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Exterior covariant derivative
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Covariant Derivatives and Christoffel Symbols
Covers accelerated and inertial coordinate systems, Jacobian, volume elements, covariant derivatives, Christoffel symbols, Lorentz case, and metric tensor properties.
Linear Shell Theory: Equilibrium and Energy
Covers the expression of the Kirchhoff-Saint Venant energy in a covariant setting and explores equilibrium equations for spherical shells and linear shell theory.
Linear Shell Theory: Equilibrium Equations
Covers the dimensional reduction of strain energy from 3D to 2D and linear shell theory equilibrium equations.
Differential Geometry of Surfaces
Covers linear pressure vessels and the basics of differential geometry of surfaces, including covariant and contravariant base vectors.
Hodge Duality and Covariant Derivatives
Introduces Hodge duality, covariant derivatives, and key 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.
Quantum Field Theory II: Maxwell Field and Symmetries
Covers the Maxwell field, symmetries, and making shifts in local and covariant derivatives.
Acceleration and geodesics
Explains acceleration along curves and geodesics on manifolds, generalizing straight lines to spheres.
Covariant derivatives along curves
Explores covariant derivatives along curves and second-order optimality conditions in vector fields and manifolds.
Differentiating Vector Fields: Definition
Introduces differentiating vector fields along curves on manifolds with connections and the unique operator satisfying specific properties.
Taylor expansions: second order
Explores Taylor expansions and retractions on Riemannian manifolds, emphasizing second-order approximations and covariant derivatives.
Parallel Transport and Geodesics
Explores parallel transport along curves and geodesics maximizing proper time between points.
Exponential Map and Curvature Tensor
Covers the exponential map, covariant derivatives, connection coefficients, and Riemann curvature tensor.
Comparing Tangent Vectors: Parallel Transport
Explores parallel transport along loops and covariant derivatives induced by metrics.
Covariant Maxwell Equations
Covers the covariant form of Maxwell equations and properties of the field strength tensor.
Differential Forms: Basics and Applications
Introduces the concept of differential forms and their applications in n-dimensional manifolds, including the Levi-Civita tensor and volume form.
Comparing Tangent Vectors: Parallel Transport
Explores the definition, existence, and uniqueness of parallel transport of tangent vectors on manifolds.
Comparing Tangent Vectors: Parallel Transport
Explores comparing tangent vectors and parallel transport 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.
Differential Forms on Manifolds
Introduces differential forms on manifolds, covering tangent bundles and intersection pairings.
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