Hodge Duality and Covariant Derivatives

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Explores Maxwell equations, Minkowski spacetime, Lorentz transformations, charge conservation, and spacetime structure in physics.

Covers tangent spaces and submersions in differential geometry, emphasizing vector spaces and differentiable structures.

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

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

Introduces one-forms, tensors, normal coordinates, and metric tensors in differential geometry.

Explores the geometry of parametric curves, covering tangent vectors, curvature, and curve smoothing techniques.

Explores tangent spaces in algebraic geometry, defining them as finite-dimensional subspaces of derivations on varieties.

Explores curves with Poritsky property, Birkhoff integrability, and Liouville nets in billiards.

Covers the differentiability of functions of multiple variables and the significance of directional derivatives and gradients.

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