Sam Walters ☕️

Maxwell's equation for the electromagnetic field in vacuum can be derived from Einstein's equations in the generic case. (Though there are some exceptions.) Here are steps for a proof (step 2 being the tricky one). #physics #generalrelativity #electromagnetism #math #geometry

Physics and geometry. The #physics: creation and annihilation operators. The #geometry: the exterior derivative. An interesting version of the #Dirac operator comes about from their interaction. #math #algebra

The Poincaré-Hopf Theorem states that the Euler characteristic of a compact smooth manifold can be found using the degrees of zeros of any vector field which has a finite number of zeros. Could the Euler char. of the Universe be found this way? #math #geometry #topology #calculus

One of my favorite Glenn Campbell songs - Gentle On My Mind. youtube.com/watch?v=lp30Z1…

This theorem says that the ratio and root limits (often used in convergence tests) are equal whenever they both exist as elements in the extended nonnegative reals [0,∞]. (Not hard to prove.) A remark is made on when one of them exists. #math #calculus #algebra #topology

The Mittag-Leffler Theorem from complex analysis. For any open set in the complex plane we can always find meromorphic functions with any pole structure that we wish. The theorem has an extension to non-compact #Riemann surfaces. #math #calculus

The fundamental theorem of #algebra for proper maps. It's an interesting generalization for non-polynomial functions. The familiar case of complex polynomials follows. #math #calculus #topology

The Euler-MacLaurin Summation Formula. Several years ago we posted one version of this formula, and here's a second involving an infinite series of odd derivatives of the underlying function. #math #calculus #MathematicalAnalysis

It's well-known that Hermitian n × n matrices H (over ℂ) have real eigenvalues. Show that the latter remains the case if H is multiplied by any positive (definite) invertible matrix P. That is, the eigenvalues of PH are all real. #math #algebra

Why do we need representation theory In #quantum theory (and even in #Relativity)? There are several reasons, but here's a basic one. #physics #math

Applying #Einstein's Equivalence Principle to each point of #spacetime we get a smooth family of inertial frames (in tangent spaces). Such frames enable us to transport equations from inertial #Minkowski spacetime to curved spacetime. E.g., the #Dirac equation. #physics #math

A description of what local and global #Lorentz frames mean in relation to #Einstein's Principle of Equivalence in #GeneralRelativity. (This is a warm-up for an upcoming post on how physicists transport #Dirac's equation for fermions to curved spacetime.) #physics #math #geometry

I just came across this interesting set of papers from a meeting just last year of some great minds on #Einstein’s theory of Relativity. Some notable papers by Wald and Unruh, among others. #physics #GeneralRelativity arxiv.org/pdf/2410.14414

The spherical Pythagorean theorem. #math #geometry #trigonometry #calculus

What an amazing story, and such courage! bbc.com/news/articles/…

The Law of Sines in the three basic geometries of Euclidean, hyperbolic, and elliptic - with constant curvature! Angles are measured in the same way in all three (in radians). #math #geometry #trigonometry

A Theorem Of Frobenius And Schur (1906). (Are you sure? Schur, I'm Schur!) #math #algebra #GroupTheory #RepresentationTheory

Guess what?! (Voted at our local Hartland Baptist Church.) #Canada #CanadaElections2025 #federalelection2025

A homework exercise for the weekend! Have a good weekend. (Not hard.) #math #calculus #algebra