Let $F(g(f))$
be a functional that sends functions of a vector variable (from $n$-dimensional vector space) to $\mathbb R$.


$g$
is some function of scalar valued functions $f$.


I'm interested in a coordinate free calculation of the first and second derivatives of functionals on the spaces of real valued functions of a vector variable, where the vector space dimension can be more than 3.


It can be [shown](https://en.m.wikipedia.org/wiki/Functional_derivative#Formula) using vector calculus that the first (order) functional derivative $\partial F$
is equal to the partial derivative of $g$
with respect to $f$
minus the divergence of the gradient of $g$.


\begin{gather\*}
F=\int {f(x)} ^{4} dx^{4} \\
\partial F= \frac{\partial{g}}{\partial{f}} - \nabla \cdot\nabla g(f).
\end{gather\*}


Divergence of the gradient should be equal to $4$, and if I plug in the above functional into the formula on Wikipedia it should work just as well for 4d and I get the result: $ \partial F=−12f^{3}$.


Can I just plug this into the formula again and get the second derivative of the functional?



Coordinate free computation of the second derivative of a functional
For
$$
F=\int d^4 z \ f^4 (z) \ ,
$$
the first functional derivative is
$$
\frac{\delta F}{\delta f(x)} = \int d^4 z \ 4 f^3 (z)\ \delta^4 (x-z)
=4 f^3 (x)
$$
and the second functional derivatives are
$$
\frac{\delta^{2} F}{\delta f(x) \delta f(y)} = 12 f^2 (x) \ \delta^4 (x-y)
$$


