If a function vanishes on an open set W, then its derivatives also vanish on W. Derivatives are "local" operators in this sense. Peetre's theorem gives us a converse: local linear operators are in fact differential operators. #math #calculus #FunctionalAnalysis

The linear #algebra behind the concept of raising and lowering indices of vectors and tensors, so often used in #Einstein's theory of Relativity. #math #physics #geometry

Liouville’s theorem for bounded entire functions on ℂ extends to harmonic functions f on n-dimensional Euclidean space ℝⁿ. Namely, if such f is bounded above, then f must be constant. (Ditto for bounded below.) #math #calculus

No matter where you are, have a Happy Canada Day!!!

Happy Canada Day🍁🇨🇦

"Every child knows that the amplitude for transmission [through a potential barrier] obeys the WKB formula ... " -- Sidney Coleman (Aspects Of Symmetry, Cambridge Univ. Press (1985; reprinted 1995), page 266). #physics

If f is any compactly supported infinitely differentiable function, its derivatives f', f'', f''', ... manifest massive fluctuations in such a way that their signed areas are always 0. Reason: Fundamental Theorem of #Calculus! Here's an illustration. #math #FunFacts

The Cauchy integral formula from complex analysis can be used to get a similar formula for 2D harmonic functions. This formula can be suitably extended to representing harmonic functions on 3D space. #math #calculus

#HappyBirthday to a dearly beloved Sister, #America, on this Fourth Of July! Cheers❤️, #Canada #HappyJuly4th #July4thCelebration #HappyFourth

A happy abstract #algebra exercise for the weekend. #HappyWeekend! #math

A matrix #algebra problem that Steven Weinberg needed in one of his papers. #math #physics

A theorem of Borel and Serre (1951) says that no n-dim'l sphere Sⁿ has an almost complex structure, beside S² and S⁶. For the 6-sphere, however, it's been known that we can write down an almost complex structure using Cayley numbers. Here's how. #math #geometry #algebra

The Schwarz-Pick lemma states that an analytic function from the open unit disk to itself never increases the hyperbolic Poincaré distance between two points. #math #calculus #ComplexAnalysis #geometry

Some examples of Calabi-Yau spaces, illustrating the 6-dimensional "curled-up dimensions" of 10-dimensional spacetimes studied in #StringTheory. They're not viable physical models because of their large (but interesting!) #Euler characteristics. #math #physics #geometry #topology