Exponential Map and Curvature Tensor

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Linear Shell Theory: Equilibrium Equations

Covers the dimensional reduction of strain energy from 3D to 2D and linear shell theory equilibrium equations.

Parallel Transport and Geodesics

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

Geodesics and Parallel Transport

Explores geodesics, parallel transport, and the Riemann tensor on two-dimensional manifolds, emphasizing fundamental concepts in differential geometry.

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 and Christoffel Symbols

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

Comparing Tangent Vectors: Parallel Transport

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

Riemannian connections

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

Hodge Duality and Covariant Derivatives

Introduces Hodge duality, covariant derivatives, and key concepts in differential geometry.

Riemann Tensor and Ricci Scalar

Covers the properties of the Riemann tensor and Ricci scalar in normal coordinates.

Symmetries and Conservation Laws

Explores Killing vectors, conserved quantities, symmetric spaces, and the Riemann curvature tensor properties.