Length of Curves and Tangent Vectors

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Explores the definition of curves, parameterized curves, and tangent vectors in relation to open intervals and continuous functions.

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

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

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

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

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

Explores parameterized curves, regularity, and geometric quantities like the carbide vector and curvature.

Explores curves in the oriented plane, discussing orientation, vector spaces, equivalence relations, and curvature of regular curves.

Explores meromorphic functions, poles, residues, orders, divisors, and the Riemann-Roch theorem.

Explores the derivative of curve lengths, fixed-end deformations, geodesics, surface point typologies, and sphere parametrization.