Differentiating vector fields along curves: Examples and finite differences

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

Introduces Manopt, a toolbox for optimization on manifolds, focusing on solving optimization problems on smooth manifolds using the Matlab version.

Explores Riemannian connections on manifolds, emphasizing smoothness and compatibility with the metric.

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

Introduces tangent spaces to submanifolds and their properties in optimization on manifolds.

Explores comparing tangent vectors and parallel transport on manifolds.

Covers optimization on manifolds, focusing on smooth manifolds and functions, and the process of gradient descent.

Covers Hessians, symmetry, and examples related to vector fields, functions, and manifolds.

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

Covers the Grassmann manifold and retractions on submanifolds.