Jacobian matrix and Gateaux derivative Let $f:\mathbb{R}^N\rightarrow\mathbb{R}^M$ be a function which is Gâteaux differentiable and let $J_f\in\mathbb{R}^{M\times N}$ be its Jacobian matrix.
Is it true that the Gâteaux derivative of $f$ along a direction $v\in\mathbb{R}^N$ is equal to the matrix-vector product $J_f \cdot v$?

As you probably already know, this is true if $f$ is Frechet differentiable.
However, there is no reason why this should be true if $f$ is only Gateaux differentiable but not Frechet differentiable, because then the directional derivatives need not be linearly related to each other, i.e. as is stated in Wikipedia, "the Gateaux differential of a function may be nonlinear".
https://en.wikipedia.org/wiki/G%C3%A2teaux_derivative
The Jacobian matrix specifies the directional derivatives in the directions of the standard basis elements (as you probably already know, these are also called partial derivatives). But if $f$ is Gateaux but not Frechet differentiable, then the directional derivative in, for example $v=e_1+e_2$ may not equal $\partial_1 f + \partial_2 f$, which means that we could not use the Jacobian matrix to compute it.
The main idea here being that matrix multiplication represents composition with a linear function, so that if the Gateaux derivative is a nonlinear function of the directional derivatives, then it cannot be represented by a matrix.
