David Templin's Blog

David Templin's Blog

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Harmonic Oscillators

Harmonic Oscillators

This post offers a brief overview of simple and damped harmonic oscillators.

Distributions

This post gives a cursory overview of the concept of distributions from functional analysis.

The Mean Value Theorem

The Mean Value Theorem

This post describes the mean value theorem and several other theorems of real analysis.

Functional Derivatives

Functional Derivatives

This post is about the concept of functional derivatives. These derivatives find important applications in physics and engineering. Note: in this post, we will use the term derivative to refer to some sort of limit or related concept, whereas we will use the term differential to refer to a sort

The Chain Rule

This post is a detailed examination of the chain rule from differential calculus.

The Fréchet and Gateaux Derivatives

The Fréchet and Gateaux Derivatives

This brief note explains the precise relationship between the two principal notions of derivatives on normed vector spaces, namely, the Fréchet and Gateaux derivatives.

Parallel Metropolis Light Transport

Parallel Metropolis Light Transport

In this post, I will describe an algorithm which I designed which I call "Parallel Metropolis Light Transport" (PMLT) which is a variant of Metropolis Light Transport which is amenable to parallelization on GPUs. A detailed overview of Metropolis Light Transport is available in the following post: Metropolis

Monads in Computer Science

Computer Science

Monads in Computer Science

This post explores the concept of a monad in computer science and its expression in various programming languages.

The Euclidean Metric in Spherical Coordinates

The Euclidean Metric in Spherical Coordinates

This post discusses the derivation of the expression for the Euclidean metric in spherical coordinates.

Universal Algebra

Universal Algebra

This post contains a collection of notes on Universal Algebra.

Large Language Models

Artificial Intelligence

Large Language Models

This post provides a semi-technical overview of large language models (LLMs), in particular, the prototypical GPT models.

General Recursive Functions

General Recursive Functions

This post motivates the definition of general recursive functions by demonstrating one method of deriving this definition.

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.