Curvilinear Integrals: Tangent Vectors and Oriented Arcs

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Covers linear and membrane theories of pressure vessels, differential geometry of surfaces, and the reduction of dimensionality from 3D to 2D.

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

Covers implicit surfaces, parametric descriptions, regular parametrization, normal vectors, and orientable surfaces.

Explores the differential geometry of parametric surfaces, covering tangent space, normal curvature, principal curvatures, and asymptotic curves.

Explores regular surfaces and painting with normal vectors, integral calculations, symmetries, and surface orientation.

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

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

Covers linear pressure vessels and the basics of differential geometry of surfaces, including covariant and contravariant base vectors.

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

Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.