TeXipedia

eth

Represents the Old English letter "eth" used in linguistics and mathematics, particularly in differential geometry.

Overview

Serves as a specialized mathematical operator and historical letter symbol with applications across multiple disciplines:

  • Common in differential geometry for denoting certain operators and derivatives
  • Used in linguistic contexts when discussing Old English or Icelandic phonetics
  • Appears in some mathematical physics equations, particularly in quantum field theory
  • Distinguished from the similar-looking partial differential symbol by its more curved appearance

Examples

Using the eth symbol in a differential operator expression common in physics.

ðf(z)=∂f∂z+1z∂f∂zˉ\eth f(z) = \frac{\partial f}{\partial z} + \frac{1}{z} \frac{\partial f}{\partial \bar{z}} 
\eth f(z) = \frac{\partial f}{\partial z} + \frac{1}{z} \frac{\partial f}{\partial \bar{z}}

Representing the Newman-Penrose eth operator in general relativity.

ðΦ=(1−∣Φ∣2)dΦdz\eth\Phi = (1-|\Phi|^2)\frac{d\Phi}{dz} 
\eth\Phi = (1-|\Phi|^2)\frac{d\Phi}{dz}

Showing the eth operator in spherical coordinates.

ð=∂∂θ+isin⁡θ∂∂ϕ\eth = \frac{\partial}{\partial \theta} + \frac{i}{\sin\theta}\frac{\partial}{\partial \phi} 
\eth = \frac{\partial}{\partial \theta} + \frac{i}{\sin\theta}\frac{\partial}{\partial \phi}