David Templin's Blog

David Templin's Blog

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The Einstein-Hilbert Action

The Einstein-Hilbert Action

This post describes how to derive the field equation of General Relativity from the Einstein-HIlbert action.

Covariant Derivatives of Tensor Fields

Differential Geometry

Covariant Derivatives of Tensor Fields

This post discusses covariant derivatives of tensor fields.

Curvature Tensors

Curvature Tensors

This post discusses the concept of curvature in manifolds and various tensor fields which measure this curvature.

Covariant Derivatives

Covariant Derivatives

This post discusses the concept of a covariant derivative on a smooth manifold.

Derivatives on Normed Vector Spaces

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.

Smooth Manifolds

Differential Geometry

Smooth Manifolds

This post examines the basic properties of smooth manifolds and gives a few detailed examples.

Differentiable Programming

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.

Parallel Metropolis-Hastings

Parallel Metropolis-Hastings

This post examines a particular method for parallelizing the Metropolis-Hastings algorithm.

Counter-Based Pseudo-Random Number Generators on the GPU

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.

Metropolis Light Transport

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.

Metric Tensor Fields

Metric Tensor Fields

This post explores the concept of metric tensor fields on smooth manifolds.

Differential Forms

Differential Forms

This post explores the basic idea of differential forms.

Exterior Algebra

Exterior Algebra

This post examines the concept of the exterior algebra of a vector space.

Differentials

This post examines the idea of the differential of a function, relating it to concepts from differential geometry.

Notes on Line and Surface Integrals

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.

Tensors

This post explores the mathematical concept of a tensor.

Cotangent Spaces

Cotangent Spaces

This post explores the concept of the cotangent space at a point on a smooth manifold.

Vector Fields

This post examines the concept of a vector field on a smooth manifold.

Topological Spaces

Topological Spaces

This post motivates the definition of a topological space using the concept of a filter.

Tangent Bundles

Tangent Bundles

This post contains a detailed explanation of the concept of the tangent bundle on a smooth manifold.

Tangent Spaces

This post contains a detailed explanation of the concept of the tangent space at a point on a smooth manifold.

Total Derivatives

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).