# Functional derivative as "partial derivatives" of a functional

To show that $$p$$ is a stationary point of a function $$f:\mathbb{R}^n\longrightarrow\mathbb{R}$$, one needs to guarantee that $$f$$'s directional derivative at $$p$$ is zero in every direction. To do so, it is sufficient to check that all n partial derivatives are zero at $$p$$. That is, $$f$$ is stationary at $$p$$ if: $$$$\frac{\partial f}{\partial x_i}\Bigg|_p=0,\;\;\mathrm{for}\;\;\;i=0,1,\cdots ,n$$$$ This comes from the fact that $$\mathrm{dom}(f)$$ having an n-element basis means that the tangent space at $$p$$ does too.

If we instead have a functional $$F$$ on some function space, say $$L^2(X)$$, we can still talk about a basis $$\{\varphi_i\}$$ for $$\mathrm{dom}(F)$$, e.g Fourier series or some orthogonal polynomial basis. I would attempt to define partial derivatives with respect to this basis as follows: $$$$\frac{\partial F}{\partial \varphi_i}\Bigg|_\rho = \lim_{\varepsilon\to 0} \frac{F(\rho + \varepsilon \varphi_i)-F(\rho)}{\varepsilon}$$$$ where $$\rho\in\mathrm{dom}(F)$$.

My first question is: if the "partial derivative" defined above is zero for all of $$\{\varphi_i\}$$, is $$\rho$$ then a stationary point of $$F$$, say for the purposes of variational calculus? More abstractly, is there a sense in which a basis for $$\mathrm{dom}(F)$$ gives us a basis for its "tangent space?"

Secondly, I'm curious to know if this is a rigorous way to define functional derivatives, since physics textbooks that I've seen are a bit handwavy about the definition, while Wiki invokes something called a Radon-Nikodym derivative which I'm not familiar with. Any insight or references to further reading would be appreciated!

I am learning about variational problems right now, so maybe my explanations can have bugs.

First question: your definition seems quite reasonable to me, and even I have written something similar to grasp the idea. But the definition that I take as a reference (in Olver86 page 245) is based on an analogy with the usual gradient and usual inner product on $$\mathbb{R}^n$$. For the case of one dependent variable:

Definition. Let $$J[u]$$ be a variational problem (for functions $$u:\mathbb{R}^p \to \mathbb{R}$$). The variational derivative of $$J$$ is the unique $$\delta J[u]$$ such that given functions $$f,\eta:\Omega \subset \mathbb{R}^p \to \mathbb{R}^q$$, $$\eta$$ with compact support, satisfies: $$\frac{d}{d\epsilon}J[f+\epsilon \eta]|_{\epsilon=0}=\int_{\Omega}\delta J[f(x)]\cdot \eta(x)dx.$$ $$\blacksquare$$

This expression is what you are denoting for $$\eta=\varphi_i$$ $$\frac{\partial J}{\partial \varphi_i}|_f.$$

Observe that this definition is analogous to the case in which $$f$$ is a function, $$x$$ a point and $$V$$ a vector:
$$\frac{d}{d\epsilon}f(x+\epsilon V)|_{\epsilon=0}=\langle (\nabla f)_x, V \rangle.$$

I don't know if there is an equivalence of your definition to the one above but I guess that maybe some technical issues can arise (for example with the domain of the $$\eta$$s). But if it is the case, and if you are able to find a suitable basis for the $$\eta$$s, using Olver's definition you can show what you want, thanks to the linearity of integral (although maybe additional hypothesis for the commutativity of $$\int$$ and $$\sum$$ would be required...).

Second question: My current reference is Olver, Aplications of Lie groups to differential equations (chapter 4), which I cited above. But it is focused on applications, it is not a book on functional analysis, so maybe it is not enough for you.

Too long for a comment but perhaps worth lookig.

At first reaction over that, I am conducted to the example of a functional of the form $$A=\int_IL(x,y,y')dx$$ which, thru the perturbation $$A_{\varepsilon}=\int_IL(x,y+\varepsilon h,y'+\varepsilon h')dx$$ one is granted with $$\dfrac{A_{\varepsilon}-A}{\varepsilon}= \int_I\left(\dfrac{\partial L}{\partial y}-\frac{d}{dx} \dfrac{\partial L}{\partial y'}\right)h\ dx+O(\varepsilon h)$$ which is got from the expansion of $$A_{\varepsilon}$$.

Now if $$h=\sum_sh^s\varphi_s$$ for some basis on admissible family of functions with a classical condition, which says, for $$I=[a,b]$$, $$h(a)=h(b)=0,$$ then that is forcing us to demand $$\varphi_s(a)=\varphi_s(b)=0$$, for all $$\varphi_s$$, to see $$\lim_{\varepsilon\to0}\dfrac{A_{\varepsilon}-A}{\varepsilon} = \int_I\left(\dfrac{\partial L}{\partial y}-\frac{d}{dx} \dfrac{\partial L}{\partial y'}\right)\sum_sh^s\varphi_s\ dx$$ $$\qquad\quad\qquad= \sum_sh^s \int_I\left(\dfrac{\partial L}{\partial y}-\frac{d}{dx} \dfrac{\partial L}{\partial y'}\right)\varphi_s\ dx,$$ a desirable linear property for derivatives.