Differentiating vector fields along curves: Examples and finite differences

About
Privacy
Disclaimer

Graph Chatbot

Related lectures (32)

Page 1 of 4

Differentiating Vector Fields: Definition

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

Comparing Tangent Vectors: Parallel Transport

Explores parallel transport along loops and covariant derivatives induced by metrics.

Shells I: Mechanics of Slender Structures

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

Covariant derivatives along curves

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

Riemannian connections: What they are and why we care

Covers Riemannian connections, emphasizing their properties and significance in geometry.

Comparing Tangent Vectors: Parallel Transport

Explores the definition, existence, and uniqueness of parallel transport of tangent vectors on manifolds.

Connections: motivation and definition

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

Geometrical Aspects of Differential Operators

Explores differential operators, regular curves, norms, and injective functions, addressing questions on curves' properties, norms, simplicity, and injectivity.

Taylor expansions: second order

Explores Taylor expansions and retractions on Riemannian manifolds, emphasizing second-order approximations and covariant derivatives.

All things Riemannian: metrics, (sub)manifolds and gradients

Covers the definition of retraction, open submanifolds, local defining functions, tangent spaces, and Riemannian metrics.