Concept

Covariant derivative

Related lectures (32)

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.

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 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.

Covariant and Contravariant Tensors: Maxwell Field Strength

Explores contravariant and covariant tensors, Maxwell field strength, and Lorentz invariance in linear transformations and metric properties.

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.

Hodge Duality and Covariant Derivatives

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

Symplectic Geometry

Covers the background on symplectic geometry, focusing on symplectic manifolds and canonical structures.

Quantum Field Theory II: Maxwell Field and Symmetries

Covers the Maxwell field, symmetries, and making shifts in local and covariant derivatives.

Christoffel Symbols and Gravity Before Einstein

Introduces Christoffel symbols and gravity concepts before Einstein, discussing mathematical tensors and the Nobel Prize in Physics.

Lorentz Transformations and Covariant Tensors

Explores Lorentz transformations, covariant tensors, rotational invariance, and linear transformations in vector spaces.

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.

Mechanics of Slender Structure: Shells III

Explores the nonlinear theory of spherical shells, including Reissner's solution and Zoelly's solution for pressure buckling.

Covariant Maxwell Equations

Covers the covariant form of Maxwell equations and properties of the field strength tensor.

Gradient: Scalar Field

Explores gradient in scalar fields, directional derivatives, and level sets.

Lorentz Invariance and Covariant Tensors

Explores Lorentz invariance, tensors in vector spaces, and electromagnetic potentials.

Descent methods and line search: Second Wolfe condition

Explores the second Wolfe condition, guiding step sizes based on the directional derivative increase.

Taylor expansions: second order

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