusses the Implicit Function Theorem and its application to tangent planes and derivatives.

Covers surface integrals with a focus on regular parametrization and the importance of understanding the normal vector.

Covers the calculation of angles between curves on regular surfaces and the concept of curvilinear abscissa.

Covers the concept of bounding the Poisson bracket invariant on surfaces, exploring joint work with A. Logunov and S. Tanny.