Lectures related to Curves: Parameterized Curves and Tangent Vectors

Differential Geometry: Parametric Curves & Surfaces

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

Curves and Surfaces

Covers the representation of plane curves and the nature of obtained curves.

Analysis Reminder: Open Sets and Denseness

Reviews open sets, denseness, real numbers, convergence, curves, continuity, and derivatives in analysis.

Curves in Space: Parametric Equations and Surfaces

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

Geometrical Aspects of Differential Operators

Explores differential operators, regular curves, norms, and injective functions, addressing questions on curves' properties, norms, simplicity, and injectivity.

Differentiability of Functions of Several Variables

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

Vector Calculus: Line Integrals

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

Parametric Curves: Descartes' Folium

Covers Descartes' Folium, a parametric curve with specific equations and properties, including domain boundaries and tangent vector analysis.

Tangent Vectors and Bézier Curves

Explains tangent vectors of curves and the construction and applications of Bézier curves in computer graphics.

Curves with Poritsky Property and Liouville Nets

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

Surfaces in R^3: Curves and Regular Surfaces

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

Curvilinear Integrals

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

Derivatives and Functions: Definitions and Interpretations

Covers the definition of derivative functions and their interpretation, focusing on differentiability, velocity, and curve lengths.

Parametric Curves: Descartes' Folium

Covers Descartes' Folium, a parametric curve with domain boundaries and tangent vector analysis.

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.

Advanced Analysis II: Isoclines Method

Explores the Isoclines Method and solutions of differential equations with tangents.

Length of Curves and Tangent Vectors

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

Curves and Isometries

Explores curves, isometries, Bezier curves, and the Frenet coordinate system.

Regular Curves and Constant Speed

Covers regular curves and constant speed, with examples of circles and helices.

Differential Geometry: Curves

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