Parametric Curves: Introduction

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Directional Derivatives

Explores directional derivatives in two-variable functions and extremum points.

Differential Geometry: Curves

Explores the geometry of parametric curves, covering tangent vectors, curvature, and curve smoothing techniques.

Curvilinear Integrals: Tangent Vectors and Oriented Arcs

Explains the tangent vectors and curvilinear integrals along oriented arcs.

From embedded to general manifolds: Why?

Explores upgrading foundations from embedded to general manifolds in optimization, discussing smooth sets and tangent vectors.

Envelope of Curves

Explores the concept of envelope of curves within families of lines, including implicit equations, super-ellipses, and curvature.

Regular Curves: Parametrization and Tangent Vectors

Explores regular curves, examples like segments and functions, and curvilinear integrals along regular curves.

Curvature and Oriented Curvature

Explores the concepts of curvature, oriented curvature, and inflection points in regular curves.

Staggered Bosons: Symmetry and Gaplessness

Explores staggered bosons, lattice realizations of gapless field theories with symmetry protection, commutation relations, and criticality.

Surfaces in R^3: Curves and Regular Surfaces

Covers curves in R^2, regular surfaces in R^3, and geometric properties of edges.

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.