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# Recent Posts

- Illustrating the correspondence between 1-forms and vectors using de Broglie waves
- Invariance under rotations in space and conservation of angular momentum
- A mathematical formulation of Feynman’s ‘mirage on a hot road’
- Notes on Sturm-Liouville theory
- Overview of the Lie theory of rotations
- A note on Green’s theorem in the plane and finding areas enclosed by parametric curves
- Study of a proof of Noether’s theorem and its application to conservation laws in physics
- Calculus derivation of a surprising result of Galileo’s
- Simple variational setups yielding Newton’s Second Law and Schrödinger’s equation
- Alternative approaches for a differential equation involving the square of a derivative
- A note on the structure of the higher-order terms in multivariate Taylor expansions
- A study of Euler’s untranslated paper `De integratione aequationum differentialium’
- On the electrodynamical part of Einstein’s paper ‘Zur Elektrodynamik bewegter Körper’
- A note on reverse engineering the Navier-Stokes equations
- A fluid-mechanical visualisation of the quantum-mechanical continuity equation
- Calculation of a quantum-mechanical commutator in three dimensions
- Derivation of Euler’s equations of motion for a perfect fluid from Newton’s second law
- Alternative approaches to formulating geodesic equations on Riemannian manifolds and proof of their equivalence
- Geometric interpretation of Christoffel symbols and some alternative approaches to calculating them
- On Lie derivatives of tensor fields
- A closer look at how the principal, orbital and magnetic quantum numbers arise in Schrödinger’s theory of the hydrogen atom
- Changing variables in the square of the quantum mechanical angular momentum operator
- A matrix calculus approach to changing variables in Laplace’s equation
- On an integral calculus approach to finding the area of a quadrarc
- A note on Bragg’s law for X-ray scattering
- On passing from the discrete to the continuous form of Slutsky’s identity in microeconomics
- A problem involving the use of exterior derivatives of differential forms to re-express the classical gradient, curl and divergence operations
- Proof that a Gauss sum associated with a quadratic character mod p is the same as a quadratic Gauss sum
- On the classification of singularities, with an application to non-rotating black holes
- Different possible branch cuts for the principal argument, principal logarithm and principal square root functions
- Advanced Number Theory Note #16: A proof of the law of quadratic reciprocity using Gauss sums and quadratic characters
- Advanced Number Theory Note #15: The Legendre symbol (a|p) as a Dirichlet character mod p
- A note on the quaternion rotation operator
- Unexpected appearance of Pythagorean triples in a homeomorphism between the 1-sphere and the extended real line
- Hyperbolic Geometry Note #2: On Möbius transformations as isometries of the upper half plane
- Advanced Number Theory Note #14: An inequality due to Pólya for the partial sums of primitive Dirichlet characters
- Hyperbolic Geometry Note #1: Strange behaviour of length calculations in the Poincaré half-plane model of hyperbolic space
- Dirichlet character tables up to mod 21
- A Euclidean-like division algorithm for converting prime reciprocals into ternary numbers
- Using Chebyshev’s inequality and the Borel-Cantelli lemma to study the variation of Brownian motion
- A note on Brownian motion and fractional Brownian motion as self-similar stochastic processes
- A note on designing and implementing richer mathematical tasks
- A variant of Einstein’s partial differential equation for Brownian motion, applied to quantum mechanics
- A note on alternative approaches to solving a variable mass problem involving accretion
- Alternative approaches to solving advanced level vector problems
- Using algebraic topology to prove the Fundamental Theorem of Algebra
- Recovering the original cubic equation from Cardano’s nested surd formula
- Polygonal numbers as quadratic sequences
- On a reduced residue system mod mn
- On linear vs statistical independence
- Note on Generalized Convolution in Number Theory
- Note on a Computer Search for Primes in Arithmetic Progression by Weintraub (1977)
- Gauss’s Flux Theorem for Gravity and Newton’s Law
- Problems Involving Maclaurin or Taylor Series Solutions of Differential Equations
- A Mechanics Problem Involving the Centre of Mass of a Uniform Trapezoidal Lamina in Equilibrium
- A Mechanics Problem Involving Suspension of a Uniform Solid Hemisphere From a Fixed Point
- A Polar Equation Problem
- A Mechanics Problem Involving Equilibrium Rotational Motion
- A Mechanics Problem Involving an Elastic String
- Advanced Number Theory Note #13: A study of primitive Dirichlet characters
- Advanced Number Theory Note #12: Separability property of Gauss sums associated with Dirichlet characters
- Advanced Number Theory Note #11: Derivation of a Möbius function formula for Ramanujan’s sums
- Topological equivalence of the 2-sphere and the extended complex plane (‘the Riemann sphere’)
- A multiplicative property of quadratic Gauss sums
- Derivation of a formula for computing sums of Ramanujan’s sums
- Proof that if a Dirichlet character has two induced moduli then their gcd must also be an induced modulus
- The p-adic number field as an ‘ultrametric completion’ of the rationals
- Advanced Number Theory Note #10: The Möbius function as a sum of primitive roots of unity
- Advanced Number Theory Note #9: Finding the finite Fourier expansion of a periodic arithmetical function
- Advanced Number Theory Notes #1 to #8
- A note on finding more base-3 palindromic primes of the form 1 + 3^k + (3^k)^2

# Categories

- Advanced Calculus and Calculus of Variations
- Advanced Number Theory
- Complex Function Theory
- Differential Geometry
- Einstein's Special and General Relativity
- Fluid Mechanics
- Galois Theory
- General Physics
- Group Theory, Lie Theory
- Hyperbolic Geometry
- Index of My Earlier Mathematical Notes
- Mathematical Chemistry and Crystallography
- Mathematical Economics, Statistics and Econometrics
- Mathematical Texts in Latin
- Miscellaneous Notes Involving Primes
- p-Adic Numbers and p-Adic Analysis
- Quantum Mechanics
- Quaternions
- Stochastic Processes
- Teaching Notes
- Topology