Differentiating vector fields along curves: Examples and finite differences

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Symmetry Property: Riemannian Connection in Geometry

Explores symmetries, Riemannian connection, vector fields, and Lie bracket in geometry.

Manifolds and Tangent Space

Introduces manifolds, charts, atlases, tangent space, tensors, and the metric in curved spaces.

Differential Geometry of Surfaces

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

Acceleration and geodesics

Explains acceleration along curves and geodesics on manifolds, generalizing straight lines to spheres.

Riemannian metrics and gradients: Examples and Riemannian submanifolds

Explores Riemannian metrics on manifolds and the concept of Riemannian submanifolds in Euclidean spaces.

Differentiation and Coordinate Changes in Multivariable Functions

Discusses differentiation of multivariable functions and coordinate transformations, including polar and cylindrical coordinates, along with the Laplacian operator and its applications.

Covariant Derivatives and Christoffel Symbols

Covers accelerated and inertial coordinate systems, Jacobian, volume elements, covariant derivatives, Christoffel symbols, Lorentz case, and metric tensor properties.

Differentiability and Partial Derivatives

Explores differentiability in two variables and the chain rule for compositions.

Riemannian metrics and gradients: Computing gradients from extensions

Explores computing gradients on Riemannian manifolds through extensions and retractions, emphasizing orthogonal projectors and smooth extensions.

Differentiating Vector Fields: How Not to Do It

Discusses the challenges in differentiating vector fields on submanifolds and the importance of choosing the right method.