Physics The Einstein-Hilbert Action This post describes how to derive the field equation of General Relativity from the Einstein-HIlbert action.
Differential Geometry Covariant Derivatives of Tensor Fields This post discusses covariant derivatives of tensor fields.
Mathematics Curvature Tensors This post discusses the concept of curvature in manifolds and various tensor fields which measure this curvature.
Mathematics Covariant Derivatives This post discusses the concept of a covariant derivative on a smooth manifold.
Mathematics Derivatives on Normed Vector Spaces This post explores the conception of derivatives as the best local linear approximation of a continuous function in the general setting of normed vector spaces, and generalizes classical results to spaces of functionals.
Differential Geometry Smooth Manifolds This post examines the basic properties of smooth manifolds and gives a few detailed examples.
Computer Science Differentiable Programming This post explains the basic idea of differentiable programming, providing sample code, and relating the concept to ideas from differential geometry.
Computer Graphics Solid Angle This post examines the concept of solid angle, the three-dimensional generalization of planar angle, and derives related mathematical formulas from scratch.
Mathematics Parallel Metropolis-Hastings This post examines a particular method for parallelizing the Metropolis-Hastings algorithm.
Computer Science Counter-Based Pseudo-Random Number Generators on the GPU This post examines the concept of a counter-based pseudo-random number generator (CBRNG), and the Squares algorithm in particular.
Computer Science Metropolis Light Transport This post offers a detailed explanation of Metropolis Light Transport in particular and path tracing in general, providing the relevant theoretical background.
Mathematics Metric Tensor Fields This post explores the concept of metric tensor fields on smooth manifolds.
Mathematics Exterior Algebra This post examines the concept of the exterior algebra of a vector space.
Mathematics Differentials This post examines the idea of the differential of a function, relating it to concepts from differential geometry.
Mathematics Notes on Line and Surface Integrals This post explores line and surface integrals, relating the classical formulations to concepts from differential geometry such as the pullback of differential forms and volume forms.
Mathematics Cotangent Spaces This post explores the concept of the cotangent space at a point on a smooth manifold.
Mathematics Topological Spaces This post motivates the definition of a topological space using the concept of a filter.
Mathematics Tangent Bundles This post contains a detailed explanation of the concept of the tangent bundle on a smooth manifold.
Mathematics Tangent Spaces This post contains a detailed explanation of the concept of the tangent space at a point on a smooth manifold.
Mathematics Total Derivatives This post contains a detailed explanation of the concept of the total derivative of a continuous function. It attempts to explain the total derivative in a way which relates it to concepts from differential geometry (such as tangent spaces).
