Lectures related to Arc Length Approximation

Curvature and Osculating Circle

Covers curvature, osculating circles, and the evolute of plane curves, with examples and equations.

Curves in Space: Parametric Equations and Surfaces

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

Geometric Areas: Integrals and Regions

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

Curves and Surfaces

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

Surfaces in Space

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

Differential Geometry: Parametric Curves & Surfaces

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

Parametrized Curves in 2D: Examples and Properties

Explores parametrized curves in 2D, including segments, straight lines, and circles with trigonometric functions.

Parametrized Curves in 2D

Covers the concept of parametrized curves in 2D and their properties.

Residue Theorem: Calculating Integrals on Closed Curves

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

Vector Calculus: Line Integrals

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

Surface of Revolution

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

Generalized Integrals: Convergence and Divergence

Explores the convergence and divergence of generalized integrals using comparison methods and variable transformations.

Derivatives and Functions: Definitions and Interpretations

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

Parametric Equations: Benchmark of Observation

Explores parametric equations and their dependence on the choice of observation benchmarks.

Continuity and Limits

Covers continuity, limits, and different approaches to approaching points in various coordinate systems.

Regular Curves and Constant Speed

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

Curves with Poritsky Property and Liouville Nets

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

Calculus Applications: Lengths and Surfaces of Revolution

Discusses the applications of calculus in calculating lengths and surfaces of revolution, emphasizing integral calculus and geometric interpretations.

Geometric Curve Reparameterization

Covers the reparameterization of geometric curves and the concept of 'same geometric curve'.

Differential Geometry: Curves

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