We consider the radial differential operator for a fixed angular mode \(n\), \[ L_n=\frac{d^2}{d r^2}+\frac{1}{r} \frac{d}{d r}-\frac{n^2}{r^2}, \quad r \in(0, \infty) \]

We collect closed-form formulas for one- and \(n\)-dimensional Gaussian integrals with linear and oscillatory terms, derive e...

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 product expansion and inequality for \(J_n(z)\).

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 recall the construction of surface elements in spherical coordinates. We also note some standard Levi-Civita identities and briefly relate tensor gradients and variational increments to familiar differential operators.

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 the relation between metric components, coordinate bases, and orthonormal frames (with spherical coordinates as a concrete example).