Parametric Curves: Descartes' Folium

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Parametric Curves: Introduction

Covers parametric curves, including examples, symmetry, tangent vectors, and derivative signs.

Curves with Poritsky Property and Liouville Nets

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

Parametric Curves: Lissajous and Folium

Explores parametric curves like Lissajous and the Folium of Descartes, discussing tangents, periodicity, and symmetry.

Curves in Space: Parametric Equations and Surfaces

Covers the equations for curves and surfaces in space, including parametric and particular surfaces.

Curvilinear Integrals

Covers curvilinear integrals in R^n, focusing on continuous functions and curves.

Surfaces in Space

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

Tangent spaces to submanifolds

Introduces tangent spaces to submanifolds and their properties in optimization on manifolds.

Tangent vectors without embedding space: Making tangent spaces linear

Explores making tangent spaces linear, defining tangent vectors without an embedding space and their operations, as well as the equivalence of different tangent space notions.

Differentiability of Functions of Several Variables

Covers the differentiability of functions of multiple variables and the significance of directional derivatives and gradients.

Length of Curves and Tangent Vectors

Covers the calculation of the length of curves and the properties of tangent vectors.