This tag is for questions regarding to the Gateaux differential or, Gateaux derivative, a generalization of the concept of directional derivative in differential calculus. It is often used to formalize the functional derivative commonly used in Physics, particularly Quantum field theory.
A function $~f~$ is said to be Gateaux differentiable at $~x~$ if there exists a bounded linear operator $~T_x ∈\mathcal B(X, Y )~$ such that $~∀ v ∈ X~$, $$\lim_{t\to 0}\dfrac{f(x+tu)-f(x)}{t}=T_xv$$The operator $~T_x~$ is called the Gateaux derivative of $~f~$ at $~x~$.
Some things to notice about the Gateaux differential:
- There is not a single Gateaux differential at each point. Rather, at each point $~x~$ there is a Gateaux differential for each direction $~u~$. In one dimension, there are two Gateaux differentials for every $~x~$: one directed “forward,” one “backward.” In two of more dimensions, there are infinitely many Gateaux differentials at each point!
- The Gateaux differential is a one-dimensional calculation along a specified direction $~u~$. Because it’s one dimensional, you can use ordinary one-dimensional calculus to compute it. Your old friends such as the chain rule work for Gateaux differentials. Thus, it’s usually easy to compute a Gateaux differential even when the space $~X~$ is infinite dimensional.
For more details please find
"Gateaux differentials and Frechet derivatives" by Kevin Long
"Gateaux and Frechet Differentiability"
"Frechet derivatives and Gateaux derivatives" by Jordan Bell
"Introduction of Frechet and Gateaux Derivative" by Daryoush Behmardi and Encyeh Dehghan Nayeri
