Differentiating vector fields: Why do it?

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Explores covariant derivatives along curves and second-order optimality conditions in vector fields and manifolds.

Explores the definition of connections for smooth vector fields on manifolds.

Explores Taylor expansions of first order in optimization on manifolds.

Introduces differentiating vector fields along curves on manifolds with connections and the unique operator satisfying specific properties.

Covers retractions, tangent bundles, and embedded submanifolds on manifolds with proofs and examples.

Explores necessary and sufficient optimality conditions for local minima on manifolds, focusing on second-order critical points.

Covers the topology of Riemann surfaces, focusing on orientation and orientability.

Explores connections on manifolds, emphasizing the axiomatic definition and properties of derivatives in differentiating vector fields.

Covers smooth maps, vector fields, and retractions on manifolds, emphasizing the importance of smoothly varying curves.

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