Mathematics

Fourier Multipliers as Operators Fourier analysis is especially useful for linear, translation-invariant operators. In physical space, such an operator may involve derivatives…

Index Notation and Embedded Surfaces Index notation and differential geometry provide compact ways to describe vectors, tensors, and curved surfaces. This short note collects a…

We show that the order- \(n\) Hankel transform in \(r\) diagonalizes the Bessel-type radial operator \(\mathscr{L}_n\), mapping the PDE ∑ Formula to the transformed equation…

We consider an axisymmetric pressure distribution \(p(r, \theta)=\bar{p} H(r-l)\) (radial dependence only) and show that its angular Fourier series contains only the \(m=0\) mode. Cons…

We consider the radial differential operator for a fixed angular mode \(n\), ∑ Formula

We collect closed-form formulas for one- and \(n\)-dimensional Gaussian integrals with linear and oscillatory terms, derive even-moment integrals, and list several useful…

We collect a few basic identities for the gamma and polygamma functions, their link to the Hurwitz zeta function, standard asymptotics for Bessel functions, and a useful Bessel…

We collect working formulas for dyadic (outer) products, the double-dot contraction of tensors, and tensor–vector cross products expressed with the Levi-Civita symbol, and we…

We review the Levi-Civita symbol in arbitrary dimension, its role in defining the cross product and wedge product, introduce the Hodge star on differential forms, and clarify…