Covariant derivative

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Differential geometry

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Related lectures (32)

Parallel Transport and Geodesics

Explores parallel transport along curves and geodesics maximizing proper time between points.

Mechanics of Slender Structure

Covers the expression of the Kirchhoff-St. Venant energy in a covariant setting and the equilibrium equations for spherical shells, among other topics.

Connections: motivation and definition

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

Differential Geometry of Surfaces

Covers the fundamentals of differential geometry of surfaces, including the equilibrium of shells, pressure vessels, and the curvature of surfaces.

Exponential Map and Curvature Tensor

Covers the exponential map, covariant derivatives, connection coefficients, and Riemann curvature tensor.

Functions with Multiple Variables

Covers functions with values in R^m, discussing limits, partial derivatives, and differentiability.

Comparing Tangent Vectors: Parallel Transport

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

Tensors and Lorentz Transformations

Covers tensors, Lorentz transformations, and electrodynamics in relativistic notation.

Directional Derivatives and Tangent Planes

Explores directional derivatives, tangent planes, and function differentiability in multivariable calculus.

Descent methods and line search: First Wolfe condition

Introduces the First Wolfe condition to ensure a proportional decrease in the objective function relative to the step length.

Adversarial Training: Mock Exam Solution

Discusses the solution to a mock exam problem on adversarial training and the uniqueness of solutions.

Shells I

Covers linear pressure vessels, thin shells, and critical buckling pressure, emphasizing the dimensional reduction from 3D to 2D.

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